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Derivatives Markets 3rd Edition by Robert L. Mcdonald

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Derivatives Markets

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Derivatives Markets
TH IRD EDITIO N

Robert L. McDonald
Northwestern University
Kellogg School of Management

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Library of Congress Cataloging-in-Publication Data
McDonald, Robert L. (Robert Lynch)
Derivatives markets / Robert L. McDonald. — 3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-321-54308-0 (hardcover)
ISBN-10: 0-321-54308-4 (hardcover)
1. Derivative securities. I. Title.
HG6024.A3M394 2013
2012029875
332.64 57—dc23
10 9 8 7 6 5 4 3 2 1
ISBN 10: 0-321-54308-4
ISBN 13: 978-0-321-54308-0

For Irene, Claire, David, and Henry

Brief
Contents

Preface

1

xxi

Introduction to Derivatives

1

PART ONE
Insurance, Hedging, and Simple Strategies

23

2

An Introduction to Forwards and Options

3

Insurance, Collars, and Other Strategies

4

Introduction to Risk Management

PART TWO
Forwards, Futures, and Swaps

25
61

89

123

5

Financial Forwards and Futures

125

6

Commodity Forwards and Futures

7

Interest Rate Forwards and Futures

8

Swaps

163
195

233

PART THREE
Options 263
9

Parity and Other Option Relationships

265

10 Binomial Option Pricing: Basic Concepts

293

11 Binomial Option Pricing: Selected Topics

323

12 The Black-Scholes Formula

349

13 Market-Making and Delta-Hedging
14 Exotic Options: I

381

409

vii

viii

Brief Contents

PART FOUR
Financial Engineering and Applications

435

15 Financial Engineering and Security Design
16 Corporate Applications 469
17 Real Options 509

PART FIVE
Advanced Pricing Theory and Applications
18 The Lognormal Distribution

437

543

545

19 Monte Carlo Valuation 573
ˆ
20 Brownian Motion and Ito’s Lemma

603

21 The Black-Scholes-Merton Equation 627
22 Risk-Neutral and Martingale Pricing 649
23 Exotic Options: II 683
24
25
26
27

Volatility 717
Interest Rate and Bond Derivatives
Value at Risk 789
Credit Risk 815

Appendix A The Greek Alphabet 851
Appendix B Continuous Compounding
Appendix C Jensen’s Inequality 859

751

853

Appendix D An Introduction to Visual Basic for Applications
Glossary

883

References
Index

915

897

863

Contents

Preface

Chapter 1
Introduction to Derivatives
1.1
1.2

1.3

1.4

1.5

PART ONE

xxi

Insurance, Hedging, and Simple
Strategies 23
1

What Is a Derivative? 2
An Overview of Financial Markets 2
Trading of Financial Assets 2
Measures of Market Size and Activity 4
Stock and Bond Markets 5
Derivatives Markets 6
The Role of Financial Markets 9
Financial Markets and the Averages 9
Risk-Sharing 10
The Uses of Derivatives 11
Uses of Derivatives 11
Perspectives on Derivatives 13
Financial Engineering and Security
Design 14
Buying and Short-Selling Financial
Assets 14
Transaction Costs and the Bid-Ask
Spread 14
Ways to Buy or Sell 15
Short-Selling 16
The Lease Rate of an Asset 18
Risk and Scarcity in Short-Selling 18
Chapter Summary 20
Further Reading 20
Problems 20

Chapter 2
An Introduction to Forwards and
Options 25
2.1

2.2

2.3

Forward Contracts 25
The Payoff on a Forward Contract 29
Graphing the Payoff on a Forward
Contract 30
Comparing a Forward and Outright
Purchase 30
Zero-Coupon Bonds in Payoff and Profit
Diagrams 33
Cash Settlement Versus Delivery 34
Credit Risk 34
Call Options 35
Option Terminology 35
Payoff and Profit for a Purchased Call
Option 36
Payoff and Profit for a Written Call
Option 38
Put Options 41
Payoff and Profit for a Purchased Put
Option 41
Payoff and Profit for a Written Put
Option 42
The “Moneyness” of an Option 44

ix

x

2.4

2.5

2.6

2.A

Contents

Summary of Forward and Option
Positions 45
Positions Long with Respect to the
Index 45
Positions Short with Respect to the
Index 46
Options Are Insurance 47
Homeowner’s Insurance Is a Put
Option 48
But I Thought Insurance Is Prudent and
Put Options Are Risky . . . 48
Call Options Are Also Insurance 49
Example: Equity-Linked CDs 50
Graphing the Payoff on the CD 50
Economics of the CD 52
Why Equity-Linked CDs? 52
Chapter Summary 53
Further Reading 54
Problems 55
More on Buying a Stock Option 57
Dividends 57
Exercise 57
Margins for Written Options 58
Taxes 58

Chapter 3
Insurance, Collars, and Other
Strategies 61
3.1

3.2

3.3

3.4

Basic Insurance Strategies 61
Insuring a Long Position: Floors 61
Insuring a Short Position: Caps 64
Selling Insurance 66
Put-Call Parity 68
Synthetic Forwards 68
The Put-Call Parity Equation 70
Spreads and Collars 71
Bull and Bear Spreads 71
Box Spreads 73
Ratio Spreads 74
Collars 74
Speculating on Volatility 79
Straddles 79
Butterfly Spreads 80
Asymmetric Butterfly Spreads 82
Chapter Summary 84
Further Reading 86
Problems 86

Chapter 4
Introduction to Risk Management
4.1

4.2

4.3

4.4

4.5

89

Basic Risk Management: The Producer’s
Perspective 89
Hedging with a Forward Contract 90
Insurance: Guaranteeing a Minimum Price with a Put Option 91
Insuring by Selling a Call 93
Adjusting the Amount of Insurance 95
Basic Risk Management: The Buyer’s
Perspective 96
Hedging with a Forward Contract 97
Insurance: Guaranteeing a Maximum Price with a Call Option 97
Why Do Firms Manage Risk? 99
An Example Where Hedging Adds
Value 100
Reasons to Hedge 102
Reasons Not to Hedge 104
Empirical Evidence on Hedging 104
Golddiggers Revisited 107
Selling the Gain: Collars 107
Other Collar Strategies 111
Paylater Strategies 111
Selecting the Hedge Ratio 112
Cross-Hedging 112
Quantity Uncertainty 114
Chapter Summary 117
Further Reading 118
Problems 118

PART TWO
Forwards, Futures, and
Swaps 123
Chapter 5
Financial Forwards and Futures
5.1
5.2

125

Alternative Ways to Buy a Stock 125
Prepaid Forward Contracts on Stock 126
Pricing the Prepaid Forward by
Analogy 127
Pricing the Prepaid Forward by Discounted
Present Value 127

Contents

5.3

5.4

5.5

5.6

5.7

5.A
5.B
5.C

Pricing the Prepaid Forward by
Arbitrage 127
Pricing Prepaid Forwards with
Dividends 129
Forward Contracts on Stock 131
Does the Forward Price Predict the Future
Spot Price? 132
Creating a Synthetic Forward
Contract 133
Synthetic Forwards in Market-Making and
Arbitrage 135
No-Arbitrage Bounds with Transaction
Costs 136
Quasi-Arbitrage 137
An Interpretation of the Forward Pricing
Formula 138
Futures Contracts 138
The S&P 500 Futures Contract 139
Margins and Marking to Market 140
Comparing Futures and Forward
Prices 143
Arbitrage in Practice: S&P 500 Index
Arbitrage 143
Quanto Index Contracts 145
Uses of Index Futures 146
Asset Allocation 146
Cross-hedging with Index Futures 147
Currency Contracts 150
Currency Prepaid Forward 150
Currency Forward 152
Covered Interest Arbitrage 152
Eurodollar Futures 153
Chapter Summary 157
Further Reading 158
Problems 158
Taxes and the Forward Rate 161
Equating Forwards and Futures 162
Forward and Futures Prices 162

Chapter 6
Commodity Forwards and
Futures 163
6.1

Introduction to Commodity
Forwards 164
Examples of Commodity Futures
Prices 164

6.2
6.3

6.4

6.5
6.6

6.7

6.8

Differences Between Commodities and
Financial Assets 166
Commodity Terminology 166
Equilibrium Pricing of Commodity
Forwards 167
Pricing Commodity Forwards by
Arbitrage 168
An Apparent Arbitrage 168
Short-selling and the Lease Rate 170
No-Arbitrage Pricing Incorporating
Storage Costs 172
Convenience Yields 174
Summary 175
Gold 175
Gold Leasing 176
Evaluation of Gold Production 177
Corn 178
Energy Markets 179
Electricity 180
Natural Gas 180
Oil 182
Oil Distillate Spreads 184
Hedging Strategies 185
Basis Risk 186
Hedging Jet Fuel with Crude Oil 187
Weather Derivatives 188
Synthetic Commodities 189
Chapter Summary 191
Further Reading 192
Problems 192

Chapter 7
Interest Rate Forwards and
Futures 195
7.1

7.2

Bond Basics 195
Zero-Coupon Bonds 196
Implied Forward Rates 197
Coupon Bonds 199
Zeros from Coupons 200
Interpreting the Coupon Rate 201
Continuously Compounded Yields 202
Forward Rate Agreements, Eurodollar
Futures, and Hedging 202
Forward Rate Agreements 203
Synthetic FRAs 204
Eurodollar Futures 206

xi

xii

7.3

7.4
7.5

7.A

Contents

Duration and Convexity 211
Price Value of a Basis Point and DV01 211
Duration 213
Duration Matching 214
Convexity 215
Treasury-Bond and Treasury-Note
Futures 217
Repurchase Agreements 220
Chapter Summary 222
Further Reading 224
Problems 225
Interest Rate and Bond Price
Conventions 228
Bonds 228
Bills 230

Chapter 8
Swaps 233
8.1

8.2

8.3

8.4

8.5
8.6

An Example of a Commodity Swap 233
Physical Versus Financial Settlement 234
Why Is the Swap Price Not $110.50? 236
The Swap Counterparty 237
The Market Value of a Swap 238
Computing the Swap Rate in General 240
Fixed Quantity Swaps 240
Swaps with Variable Quantity and
Price 241
Interest Rate Swaps 243
A Simple Interest Rate Swap 243
Pricing and the Swap Counterparty 244
Swap Rate and Bond Calculations 246
The Swap Curve 247
The Swap’s Implicit Loan Balance 248
Deferred Swaps 249
Related Swaps 250
Why Swap Interest Rates? 251
Amortizing and Accreting Swaps 252
Currency Swaps 252
Currency Swap Formulas 255
Other Currency Swaps 256
Swaptions 256
Total Return Swaps 257
Chapter Summary 259
Further Reading 260
Problems 261

PART THREE
Options

263

Chapter 9
Parity and Other Option
Relationships 265
9.1

9.2

9.3

9.A
9.B

Put-Call Parity 265
Options on Stocks 266
Options on Currencies 269
Options on Bonds 269
Dividend Forward Contracts 269
Generalized Parity and Exchange
Options 270
Options to Exchange Stock 272
What Are Calls and Puts? 272
Currency Options 273
Comparing Options with Respect to Style,
Maturity, and Strike 275
European Versus American Options 276
Maximum and Minimum Option
Prices 276
Early Exercise for American Options 277
Time to Expiration 280
Different Strike Prices 281
Exercise and Moneyness 286
Chapter Summary 286
Further Reading 287
Problems 288
Parity Bounds for American Options 291
Algebraic Proofs of Strike-Price
Relations 292

Chapter 10
Binomial Option Pricing: Basic
Concepts 293
10.1 A One-Period Binomial Tree 293
Computing the Option Price 294
The Binomial Solution 295
Arbitraging a Mispriced Option 297
A Graphical Interpretation of the Binomial
Formula 298
Risk-Neutral Pricing 299
10.2 Constructing a Binomial Tree 300

Contents

10.3

10.4
10.5
10.6

10.A

Continuously Compounded Returns 301
Volatility 302
Constructing u and d 303
Estimating Historical Volatility 303
One-Period Example with a Forward
Tree 305
Two or More Binomial Periods 306
A Two-Period European Call 306
Many Binomial Periods 308
Put Options 309
American Options 310
Options on Other Assets 312
Option on a Stock Index 312
Options on Currencies 312
Options on Futures Contracts 314
Options on Commodities 315
Options on Bonds 316
Summary 317
Chapter Summary 318
Further Reading 319
Problems 319
Taxes and Option Prices 322

Chapter 11
Binomial Option Pricing: Selected
Topics 323
11.1 Understanding Early Exercise 323
11.2 Understanding Risk-Neutral Pricing 326
The Risk-Neutral Probability 326
Pricing an Option Using Real
Probabilities 327
11.3 The Binomial Tree and Lognormality 330
The Random Walk Model 330
Modeling Stock Prices as a Random
Walk 331
The Binomial Model 332
Lognormality and the Binomial Model 333
Alternative Binomial Trees 335
Is the Binomial Model Realistic? 336
11.4 Stocks Paying Discrete Dividends 336
Modeling Discrete Dividends 337
Problems with the Discrete Dividend
Tree 337
A Binomial Tree Using the Prepaid
Forward 339

xiii

Chapter Summary 340
Further Reading 341
Problems 341
11.A Pricing Options with True
Probabilities 343
11.B Why Does Risk-Neutral Pricing
Work? 344
Utility-Based Valuation 344
Standard Discounted Cash Flow 345
Risk-Neutral Pricing 345
Physical vs. Risk-Neutral Probabilities 346
Example 347

Chapter 12
The Black-Scholes Formula

349

12.1 Introduction to the Black-Scholes
Formula 349
Call Options 349
Put Options 352
When Is the Black-Scholes Formula
Valid? 352
12.2 Applying the Formula to Other
Assets 353
Options on Stocks with Discrete
Dividends 354
Options on Currencies 354
Options on Futures 355
12.3 Option Greeks 356
Definition of the Greeks 356
Greek Measures for Portfolios 361
Option Elasticity 362
12.4 Profit Diagrams Before Maturity 366
Purchased Call Option 366
Calendar Spreads 367
12.5 Implied Volatility 369
Computing Implied Volatility 369
Using Implied Volatility 370
12.6 Perpetual American Options 372
Valuing Perpetual Options 373
Barrier Present Values 374
Chapter Summary 374
Further Reading 375
Problems 375
12.A The Standard Normal Distribution 378

xiv

Contents

12.B Formulas for Option Greeks
Delta ( ) 379
Gamma ( ) 379
Theta (θ) 379
Vega 380
Rho (ρ) 380
Psi (ψ) 380

379

Chapter 13
Market-Making and DeltaHedging 381
13.1 What Do Market-Makers Do? 381
13.2 Market-Maker Risk 382
Option Risk in the Absence of
Hedging 382
Delta and Gamma as Measures of
Exposure 383
13.3 Delta-Hedging 384
An Example of Delta-Hedging for 2
Days 385
Interpreting the Profit Calculation 385
Delta-Hedging for Several Days 387
A Self-Financing Portfolio: The Stock
Moves One σ 389
13.4 The Mathematics of Delta-Hedging 389
Using Gamma to Better Approximate the
Change in the Option Price 390
Delta-Gamma Approximations 391
Theta: Accounting for Time 392
Understanding the Market-Maker’s
Profit 394
13.5 The Black-Scholes Analysis 395
The Black-Scholes Argument 396
Delta-Hedging of American Options 396
What Is the Advantage to Frequent
Re-Hedging? 397
Delta-Hedging in Practice 398
Gamma-Neutrality 399
13.6 Market-Making as Insurance 402
Insurance 402
Market-Makers 403
Chapter Summary 403
Further Reading 404
Problems 404
13.A Taylor Series Approximations 406
13.B Greeks in the Binomial Model 407

Chapter 14
Exotic Options: I

409

14.1 Introduction 409
14.2 Asian Options 410
XYZ’s Hedging Problem 411
Options on the Average 411
Comparing Asian Options 412
An Asian Solution for XYZ 413
14.3 Barrier Options 414
Types of Barrier Options 415
Currency Hedging 416
14.4 Compound Options 418
Compound Option Parity 419
Options on Dividend-Paying Stocks 419
Currency Hedging with Compound
Options 421
14.5 Gap Options 421
14.6 Exchange Options 424
European Exchange Options 424
Chapter Summary 425
Further Reading 426
Problems 426
14.A Pricing Formulas for Exotic Options 430
Asian Options Based on the Geometric
Average 430
Compound Options 431
Infinitely Lived Exchange Option 432

PART FOUR
Financial Engineering and
Applications 435
Chapter 15
Financial Engineering and Security
Design 437
15.1 The Modigliani-Miller Theorem 437
15.2 Structured Notes without Options 438
Single Payment Bonds 438
Multiple Payment Bonds 441
15.3 Structured Notes with Options 445
Convertible Bonds 446
Reverse Convertible Bonds 449
Tranched Payoffs 451

Contents

Variable Prepaid Forwards 452
15.4 Strategies Motivated by Tax and
Regulatory Considerations 453
Capital Gains Deferral 454
Marshall & Ilsley SPACES 458
15.5 Engineered Solutions for
Golddiggers 460
Gold-Linked Notes 460
Notes with Embedded Options 462
Chapter Summary 463
Further Reading 464
Problems 464

17.2

17.3

17.4

Chapter 16
Corporate Applications

469

16.1 Equity, Debt, and Warrants 469
Debt and Equity as Options 469
Leverage and the Expected Return on Debt and Equity 472
Multiple Debt Issues 477
Warrants 478
Convertible Bonds 479
Callable Bonds 482
Bond Valuation Based on the Stock
Price 485
Other Bond Features 485
Put Warrants 486
16.2 Compensation Options 487
The Use of Compensation Options 487
Valuation of Compensation Options 489
Repricing of Compensation Options 492
Reload Options 493
Level 3 Communications 495
16.3 The Use of Collars in Acquisitions 499
The Northrop Grumman—TRW merger 499
Chapter Summary 502
Further Reading 503
Problems 503
16.A An Alternative Approach to Expensing
Option Grants 507

Chapter 17
Real Options

509

17.1 Investment and the NPV Rule
Static NPV 510

509

17.5

17.A
17.B

xv

The Correct Use of NPV 511
The Project as an Option 511
Investment under Uncertainty 513
A Simple DCF Problem 513
Valuing Derivatives on the Cash Flow 514
Evaluating a Project with a 2-Year
Investment Horizon 515
Evaluating the Project with an Infinite
Investment Horizon 519
Real Options in Practice 519
Peak-Load Electricity Generation 519
Research and Development 523
Commodity Extraction as an Option 525
Single-Barrel Extraction under
Certainty 525
Single-Barrel Extraction under
Uncertainty 528
Valuing an Infinite Oil Reserve 530
Commodity Extraction with Shutdown and Restart Options 531
Permanent Shutting Down 533
Investing When Shutdown Is Possible 535
Restarting Production 536
Additional Options 537
Chapter Summary 538
Further Reading 538
Problems 538
Calculation of Optimal Time to Drill an
Oil Well 541
The Solution with Shutting Down and
Restarting 541

PART FIVE
Advanced Pricing Theory and
Applications 543
Chapter 18
The Lognormal Distribution

545

18.1 The Normal Distribution 545
Converting a Normal Random Variable to
Standard Normal 548
Sums of Normal Random Variables 549
18.2 The Lognormal Distribution 550
18.3 A Lognormal Model of Stock Prices 552

xvi

Contents

18.4 Lognormal Probability Calculations 556
Probabilities 556
Lognormal Prediction Intervals 557
The Conditional Expected Price 559
The Black-Scholes Formula 561
18.5 Estimating the Parameters of a Lognormal
Distribution 562
18.6 How Are Asset Prices Distributed? 564
Histograms 564
Normal Probability Plots 566
Chapter Summary 569
Further Reading 569
Problems 570
18.A The Expectation of a Lognormal
Variable 571
18.B Constructing a Normal Probability
Plot 572

Chapter 19
Monte Carlo Valuation

573

19.1 Computing the Option Price as a
Discounted Expected Value 573
Valuation with Risk-Neutral
Probabilities 574
Valuation with True Probabilities 575
19.2 Computing Random Numbers 577
19.3 Simulating Lognormal Stock Prices 578
Simulating a Sequence of Stock Prices 578
19.4 Monte Carlo Valuation 580
Monte Carlo Valuation of a European
Call 580
Accuracy of Monte Carlo 581
Arithmetic Asian Option 582
19.5 Efficient Monte Carlo Valuation 584
Control Variate Method 584
Other Monte Carlo Methods 587
19.6 Valuation of American Options 588
19.7 The Poisson Distribution 591
19.8 Simulating Jumps with the Poisson
Distribution 593
Simulating the Stock Price with
Jumps 593
Multiple Jumps 596
19.9 Simulating Correlated Stock Prices 597
Generating n Correlated Lognormal
Random Variables 597

Chapter Summary 599
Further Reading 599
Problems 599
19.A Formulas for Geometric Average
Options 602

Chapter 20
ˆ
Brownian Motion and Ito’s
Lemma 603
20.1 The Black-Scholes Assumption about
Stock Prices 603
20.2 Brownian Motion 604
Definition of Brownian Motion 604
Properties of Brownian Motion 606
Arithmetic Brownian Motion 607
The Ornstein-Uhlenbeck Process 608
20.3 Geometric Brownian Motion 609
Lognormality 609
Relative Importance of the Drift and Noise
Terms 610
Multiplication Rules 610
Modeling Correlated Asset Prices 612
ˆ
20.4 Ito’s Lemma 613
ˆ
Functions of an Ito Process 614
ˆ
Multivariate Ito’s Lemma 616
20.5 The Sharpe Ratio 617
20.6 Risk-Neutral Valuation 618
A Claim That Pays S(T )a 619
Specific Examples 620
Valuing a Claim on S a Qb 621
20.7 Jumps in the Stock Price 623
Chapter Summary 624
Further Reading 624
Problems 624
20.A Valuation Using Discounted Cash
Flow 626

Chapter 21
The Black-Scholes-Merton
Equation 627
21.1 Differential Equations and Valuation under Certainty 627
The Valuation Equation 628
Bonds 628
Dividend-Paying Stocks 629

Contents

21.2

21.3

21.4
21.5

21.A
21.B
21.C

The General Structure 629
The Black-Scholes Equation 629
Verifying the Formula for a Derivative 631
The Black-Scholes Equation and
Equilibrium Returns 634
What If the Underlying Asset Is Not an
Investment Asset? 635
Risk-Neutral Pricing 637
Interpreting the Black-Scholes
Equation 637
The Backward Equation 637
Derivative Prices as Discounted Expected
Cash Flows 638
Changing the Numeraire 639
Option Pricing When the Stock Price Can
Jump 642
Merton’s Solution for Diversifiable
Jumps 643
Chapter Summary 644
Further Reading 644
Problems 645
Multivariate Black-Scholes Analysis 646
Proof of Proposition 21.1 646
Solutions for Prices and Probabilities 647

Chapter 22
Risk-Neutral and Martingale
Pricing 649
22.1 Risk Aversion and Marginal Utility 650
22.2 The First-Order Condition for Portfolio
Selection 652
22.3 Change of Measure and Change of
Numeraire 654
Change of Measure 655
The Martingale Property 655
Girsanov’s Theorem 657
22.4 Examples of Numeraire and Measure
Change 658
The Money-Market Account as Numeraire
(Risk-Neutral Measure) 659
Risky Asset as Numeraire 662
Zero Coupon Bond as Numeraire (Forward
Measure) 662
22.5 Examples of Martingale Pricing 663
Cash-or-Nothing Call 663

22.6

22.A

22.B

22.C

xvii

Asset-or-Nothing Call 665
The Black-Scholes Formula 666
European Outperformance Option 667
Option on a Zero-Coupon Bond 667
Example: Long-Maturity Put Options 667
The Black-Scholes Put Price
Calculation 668
Is the Put Price Reasonable? 669
Discussion 671
Chapter Summary 671
Further Reading 673
Problems 673
The Portfolio Selection Problem 676
The One-Period Portfolio Selection
Problem 676
The Risk Premium of an Asset 678
Multiple Consumption and Investment
Periods 679
Girsanov’s Theorem 679
The Theorem 679
Constructing Multi-Asset Processes from
Independent Brownian Motions 680
Risk-Neutral Pricing and Marginal Utility in the Binomial Model 681

Chapter 23
Exotic Options: II

683

23.1 All-or-Nothing Options 683
Terminology 683
Cash-or-Nothing Options 684
Asset-or-Nothing Options 685
Ordinary Options and Gap Options 686
Delta-Hedging All-or-Nothing
Options 687
23.2 All-or-Nothing Barrier Options 688
Cash-or-Nothing Barrier Options 690
Asset-or-Nothing Barrier Options 694
Rebate Options 694
Perpetual American Options 695
23.3 Barrier Options 696
23.4 Quantos 697
The Yen Perspective 698
The Dollar Perspective 699
A Binomial Model for the DollarDenominated Investor 701

xviii

Contents

23.5 Currency-Linked Options 704
Foreign Equity Call Struck in Foreign
Currency 705
Foreign Equity Call Struck in Domestic
Currency 706
Fixed Exchange Rate Foreign Equity
Call 707
Equity-Linked Foreign Exchange Call 707
23.6 Other Multivariate Options 708
Options on the Best of Two Assets 709
Basket Options 710
Chapter Summary 711
Further Reading 711
Problems 712
23.A The Reflection Principle 715

Chapter 24
Volatility 717
24.1 Implied Volatility 718
24.2 Measurement and Behavior of
Volatility 720
Historical Volatility 720
Exponentially Weighted Moving
Average 721
Time-Varying Volatility: ARCH 723
The GARCH Model 727
Realized Quadratic Variation 729
24.3 Hedging and Pricing Volatility 731
Variance and Volatility Swaps 731
Pricing Volatility 733
24.4 Extending the Black-Scholes Model 736
Jump Risk and Implied Volatility 737
Constant Elasticity of Variance 737
The Heston Model 740
Evidence 742
Chapter Summary 745
Further Reading 745
Problems 746

Chapter 25
Interest Rate and Bond
Derivatives 751
25.1 An Introduction to Interest Rate
Derivatives 752
Bond and Interest Rate Forwards 752

25.2

25.3

25.4

25.5

25.A

Options on Bonds and Rates 753
Equivalence of a Bond Put and an Interest
Rate Call 754
Taxonomy of Interest Rate Models 754
Interest Rate Derivatives and the
Black-Scholes-Merton Approach 756
An Equilibrium Equation for Bonds 757
Continuous-Time Short-Rate Models 760
The Rendelman-Bartter Model 760
The Vasicek Model 761
The Cox-Ingersoll-Ross Model 762
Comparing Vasicek and CIR 763
Duration and Convexity Revisited 764
Short-Rate Models and Interest Rate
Trees 765
An Illustrative Tree 765
The Black-Derman-Toy Model 769
Hull-White Model 773
Market Models 780
The Black Model 780
LIBOR Market Model 781
Chapter Summary 783
Further Reading 784
Problems 784
Constructing the BDT Tree 787

Chapter 26
Value at Risk

789

26.1 Value at Risk 789
Value at Risk for One Stock 793
VaR for Two or More Stocks 795
VaR for Nonlinear Portfolios 796
VaR for Bonds 801
Estimating Volatility 805
Bootstrapping Return Distributions 806
26.2 Issues with VaR 807
Alternative Risk Measures 807
VaR and the Risk-Neutral Distribution 810
Subadditive Risk Measures 811
Chapter Summary 812
Further Reading 813
Problems 813

Chapter 27
Credit Risk

815

27.1 Default Concepts and Terminology

815

Contents

27.2 The Merton Default Model 817
Default at Maturity 817
Related Models 819
27.3 Bond Ratings and Default
Experience 821
Rating Transitions 822
Recovery Rates 824
Reduced Form Bankruptcy Models 824
27.4 Credit Default Swaps 826
Single-Name Credit Default Swaps 826
Pricing a Default Swap 828
CDS Indices 832
Other Credit-Linked Structures 834
27.5 Tranched Structures 834
Collateralized Debt Obligations 836
CDO-Squareds 840
Nth to default baskets 842
Chapter Summary 844
Further Reading 846
Problems 846

Appendix A
The Greek Alphabet

851

Appendix B
Continuous Compounding
B.1
B.2

Appendix C
Jensen’s Inequality
C.1
C.2
C.3

853

The Language of Interest Rates 853
The Logarithmic and Exponential
Functions 854
Changing Interest Rates 855
Symmetry for Increases and Decreases 855
Problems 856

859

Example: The Exponential Function
Example: The Price of a Call 860
Proof of Jensen’s Inequality 861
Problems 862

859

D.2
D.3

How to Learn VBA 864
Calculations with VBA 864
Creating a Simple Function 864
A Simple Example of a Subroutine 865
Creating a Button to Invoke a
Subroutine 866
Functions Can Call Functions 867
Illegal Function Names 867
Differences between Functions and
Subroutines 867
D.4 Storing and Retrieving Variables in a
Worksheet 868
Using a Named Range to Read and Write
Numbers from the Spreadsheet 868
Reading and Writing to Cells That Are Not
Named 869
Using the Cells Function to Read and
Write to Cells 870
Reading from within a Function 870
D.5 Using Excel Functions from within
VBA 871
Using VBA to Compute the Black-Scholes
Formula 871
The Object Browser 872
D.6 Checking for Conditions 873
D.7 Arrays 874
Defining Arrays 874
D.8 Iteration 875
A Simple for Loop 876
Creating a Binomial Tree 876
Other Kinds of Loops 877
D.9 Reading and Writing Arrays 878
Arrays as Output 878
Arrays as Inputs 879
D.10 Miscellany 880
Getting Excel to Generate Macros for
You 880
Using Multiple Modules 881
Recalculation Speed 881
Debugging 882
Creating an Add-In 882
Glossary

Appendix D
An Introduction to Visual Basic for
Applications 863
D.1

Calculations without VBA

863

xix

883

References
Index

915

897

Preface

Y

ou cannot understand modern finance and financial markets without understanding derivatives. This book will help you to understand the derivative instruments that exist, how they are used, how they are priced, and how the tools and concepts underlying derivatives are useful more broadly in finance.
Derivatives are necessarily an analytical subject, but I have tried throughout to emphasize intuition and to provide a common sense way to think about the formulas. I do assume that a reader of this book already understands basic financial concepts such as present value, and elementary statistical concepts such as mean and standard deviation. In order to make the book accessible to readers with widely varying backgrounds and experiences, I use a
“tiered” approach to the mathematics. Chapters 1–9 emphasize present value calculations, and there is almost no calculus until Chapter 18.
The last part of the book develops the Black-Scholes-Merton approach to pricing derivatives and presents some of the standard mathematical tools used in option pricing, such as Itˆ ’s Lemma. There are also chapters dealing with applications to corporate finance, o financial engineering, and real options.
Most of the calculations in this book can be replicated using Excel spreadsheets on the CD-ROM that comes with the book.1 These allow you to experiment with the pricing models and build your own spreadsheets. The spreadsheets on the CD-ROM contain option pricing functions written in Visual Basic for Applications, the macro language in Excel.
You can incorporate these functions into your own spreadsheets. You can also examine and modify the Visual Basic code for the functions. Appendix D explains how to write such functions in Excel, and documentation on the CD-ROM lists the option pricing functions that come with the book. Relevant Excel functions are also mentioned throughout the book.

WHAT IS NEW IN THE THIRD EDITION
The reader familiar with the previous editions will find the same overall plan, but will discover many changes. Some are small, some are major. In general:

1. Some of the advanced calculations are not easy in Excel, for example the Heston option pricing calculation. As an alternative to Excel I used R (http://r-project.org) to prepare many of the new graphs and calculations. In the near future I hope to provide an R tutorial for the interested reader.

xxi

xxii

Preface

.

Many examples have been updated.

.

There are numerous changes to streamline and clarify exposition.

.

.

There are connections throughout to events during the financial crisis and to the DoddFrank financial reform act.
New boxes cover Bernie Madoff, Mexico’s oil hedge, oil arbitrage, LIBOR during the financial crisis, Islamic finance, Bank capital, Google and compensation options,
Abacus and Magnetar, and other topics.
Several chapters have also been extensively revised:

.

.

.

.

.

Chapter 1 has a new discussion of clearing and the organization and measurement of markets. The chapter on commodities, Chapter 6, has been reorganized. There is a new introductory discussion and overview of differences between commodities and financial assets, a discussion of commodity arbitrage using copper, a discussion of commodity indices, and boxes on tanker-based oil-market arbitrage and illegal futures contracts.
Chapter 15 has a revamped discussion of structures, a new discussion of reverse convertibles, and a new discussion of tranching.
Chapter 25 has been heavily revised. There is a discussion of the taxonomy of fixed income models, distinguishing short-rate models and market models. New sections on the Hull-White and LIBOR market models have been added.
Chapter 27 also has been heavily revised. One of the most important structuring issues highlighted by the financial crisis is the behavior of tranched claims that are themselves based on tranched claims. Many collateralized debt obligations satisfy this description, as do so-called CDO-squared contracts. There is a section on CDOsquareds and a box on Goldman Sach’s Abacus transaction and the hedge fund
Magnetar. The 2009 standardization of CDS contracts is discussed.

Finally, Chapter 22 is new in this edition, focusing on the martingale approach to pricing derivatives. The chapter explains the important connection between investor portfolio decisions and derivatives pricing models. In this context, it provides the rationale for risk-neutral pricing and for different classes of fixed income pricing models. The chapter discusses Warren Buffett’s critique of the Black-Scholes put pricing formula. You can skip this chapter and still understand the rest of the book, but the material in even the first few sections will deepen your understanding of the economic underpinnings of the models.

PLAN OF THE BOOK
This book grew from my teaching notes for two MBA derivatives courses at Northwestern
University’s Kellogg School of Management. The two courses roughly correspond to the first two-thirds and last third of the book. The first course is a general introduction to derivative products (principally futures, options, swaps, and structured products), the markets in which they trade, and applications. The second course is for those wanting a deeper understanding of the pricing models and the ability to perform their own analysis. The advanced course assumes that students know basic statistics and have seen calculus, and from that point develops the Black-Scholes option-pricing framework. A 10-week MBA-level course

Preface

will not produce rocket scientists, but mathematics is the language of derivatives and it would be cheating students to pretend otherwise.
I wrote chapters to allow flexible use of the material, with suggested possible paths through the material below. In many cases it is possible to cover chapters out of order. For example, I wrote the book anticipating that the chapters on lognormality and Monte Carlo simulation might be used in a first derivatives course.
The book has five parts plus appendixes. Part 1 introduces the basic building blocks of derivatives: forward contracts and call and put options. Chapters 2 and 3 examine these basic instruments and some common hedging and investment strategies. Chapter 4 illustrates the use of derivatives as risk management tools and discusses why firms might care about risk management. These chapters focus on understanding the contracts and strategies, but not on pricing.
Part 2 considers the pricing of forward, futures, and swaps contracts. In these contracts, you are obligated to buy an asset at a pre-specified price, at a future date. What is the pre-specified price, and how is it determined? Chapter 5 examines forwards and futures on financial assets, Chapter 6 discusses commodities, and Chapter 7 looks at bond and interest rate forward contracts. Chapter 8 shows how swap prices can be deduced from forward prices. Part 3 studies option pricing. Chapter 9 develops intuition about options prior to delving into the mechanics of option pricing. Chapters 10 and 11 cover binomial option pricing and Chapter 12, the Black-Scholes formula and option Greeks. Chapter 13 explains delta-hedging, which is the technique used by market-makers when managing the risk of an option position, and how hedging relates to pricing. Chapter 14 looks at a few important exotic options, including Asian options, barrier options, compound options, and exchange options. The techniques and formulas in earlier chapters are applied in Part 4. Chapter 15 covers financial engineering, which is the creation of new financial products from the derivatives building blocks in earlier chapters. Debt and equity pricing, compensation options, and mergers are covered in Chapter 16. Chapter 17 studies real options—the application of derivatives models to the valuation and management of physical investments.
Finally, Part 5 explores pricing and hedging in depth. The material in this part explains in more detail the structure and assumptions underlying the standard derivatives models.
Chapter 18 covers the lognormal model and shows how the Black-Scholes formula is a discounted expected value. Chapter 19 discusses Monte Carlo valuation, a powerful and commonly used pricing technique. Chapter 20 explains what it means to say that stock prices follow a diffusion process, and also covers Itˆ ’s Lemma, which is a key result in o the study of derivatives. (At this point you will discover that Itˆ ’s Lemma has already been o developed intuitively in Chapter 13, using a simple numerical example.)
Chapter 21 derives the Black-Scholes-Merton partial differential equation (PDE).
Although the Black-Scholes formula is famous, the Black-Scholes-Merton equation, discussed in this chapter, is the more profound result. The martingale approach to pricing is covered in Chapter 22. We obtain the same pricing formulas as with the PDE, of course, but the perspective is different and helps to lay groundwork for later fixed income discussions.
Chapter 23 covers exotic options in more detail than Chapter 14, including digital barrier options and quantos. Chapter 24 discusses volatility estimation and stochastic volatility pricing models. Chapter 25 shows how the Black-Scholes and binomial analysis apply to bonds and interest rate derivatives. Chapter 26 covers value-at-risk, and Chapter 27 discusses credit products. xxiii

xxiv

Preface

NAVIGATING THE MATERIAL
The material is generally presented in order of increasing mathematical and conceptual difficulty, which means that related material is sometimes split across distant chapters. For example, fixed income is covered in Chapters 7 and 25, and exotic options in Chapters 14 and 23. As an illustration of one way to use the book, here is a rough outline of material
I cover in the courses I teach (within the chapters, I skip specific topics due to time constraints): .

Introductory course: 1–6, 7.1, 8–10, 12, 13.1–13.3, 14, 16, 17.1, 17.3.

.

Advanced course: 13, 18–22, 7, 8, 15, 23–27.

Table P.1 outlines some possible sets of chapters to use in courses that have different emphases. There are a few sections of the book that provide background on topics every reader should understand. These include short-sales (Section 1.4), continuous compounding
(Appendix B), prepaid forward contracts (Sections 5.1 and 5.2), and zero-coupon bonds and implied forward rates (Section 7.1).

A NOTE ON EXAMPLES
Many of the numerical examples in this book display intermediate steps to assist you in following the logic and steps of a calculation. Numbers displayed in the text necessarily are rounded to three or four decimal points, while spreadsheet calculations have many more significant digits. This creates a dilemma: Should results in the book match those you would obtain using a spreadsheet, or those you would obtain by computing the displayed equations? As a general rule, the numerical examples in the book will provide the results you would obtain by entering the equations directly in a spreadsheet. Due to rounding, the displayed equations will not necessarily produce the correct result.

SUPPLEMENTS
A robust package of ancillary materials for both instructors and students accompanies the text. Instructor’s Resources
For instructors, an extensive set of online tools is available for download from the catalog page for Derivatives Markets at www.pearsonhighered.com/mcdonald.
´
An online Instructor’s Solutions Manual by R¨ diger Fahlenbrach, Ecole Polytechu nique F´ d´ rale de Lausanne, contains complete solutions to all end-of-chapter problems in e e the text and spreadsheet solutions to selected problems.
The online Test Bank by Matthew W. Will, University of Indianapolis, features approximately ten to fifteen multiple-choice questions, five short-answer questions, and one longer essay question for each chapter of the book.
The Test Bank is available in several electronic formats, including Windows and
Macintosh TestGen files and Microsoft Word files. The TestGen and Test Bank are available online at www.pearsonhighered.com/irc.

Preface

TABLE P.1

xxv

Possible chapters for different courses. Chapters marked with a “Y” are strongly recommended, those marked with a “*” are recommended, and those with a “†” fit with the track but are optional. The advanced course assumes students have already taken a basic course. Sections 1.4, 5.1, 5.2, 7.1, and
Appendix B are recommended background for all introductory courses.

Introductory
Chapter
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.

Introduction
Intro. to Forwards and Options
Insurance, Collars, and Other Strategies
Intro. to Risk Management
Financial Forwards and Futures
Commodity Forwards and Futures
Interest Rate Forwards and Futures
Swaps
Parity and Other Option Relationships
Binomial Option Pricing: I
Binomial Option Pricing: II
The Black-Scholes Formula
Market-Making and Delta-Hedging
Exotic Options: I
Financial Engineering
Corporate Applications
Real Options
The Lognormal Distribution
Monte Carlo Valuation
Brownian Motion and Itˆ ’s Lemma o The Black-Scholes Equation
Risk-neutral and Martingale Pricing
Exotic Options: II
Volatility
Interest Rate Models
Value at Risk
Credit Risk

Risk
General Futures Options Management Advanced
Y
Y
Y
*
Y
*
*
Y
*
Y
*
Y


*





Y
Y
Y
*
Y
Y
Y
Y

*
*

*

Y
Y
Y
Y
Y


Y
Y
*
Y
Y
Y
*
*
*
*
*

Y
Y
Y
Y
Y
*
*
Y

Y
Y
*
*
Y
*
*
*
*

Y
*

Y
Y

Y
*

Y
Y
Y
Y

Y
Y
Y
Y
Y

xxvi

Preface

Online PowerPoint slides, developed by Peter Childs, University of Kentucky, provide lecture outlines and selected art from the book. Copies of the slides can be downloaded and distributed to students to facilitate note taking during class.

Student Resources
´
A printed Student Solutions Manual by R¨ diger Fahlenbrach, Ecole Polytechnique F´ d´ rale u e e de Lausanne, provides answers to all the even-numbered problems in the textbook.
A printed Student Problems Manual, by R¨ diger Fahlenbrach, contains additional u problems and worked-out solutions for each chapter of the textbook.
Spreadsheets with user-defined option pricing functions in Excel are included on a
CD-ROM packaged with the book. These Excel functions are written in VBA, with the code accessible and modifiable via the Visual Basic editor built into Excel. These spreadsheets and any updates are also posted on the book’s website.

ACKNOWLEDGMENTS
Kellogg student Tejinder Singh catalyzed the book in 1994 by asking that the Kellogg
Finance Department offer an advanced derivatives course. Kathleen Hagerty and I initially co-taught that course, and my part of the course notes (developed with Kathleen’s help and feedback) evolved into the last third of this book.
In preparing this revision, I once again received invaluable assistance from R¨ diger u ´
Fahlenbrach, Ecole Polytechnique F´ d´ rale de Lausanne, who read the manuscript and e e offered thoughtful suggestions, comments, and corrections. I received helpful feedback and suggestions from Akash Bandyopadhyay, Northwestern University; Snehal Banerjee,
Northwestern University; Kathleen Hagerty, Northwestern University; Ravi Jagannathan,
Northwestern University; Arvind Krishnamurthy, Northwestern University; Deborah Lucas, MIT; Alan Marcus, Boston College; Samuel Owen; Sergio Rebelo, Northwestern
University; and Elias Shu, University of Iowa. I would like to thank the following reviewers for their helpful feedback for the third edition: Tim Adam, Humboldt University of
Berlin; Philip Bond, University of Minnesota; Jay Coughenour, University of Delaware;
Jefferson Duarte, Rice University; Shantaram Hedge, University of Connecticut; Christine
X. Jiang, University of Memphis; Gregory LaFlame, Kent State University; Minqiang Li,
Bloomberg L.P.; D.K. Malhotra, Philadelphia University; Clemens Sialm, University of
Texas at Austin; Michael J. Tomas III, University of Vermont; and Eric Tsai, SUNY Oswego. Among the many readers who contacted me about errors and with suggestions, I would like to especially acknowledge Joe Francis and Abraham Weishaus.
I am grateful to Kellogg’s Zell Center for Risk Research for financial support. A special note of thanks goes to David Hait, president of OptionMetrics, for permission to include options data on the CD-ROM.
I would be remiss not to acknowledge those who assisted with previous editions, including George Allayanis, University of Virginia; Torben Andersen, Northwestern University; Tom Arnold, Louisiana State University; Turan Bali, Baruch College, City University of New York; David Bates, University of Iowa; Luca Benzoni, Federal Reserve Bank of
Chicago; Philip Bond, University of Minnesota; Michael Brandt, Duke University; Mark
Broadie, Columbia University; Jeremy Bulow, Stanford University; Charles Cao, Pennsylvania State University; Mark A. Cassano, University of Calgary; Mikhail Chernov, LSE;

Preface

George M. Constantinides, University of Chicago; Kent Daniel, Columbia University; Darrell Duffie, Stanford University; Jan Eberly, Northwestern University; Virginia France, University of Illinois; Steven Freund, Suffolk University; Rob Gertner, University of Chicago;
Bruce Grundy, University of Melbourne; Raul Guerrero, Dynamic Decisions; Kathleen
Hagerty, Northwestern University; David Haushalter, University of Oregon; Shantaram
Hegde, University of Connecticut; James E. Hodder, University of Wisconsin–Madison;
Ravi Jagannathan, Northwestern University; Avraham Kamara, University of Washington;
Darrell Karolyi, Compensation Strategies, Inc.; Kenneth Kavajecz, University of Wisconsin; Arvind Krishnamurthy, Northwestern University; Dennis Lasser, State University of
New York at Binghamton; C. F. Lee, Rutgers University; Frank Leiber, Bell Atlantic; Cornelis A. Los, Kent State University; Deborah Lucas, MIT; Alan Marcus, Boston College;
David Nachman, University of Georgia; Mitchell Petersen, Northwestern University; Todd
Pulvino, Northwestern University; Ehud Ronn, University of Texas, Austin; Ernst Schaumburg, Federal Reserve Bank of New York; Eduardo Schwartz, University of California–Los
Angeles; Nejat Seyhun, University of Michigan; David Shimko, Risk Capital Management
Partners, Inc.; Anil Shivdasani, University of North Carolina-Chapel Hill; Costis Skiadas,
Northwestern University; Donald Smith, Boston University; John Stansfield, University of
Missouri, Columbia; Christopher Stivers, University of Georgia; David Stowell, Northwestern University; Alex Triantis, University of Maryland; Joel Vanden, Dartmouth College; and Zhenyu Wang, Indiana University. The following served as software reviewers: James
Bennett, University of Massachusetts–Boston; Gordon H. Dash, University of Rhode Island; Adam Schwartz, University of Mississippi; Robert E. Whaley, Duke University; and
Nicholas Wonder, Western Washington University.
I thank R¨ diger Fahlenbrach, Matt Will, and Peter Childs for their excellent work on u the ancillary materials for this book. In addition, R¨ diger Fahlenbrach, Paskalis Glabadaniu dis, Jeremy Graveline, Dmitry Novikov, and Krishnamurthy Subramanian served as accuracy checkers for the first edition, and Andy Kaplin provided programming assistance.
Among practitioners who helped, I thank Galen Burghardt of Carr Futures, Andy
Moore of El Paso Corporation, Brice Hill of Intel, Alex Jacobson of the International
Securities Exchange, and Blair Wellensiek of Tradelink, L.L.C.
With any book, there are many long-term intellectual debts. From the many, I want to single out two. I had the good fortune to take several classes from Robert Merton at MIT while I was a graduate student. His classic papers from the 1970s are as essential today as they were 30 years ago. I also learned an enormous amount working with Dan Siegel, with whom I wrote several papers on real options. Dan’s death in 1991, at the age of 35, was a great loss to the profession, as well as to me personally.
The editorial and production team at Pearson has always supported the goal of producing a high-quality book. I was lucky to have the project overseen by Pearson’s talented and tireless Editor in Chief, Donna Battista. Project Manager Jill Kolongowski sheparded the revision, Development Editor Mary Clare McEwing expertly kept track of myriad details and offered excellent advice when I needed a sounding board. Production Project Manager
Carla Thompson marshalled forces to turn manuscript into a physical book and managed supplement production. Paul Anagnostopoulos of Windfall Software was a pleasure to work with. His ZzTEX macro package was used to typeset the book.
I received numerous compliments on the design of the first edition, which has been carried through ably into this edition. Kudos are due to Gina Kolenda Hagen and Jayne
Conte for their creativity in text and cover design.

xxvii

xxviii

Preface

The Pearson team and I have tried hard to minimize errors, including the use of the accuracy checkers noted above. Nevertheless, of course, I alone bear responsibility for remaining errors. Errata and software updates will be available at www.pearsonhighered
.com/mcdonald. Please let us know if you do find errors so we can update the list. a I produced drafts with Gnu Emacs, L TEX, Octave, and R, extraordinarily powerful and robust tools. I am deeply grateful to the worldwide community that produces and supports this extraordinary software.
My deepest and most heartfelt thanks go to my family. Through three editions I have relied heavily on their understanding, love, support, and tolerance. This book is dedicated to my wife, Irene Freeman, and children, Claire, David, and Henry.
RLM, June 2012

Robert L. McDonald is Erwin P. Nemmers Professor of Finance at Northwestern University’s Kellogg School of Management, where he has taught since 1984. He has been
Co-Editor of the Review of Financial Studies and Associate Editor of the Journal of Finance, Journal of Financial and Quantitative Analysis, Management Science, and other journals, and a director of the American Finance Association. He has a BA in Economics from the University of North Carolina at Chapel Hill and a Ph.D. in Economics from MIT.

Derivatives Markets

1

Introduction to
Derivatives

T

he world of finance has changed dramatically in recent decades. Electronic processing, globalization, and deregulation have all transformed markets, with many of the most important changes involving derivatives. The set of financial claims traded today is quite different than it was in 1970. In addition to ordinary stocks and bonds, there is now a wide array of products collectively referred to as financial derivatives: futures, options, swaps, credit default swaps, and many more exotic claims.
Derivatives sometimes make headlines. Prior to the financial crisis in 2008, there were a number of well-known derivatives-related losses: Procter & Gamble lost $150 million in
1994, Barings Bank lost $1.3 billion in 1995, Long-Term Capital Management lost $3.5
C
billion in 1998, the hedge fund Amaranth lost $6 billion in 2006, Soci´ t´ G´ n´ rale lost = 5 ee e e billion in 2008. During the crisis in 2008 the Federal Reserve loaned $85 billion to AIG in conjunction with AIG’s losses on credit default swaps. In the wake of the financial crisis, a significant portion of the Dodd-Frank Wall Street Reform and Consumer Protection Act of
2010 pertained to derivatives.
What is not in the headlines is the fact that, most of the time, for most companies and most users, these financial products are a useful and everyday part of business. Just as companies routinely issue debt and equity, they also routinely use swaps to fix the cost of production inputs, futures contracts to hedge foreign exchange risk, and options to compensate employees, to mention just a few examples.
Besides their widespread use, another important reason to understand derivatives is that the theory underlying financial derivatives provides a language and a set of analytical techniques that is fundamental for thinking about risk and valuation. It is almost impossible to discuss or perform asset management, risk management, credit evaluation, or capital budgeting without some understanding of derivatives and derivatives pricing.
This book provides an introduction to the products and concepts underlying derivatives. In this first chapter, we introduce some important concepts and provide some background to place derivatives in context. We begin by defining a derivative. We will then briefly examine financial markets, and see that derivatives markets have become increasingly important in recent years. The size of these markets may leave you wondering exactly what functions they serve. We next discuss the role of financial markets in our lives, and the importance of risk sharing. We also discuss different perspectives on derivatives. Finally, we will discuss how trading occurs, providing some basic concepts and language that will be useful in later chapters.

1

2

Chapter 1. Introduction to Derivatives

1.1 WHAT IS A DERIVATIVE?
A derivative is a financial instrument that has a value determined by the price of something else. Options, futures, and swaps are all examples of derivatives. A bushel of corn is not a derivative; it is a commodity with a value determined in the corn market. However, you could enter into an agreement with a friend that says: If the price of a bushel of corn in 1 year is greater than $3, you will pay the friend $1. If the price of corn is less than $3, the friend will pay you $1. This is a derivative in the sense that you have an agreement with a value depending on the price of something else (corn, in this case).
You might think: “That doesn’t sound like it’s a derivative; that’s just a bet on the price of corn.” Derivatives can be thought of as bets on the price of something, but the term
“bet” is not necessarily pejorative. Suppose your family grows corn and your friend’s family buys corn to mill into cornmeal. The bet provides insurance: You earn $1 if your family’s corn sells for a low price; this supplements your income. Your friend earns $1 if the corn his family buys is expensive; this offsets the high cost of corn. Viewed in this light, the bet hedges you both against unfavorable outcomes. The contract has reduced risk for both of you.
Investors who do not make a living growing or processing corn could also use this kind of contract simply to speculate on the price of corn. In this case the contract does not serve as insurance; it is simply a bet. This example illustrates a key point: It is not the contract itself, but how it is used, and who uses it, that determines whether or not it is risk-reducing.
Context is everything.
If you are just learning about derivatives, the implications of the definition will not be obvious right away. You will come to a deeper understanding of derivatives as we progress through the book, studying different products and their underlying economics.

1.2 AN OVERVIEW OF FINANCIAL MARKETS
In this section we will discuss the variety of markets and financial instruments that exist.
You should bear in mind that financial markets are rapidly evolving and that any specific description today may soon be out-of-date. Nevertheless, though the specific details may change, the basic economic functions associated with trading will continue to be necessary.

Trading of Financial Assets
The trading of a financial asset—i.e., the process by which an asset acquires a new owner— is more complicated than you might guess and involves at least four discrete steps. To understand the steps, consider the trade of a stock:
1. The buyer and seller must locate one another and agree on a price. This process is what most people mean by “trading” and is the most visible step. Stock exchanges, derivatives exchanges, and dealers all facilitate trading, providing buyers and sellers a means to find one another.
2. Once the buyer and seller agree on a price, the trade must be cleared, i.e., the obligations of each party are specified. In the case of a stock transaction, the buyer

1.2 An Overview of Financial Markets

will be required to deliver cash and the seller to deliver the stock. In the case of some derivatives transactions, both parties must post collateral.1
3. The trade must be settled, that is, the buyer and seller must deliver in the required period of time the cash or securities necessary to satisfy their obligations.
4. Once the trade is complete, ownership records are updated.
To summarize, trading involves striking a deal, clearing, settling, and maintaining records.
Different entities can be involved in these different steps.
Much trading of financial claims takes place on organized exchanges. An exchange is an organization that provides a venue for trading, and that sets rules governing what is traded and how trading occurs. A given exchange will trade a particular set of financial instruments. The New York Stock Exchange, for example, lists several thousand firms, both
U.S. and non-U.S., for which it provides a trading venue. Once upon a time, the exchange was solely a physical location where traders would stand in groups, buying and selling by talking, shouting, and gesturing. However, such in-person trading venues have largely been replaced by electronic networks that provide a virtual trading venue.2
After a trade has taken place, a clearinghouse matches the buyers and sellers, keeping track of their obligations and payments. The traders who deal directly with a clearinghouse are called clearing members. If you buy a share of stock as an individual, your transaction ultimately is cleared through the account of a clearing member.
For publicly traded securities in the United States, the Depository Trust and Clearing Corporation (DTCC) and its subsidiary, the National Securities Clearing Corporation
(NSCC), play key roles in clearing and settling virtually every stock and bond trade that occurs in the U.S. Other countries have similar institutions. Derivatives exchanges are always associated with a clearing organization because such trades must also be cleared and settled. Examples of derivatives clearinghouses are CME Clearing, which is associated with the CME Group (formerly the Chicago Mercantile Exchange), and ICE Clear U.S., which is associated with the Intercontinental Exchange (ICE).
With stock and bond trades, after the trade has cleared and settled, the buyer and seller have no continuing obligations to one another. However, with derivatives trades, one party may have to pay another in the future. To facilitate these payments and to help manage credit risk, a derivatives clearinghouse typically interposes itself in the transaction, becoming the buyer to all sellers and the seller to all buyers. This substitution of one counterparty for another is known as novation.
It is possible for large traders to trade many financial claims directly with a dealer, bypassing organized exchanges. Such trading is said to occur in the over-the-counter (OTC)

1. A party “posting collateral” is turning assets over to someone else to ensure that they will be able to meet their obligations. The posting of collateral is a common practice in financial markets.
2. When trading occurs in person, it is valuable for a trader to be physically close to other traders. With certain kinds of automated electronic trading, it is valuable for a trader’s computer to be physically close to the computers of an exchange. Traders make large investments to gain such speed advantages. One group is spending $300 million for an undersea cable in order to reduce communication time between New York and London by 5 milliseconds (Philips, 2012).

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market.3 There are several reasons why buyers and sellers might transact directly with dealers rather than on an exchange. First, it can be easier to trade a large quantity directly with another party. A seller of fifty thousand shares of IBM may negotiate a single price with a dealer, avoiding exchange fees as well as the market tumult and uncertainty about price that might result from simply announcing a fifty-thousand-share sale. Second, we might wish to trade a custom financial claim that is not available on an exchange. Third, we might wish to trade a number of different financial claims at once. A dealer could execute the entire trade as a single transaction, compared to the alternative of executing individual orders on a variety of different markets.
Most of the trading volume numbers you see reported in the newspaper pertain to exchange-based trading. Exchange activity is public and highly regulated. Over-the-counter trading is not easy to observe or measure and is generally less regulated. For many categories of financial claims, the value of OTC trading is greater than the value traded on exchanges.
Financial institutions are rapidly evolving and consolidating, so any description of the industry is at best a snapshot. Familiar names have melded into single entities. In recent years, for example, the New York Stock Exchange merged with Euronext, a group of
European exchanges, to form NYSE Euronext, which in turn bought the American Stock
Exchange (AMEX). The Chicago Mercantile Exchange merged with the Chicago Board of Trade and subsequently acquired the New York Mercantile Exchange, forming CME
Group.

Measures of Market Size and Activity
Before we discuss specific markets, it will be helpful to explain some ways in which the size of a market and its activity can be measured. There are at least four different measures that you will see mentioned in the press and on financial websites. No one measure is “correct” or best, but some are more applicable to stock and bond markets, others to derivatives markets.
The different measures count the number of transactions that occur daily (trading volume), the number of positions that exist at the end of a day (open interest), and the value (market value) and size (notional value) of these positions. Here are more detailed definitions:
Trading volume. This measure counts the number of financial claims that change hands daily or annually. Trading volume is the number commonly emphasized in press coverage, but it is a somewhat arbitrary measure because it is possible to redefine the meaning of a financial claim. For example, on a stock exchange, trading volume refers to the number of shares traded. On an options exchange, trading volume refers to the number of options traded, but each option on an individual stock covers 100 shares of stock.4
Market value. The market value (or “market cap”) of the listed financial claims on an exchange is the sum of the market value of the claims that could be traded, without

3. In an OTC trade, the dealer serves the economic function of a clearinghouse, effectively serving as counterparty to a large number of investors. Partly because of concerns about the fragility of a system where dealers also play the role of clearinghouses, the Dodd-Frank Act in 2010 required that, where feasible, derivatives transactions be cleared through designated clearinghouses. Duffie and Zhu (2011) discuss the costs and benefits of such a central clearing mandate.
4. When there are stock splits or mergers, individual stock options will sometimes cover a different number of shares.

1.2 An Overview of Financial Markets

regard to whether they have traded. A firm with 1 million shares and a share price of $50 has a market value of $50 million.5 Some derivative claims can have a zero market value; for such claims, this measure tells us nothing about activity at an exchange. Notional value. Notional value measures the scale of a position, usually with reference to some underlying asset. Suppose the price of a stock is $100 and that you have a derivative contract giving you the right to buy 100 shares at a future date. We would then say that the notional value of one such contract is 100 shares, or $10,000. The concept of notional value is especially important in derivatives markets. Derivatives exchanges frequently report the notional value of contracts traded during a period of time. Open interest. Open interest measures the total number of contracts for which counterparties have a future obligation to perform. Each contract will have two counterparties.
Open interest measures contracts, not counterparties. Open interest is an important statistic in derivatives markets.

Stock and Bond Markets
Companies often raise funds for purposes such as financing investments. Typically they do so either by selling ownership claims on the company (common stock) or by borrowing money (obtaining a bank loan or issuing a bond). Such financing activity is a routine part of operating a business. Virtually every developed country has a market in which investors can trade with each other the stocks that firms have issued.
Securities exchanges facilitate the exchange of ownership of a financial asset from one party to another. Some exchanges, such as the NYSE, designate market-makers, who stand ready to buy or sell to meet customer demand. Other exchanges, such as NASDAQ, rely on a competitive market among many traders to provide fair prices. In practice, most investors will not notice these distinctions.
The bond market is similar in size to the stock market, but bonds generally trade through dealers rather than on an exchange. Most bonds also trade much less frequently than stocks.
Table 1.1 shows the market capitalization of stocks traded on the six largest stock exchanges in the world in 2011. To provide some perspective, the aggregate value of publicly traded common stock in the U.S. was about $20 trillion at the end of 2011. Total corporate debt was about $10 trillion, and borrowings of federal, state, and local governments in the
U.S. was about $18 trillion. By way of comparison, the gross domestic product (GDP) of the U.S. in 2011 was $15.3 trillion.6

5. For example, in early 2012 IBM had a share price of about $180 and about 1.15 billion shares outstanding.
The market value was thus about $180 × 1.15 billion = $207 billion. Market value changes with the price of the underlying shares.
6. To be clear about the comparison: The values of securities represent the outstanding amount at the end of the year, irrespective of the year in which the securities were first issued. GDP, by contrast, represents output produced in the U.S. during the year. The market value and GDP numbers are therefore not directly comparable. The comparison is nonetheless frequently made.

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Chapter 1. Introduction to Derivatives

TABLE 1.1

The six largest stock exchanges in the world, by market capitalization (in billions of US dollars) in 2011.

Rank

Exchange

Market Cap (Billions of U.S. $)

1
3
2
4
5
6

NYSE Euronext (U.S.)
Nasdaq OMX
Tokyo Stock Exchange
London Stock Exchange
NYSE Euronext (Europe)
Shanghai Stock Exchange

11,796
3,845
3,325
3,266
2,447
2,357
Source: http://www.world-exchanges.org/.

Derivatives Markets
Because a derivative is a financial instrument with a value determined by the price of something else, there is potentially an unlimited variety of derivative products. Derivatives exchanges trade products based on a wide variety of stock indexes, interest rates, commodity prices, exchange rates, and even nonfinancial items such as weather. A given exchange may trade futures, options, or both. The distinction between exchanges that trade physical stocks and bonds, as opposed to derivatives, has largely been due to regulation and custom, and is eroding. The introduction and use of derivatives in a market often coincides with an increase in price risk in that market. Currencies were permitted to float in 1971 when the gold standard was officially abandoned. The modern market in financial derivatives began in
1972, when the Chicago Mercantile Exchange (CME) started trading futures on seven currencies. OPEC’s 1973 reduction in the supply of oil was followed by high and variable oil prices. U.S. interest rates became more volatile following inflation and recessions in the
1970s. The market for natural gas has been deregulated gradually since 1978, resulting in a volatile market and the introduction of futures in 1990. The deregulation of electricity began during the 1990s.
To illustrate the increase in variability since the early 1970s, panels (a)–(c) in Figure
1.1 show monthly changes for the 3-month Treasury bill rate, the dollar-pound exchange rate, and a benchmark spot oil price. The link between price variability and the development of derivatives markets is natural—there is no need to manage risk when there is no risk.7
When risk does exist, we would expect that markets will develop to permit efficient risksharing. Investors who have the most tolerance for risk will bear more of it, and risk-bearing will be widely spread among investors.

7. It is sometimes argued that the existence of derivatives markets can increase the price variability of the underlying asset or commodity. However, the introduction of derivatives can also be a response to increased price variability.

1.2 An Overview of Financial Markets

7

FIGURE 1.1

% Change in dollar-pound exchange rate

Change in 3-month T-bill rate

(a) The monthly change in the 3-month Treasury bill rate, 1947–2011. (b) The monthly percentage change in the dollar-pound exchange rate, 1947–2011. (c) The monthly percentage change in the
West Texas Intermediate (WTI) spot oil price, 1947–2011. (d) Millions of futures contracts traded annually at the Chicago Board of Trade (CBT), Chicago Mercantile Exchange (CME), and the New
York Mercantile Exchange (NYMEX), 1970–2011.

2
0
–2
–4
1950

1970

1990

0.10
0.05
0.00
–0.05
–0.10
–0.15
–0.20
1950

2010

1970

0.8

Trading volume
(millions of contracts)

% Change in WTI spot price

Date
(a)

0.6
0.4
0.2
0.0
–0.2

1990
Date
(b)

2010

2500
2000
1500
1000
500
0

–0.4
1950

1970

1990
Date
(c)

2010

1980

1990
Date
(d)

2000

2010

Sources: (a) St. Louis Fed; (b) DRI and St. Louis Fed; (c) St. Louis Fed; (d) CRB Yearbook.

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Chapter 1. Introduction to Derivatives

TABLE 1.2

Examples of underlying assets on which futures contracts are traded.

Category

Description

Stock index

S&P 500 index, Euro Stoxx 50 index, Nikkei 225, DowJones Industrials, Dax, NASDAQ, Russell 2000, S&P Sectors
(healthcare, utilities, technology, etc.)
30-year U.S. Treasury bond, 10-year U.S. Treasury notes, Fed funds rate, Euro-Bund, Euro-Bobl, LIBOR, Euribor
Euro, Japanese yen, British pound, Swiss franc, Australian dollar,
Canadian dollar, Korean won
Oil, natural gas, gold, copper, aluminum, corn, wheat, lumber, hogs, cattle, milk
Heating and cooling degree-days, credit, real estate

Interest rate
Foreign exchange
Commodity
Other

Table 1.2 provides examples of some of the specific prices and items upon which futures contracts are based.8 Some of the names may not be familiar to you, but most will appear later in the book.9
Panel (d) of Figure 1.1 depicts combined futures contract trading volume for the three largest U.S. futures exchanges over the last 40 years. The point of this graph is that trading activity in futures contracts has grown enormously over this period. Derivatives exchanges in other countries have generally experienced similar growth. Eurex, the European electronic exchange, traded over 2 billion contracts in 2011. There are many other important derivatives exchanges, including the Chicago Board Options Exchange, the International Securities
Exchange (an electronic exchange headquartered in the U.S.), the London International
Financial Futures Exchange, and exchanges headquartered in Australia, Brazil, China,
Korea, and Singapore, among many others.
The OTC markets have also grown rapidly over this period. Table 1.3 presents an estimated annual notional value of swaps in five important categories. The estimated yearend outstanding notional value of interest rate and currency swaps in 2010 was an eyepopping $523 trillion. For a variety of reasons the notional value number can be difficult to interpret, but the enormous growth in these contracts in recent years is unmistakable.

8. It is instructive to browse the websites of derivatives exchanges. For example, the CME Group open interest report for April 2012 reports positive open interest for 16 different interest rate futures contracts,
26 different equity index contracts, 15 metals, hundreds of different energy futures contracts, and over 40 currencies. Many of these contracts exist to handle specialized requirements.
9. German government bonds are known as “Bubills” (bonds with maturity of less than 1 year), “Schaetze”
(maturity of 2 years), “Bobls” (5 years), and “Bunds” (10 and 30 years). Futures contracts also trade on
Japanese and UK government bonds (“gilt”).

1.3 The Role of Financial Markets

TABLE 1.3

Estimated year-end notional value of outstanding derivative contracts, by category, in billions of dollars.

Foreign
Exchange
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010

Interest
Rate

Equity

Commodity

Credit
Default

Total

18011
14344
15665
16747
18447
24475
29288
31360
40270
56238
50042
49181
57795

50014
60090
64667
77567
101657
141990
190501
211970
291581
393138
432657
449874
465259

1488
1809
1890
1880
2308
3787
4384
5793
7487
8469
6471
5937
5634

408
548
662
598
923
1405
1443
5434
7115
8455
4427
2944
2921







6395
13908
28650
58243
41882
32692
29897

80309
88201
95199
111177
141665
197166
258627
299260
418131
585932
598147
603899
601046

Source: Bank of International Settlements.

1.3 THE ROLE OF FINANCIAL MARKETS
Stock, bond, and derivatives markets are large and active, but what role do financial markets play in the economy and in our lives? We routinely see headlines stating that the Dow Jones
Industrial Average has gone up 100 points, the dollar has fallen against the euro, and interest rates have risen. But why do we care about these things? In this section we will examine how financial markets affect our lives.

Financial Markets and the Averages
To understand how financial markets affect us, consider the Average family, living in
Anytown. Joe and Sarah Average have children and both work for the XYZ Co., the dominant employer in Anytown. Their income pays for their mortgage, transportation, food, clothing, and medical care. Remaining income goes into savings earmarked for their children’s college tuition and their own retirement.
The Averages are largely unaware of the ways in which global financial markets affect their lives. Here are a few:
.

The Averages invest their savings in mutual funds that own stocks and bonds from companies around the world. The transaction cost of buying stocks and bonds in this

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Chapter 1. Introduction to Derivatives

way is low. Moreover, the Averages selected mutual funds that provide diversified investments. As a result, the Averages are not heavily exposed to any one company.10
.

.

.

.

The Averages live in an area susceptible to tornadoes and insure their home. The local insurance company reinsures tornado risk in global markets, effectively pooling
Anytown tornado risk with Japan earthquake risk and Florida hurricane risk. This pooling makes insurance available at lower rates and protects the Anytown insurance company. The Averages financed their home with a mortgage from Anytown bank. The bank in turn sold the mortgage to other investors, freeing itself from interest rate and default risk associated with the mortgage. Because the risks of their mortgage is borne by those willing to pay the highest price for it, the Averages get the lowest possible mortgage rate.
The Average’s employer, XYZ Co., can access global markets to raise money. Investors in Asia, for example, may thereby finance an improvement to the Anytown factory. XYZ Co. insures itself against certain risks. In addition to having property and casualty insurance for its buildings, it uses global derivatives markets to protect itself against adverse currency, interest rate, and commodity price changes. By being able to manage these risks, XYZ is less likely to go into bankruptcy, and the Averages are less likely to become unemployed.

In all of these examples, particular financial functions and risks have been split up and parceled out to others. A bank that sells a mortgage does not have to bear the risk of the mortgage. A single insurance company does not bear the entire risk of a regional disaster.
Risk-sharing is one of the most important functions of financial markets.

Risk-Sharing
Risk is an inevitable part of our lives and all economic activity. As we’ve seen in the example of the Averages, financial markets enable at least some of these risks to be shared. Risk arises from natural events, such as earthquakes, floods, and hurricanes, and from unnatural events such as wars and political conflicts. Drought and pestilence destroy agriculture every year in some part of the world. Some economies boom as others falter. On a more personal scale, people are born, die, retire, find jobs, lose jobs, marry, divorce, and become ill.
Given that risk exists, it is natural to have arrangements where the lucky share with the unlucky. There are both formal and informal risk-sharing arrangements. On the formal level, the insurance market is a way to share risk. Buyers pay a premium to obtain various kinds of insurance, such as homeowner’s insurance. Total collected premiums are then available to help those whose houses burn down. The lucky, meanwhile, did not need insurance and have lost their premium. This market makes it possible for the lucky to help the unlucky.
On the informal level, risk-sharing also occurs in families and communities, where those encountering misfortune are helped by others.

10. There is one important risk that the Averages cannot easily avoid. Since both Averages work at XYZ, they run the risk that if XYZ falls on hard times they will lose their jobs. This could be an important reason for the Averages to avoid investing in XYZ.

1.4 The Uses of Derivatives

In the business world, changes in commodity prices, exchange rates, and interest rates can be the financial equivalent of a house burning down. If the dollar becomes expensive relative to the yen, some companies are helped and others are hurt. It makes sense for there to be a mechanism enabling companies to exchange this risk, so that the lucky can, in effect, help the unlucky.
Even insurers need to share risk. Consider an insurance company that provides earthquake insurance for California residents. A large earthquake could generate claims sufficient to bankrupt a stand-alone insurance company. Thus, insurance companies often use the reinsurance market to buy, from reinsurers, insurance against large claims. Reinsurers pool different kinds of risks, thereby enabling insurance risks to become more widely held.
In some cases, reinsurers further share risks by issuing catastrophe bonds—bonds that the issuer need not repay if there is a specified event, such as a large earthquake, causing large insurance claims. Bondholders willing to accept earthquake risk can buy these bonds, in exchange for greater interest payments on the bond if there is no earthquake. An earthquake bond allows earthquake risk to be borne by exactly those investors who wish to bear it.
You might be wondering what this discussion has to do with the notions of diversifiable and nondiversifiable risk familiar from portfolio theory. Risk is diversifiable risk if it is unrelated to other risks. The risk that a lightning strike will cause a factory to burn down, for example, is idiosyncratic and hence diversifiable. If many investors share a small piece of this risk, it has no significant effect on anyone. Risk that does not vanish when spread across many investors is nondiversifiable risk. The risk of a stock market crash, for example, is nondiversifiable.
Financial markets in theory serve two purposes. Markets permit diversifiable risk to be widely shared. This is efficient: By definition, diversifiable risk vanishes when it is widely shared. At the same time, financial markets permit nondiversifiable risk, which does not vanish when shared, to be held by those most willing to hold it. Thus, the fundamental economic idea underlying the concepts and markets discussed in this book is that the existence of risk-sharing mechanisms benefits everyone.
Derivatives markets continue to evolve. A recent development has been the growth in prediction markets, discussed in the box on page 12.

1.4 THE USES OF DERIVATIVES
We can think about derivatives and other financial claims in different ways. One is a functional perspective: Who uses them and why? Another is an analytical perspective: When we look at financial markets, how do we interpret the activity that we see? In this section, we discuss these different perspectives.

Uses of Derivatives
What are reasons someone might use derivatives? Here are some motives:
Risk management. Derivatives are a tool for companies and other users to reduce risks.
With derivatives, a farmer—a seller of corn—can enter into a contract that makes a payment when the price of corn is low. This contract reduces the risk of loss for the farmer, who we therefore say is hedging. Every form of insurance is a derivative: For

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BOX

P

Chapter 1. Introduction to Derivatives

1.1: Prediction Markets

rediction markets are derivatives markets in which the value of traded claims depends on the outcome of events. With one common contract, an investor can own a claim that receives $1 if the event occurs, or sell a claim that requires the investor to pay $1 if the event occurs. Such markets can be used to speculate on presidential elections, the winner of an Olympic event, the occurrence of a natural disaster, or the value of a government statistic such as employment or the consumer price index.
Much of the interest in prediction markets stems from the idea that the price of a contract will aggregate individual information that could not otherwise be observed. For years, the
Iowa Electronic Markets (http://tippie.uiowa.edu/ iem/markets/) has permitted speculation on the outcome of presidential and other elections. The
“vote share” contracts pay in cents the percentage of the popular vote received by the official candidate of a party. For example, if the Re-

publican were to receive 48% of the vote, the
Republican vote share contract on election day would be worth $0.48. The Democratic and Republican “winner-take-all” contracts pay $1 if that party wins the election, and zero otherwise.
There are also party nomination contracts that allow an investor to bet on a specific candidate winning the party nomination. (See the box on page 17.)
Intrade.com allows political bets along with a wide variety of others. One contract stated its event as “Higgs Boson Particle to be observed on/before 31 Dec 2014,” while another stated
“Highest Marginal Single-Filer Fed Income Tax
Rate to be Equal or Greater than 34% in 2012.”
Prediction markets face significant regulatory hurdles in the U.S. As of this writing, a 2006 law prohibiting unauthorized internet gambling made it illegal for U.S. investors to use any prediction market not authorized by the Commodity Futures
Trading Commission.

example, automobile insurance pays if you have an accident. If you destroy your car in an accident, your insurance is valuable; if the car remains undamaged, it is not.
Speculation. Derivatives can serve as investment vehicles. As you will see later in the book, derivatives can provide a way to make bets that are highly leveraged (that is, the potential gain or loss on the bet can be large relative to the initial cost of making the bet) and tailored to a specific view. For example, if you want to bet that the S&P
500 stock index will be between 1300 and 1400 1 year from today, derivatives can be constructed to let you do that.
Reduced transaction costs. Sometimes derivatives provide a lower-cost way to undertake a particular financial transaction. For example, the manager of a mutual fund may wish to sell stocks and buy bonds. Doing this entails paying fees to brokers and paying other trading costs, such as the bid-ask spread, which we will discuss later. It is possible to trade derivatives instead and achieve the same economic effect as if stocks had actually been sold and replaced by bonds. Using the derivative might result in lower transaction costs than actually selling stocks and buying bonds.
Regulatory arbitrage. It is sometimes possible to circumvent regulatory restrictions, taxes, and accounting rules by trading derivatives. Derivatives are often used, for example, to achieve the economic sale of stock (receive cash and eliminate the risk

1.4 The Uses of Derivatives

of holding the stock) while still maintaining physical possession of the stock. This transaction may allow the owner to defer taxes on the sale of the stock, or retain voting rights, without the risk of holding the stock.
These are common reasons for using derivatives. The general point is that derivatives provide an alternative to a simple sale or purchase, and thus increase the range of possibilities for an investor or manager seeking to accomplish some goal. Obviously, for society as a whole, hedging may be desirable while regulatory arbitrage is not. But it is difficult to control trading on the basis of motive.
In recent years the U.S. Securities and Exchange Commission (SEC), the Financial
Accounting Standards Board (FASB), and the International Accounting Standard Board
(IASB) have increased the requirements for corporations to report on their use of derivatives.
Nevertheless, surprisingly little is known about how companies actually use derivatives to manage risk. The basic strategies companies use are well-understood—and will be described in this book—but it is not known, for example, what fraction of perceived risk is hedged by a given company, or by all companies in the aggregate. We frequently do not know a company’s specific rationale for hedging or not hedging.
We would expect the use of derivatives to vary by type of firm. For example, financial firms, such as banks, are highly regulated and have capital requirements. They may have assets and liabilities in different currencies, with different maturities, and with different credit risks. Hence banks could be expected to use interest rate derivatives, currency derivatives, and credit derivatives to manage risks in those areas. Manufacturing firms that buy raw materials and sell in global markets might use commodity and currency derivatives, but their incentives to manage risk are less clear-cut because they are not regulated in the same ways as financial firms.

Perspectives on Derivatives
How you think about derivatives depends on who you are. In this book we will think about three distinct perspectives on derivatives:
The end-user perspective. End-users are the corporations, investment managers, and investors who enter into derivative contracts for the reasons listed in the previous section: to manage risk, speculate, reduce costs, or avoid a rule or regulation. Endusers have a goal (for example, risk reduction) and care about how a derivative helps to meet that goal.
The market-maker perspective. Market-makers are intermediaries, traders who will buy derivatives from customers who wish to sell, and sell derivatives to customers who wish to buy. In order to make money, market-makers charge a spread: They buy at a low price and sell at a high price. In this respect market-makers are like grocers, who buy at the low wholesale price and sell at the higher retail price. Market-makers are also like grocers in that their inventory reflects customer demands rather than their own preferences: As long as shoppers buy paper towels, the grocer doesn’t care whether they buy the decorative or super-absorbent style. After dealing with customers, market-makers are left with whatever position results from accommodating customer demands. Market-makers typically hedge this risk and thus are deeply concerned about the mathematical details of pricing and hedging.

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The economic observer. Finally, we can look at the use of derivatives, the activities of the market-makers, the organization of the markets, and the logic of the pricing models and try to make sense of everything. This is the activity of the economic observer.
Regulators must often don their economic observer hats when deciding whether and how to regulate a certain activity or market participant.
These three perspectives are intertwined throughout the book, with different degrees of emphasis.

Financial Engineering and Security Design
One of the major ideas in derivatives—perhaps the major idea—is that it is generally possible to create a given payoff in multiple ways. The construction of a given financial product from other products is sometimes called financial engineering. The fact that this is possible has several implications. First, since market-makers need to hedge their positions, this idea is central in understanding how market-making works. The market-maker sells a contract to an end-user, and then creates an offsetting position that pays him if it is necessary to pay the customer. This creates a hedged position.
Second, the idea that a given contract can be replicated often suggests how it can be customized. The market-maker can, in effect, turn dials to change the risk, initial premium, and payment characteristics of a derivative. These changes permit the creation of a product that is more appropriate for a given situation.
Third, it is often possible to improve intuition about a given derivative by realizing that it is equivalent to something we already understand.
Finally, because there are multiple ways to create a payoff, the regulatory arbitrage discussed above can be difficult to stop. Distinctions existing in the tax code, or in regulations, may not be enforceable, since a particular security or derivative that is regulated or taxed may be easily replaced by one that is treated differently but has the same economic profile. 1.5 BUYING AND SHORT-SELLING FINANCIAL ASSETS
Throughout this book we will talk about buying, selling, and short-selling assets such as stocks. These basic transactions are so important that it is worth describing the details.
First, it is important to understand the costs associated with buying and selling. Second, it is helpful to understand the mechanisms one can use to buy or sell. Third, a very important idea used throughout the book is that of short-sales. The concept of short-selling seems as if it should be intuitive—a short-sale is just the opposite of a purchase—but for almost everyone it is hard to grasp at first. Even if you are familiar with short-sales, you should spend a few minutes reading this section.
Although we will use shares of stock to illustrate the mechanics of buying and selling, there are similar issues associated with buying any asset.

Transaction Costs and the Bid-Ask Spread
Suppose you want to buy shares of XYZ stock. We can calculate the cost to own 100 shares:
If the price to buy the stock is $50, 100 shares will cost $50 × 100 = $5000. However, we must also account for transaction costs.

1.5 Buying and Short-Selling Financial Assets

First, there is a commission, which is a transaction fee you pay your broker. A commission for the above order could be $15, or 0.3% of the purchase price.
Second, the term “stock price” is, surprisingly, imprecise. There are in fact two prices, a price at which you can buy, and a price at which you can sell. The price at which you can buy is called the offer price or ask price, and the price at which you can sell is called the bid price. To understand these terms, consider the position of the broker.
To buy stock, you contact a broker. Suppose that you wish to buy immediately at the best available price. If the stock is not too obscure and your order is not too large, your purchase will probably be completed in a matter of seconds. Where does the stock that you have just bought come from? It is possible that at the exact same moment, another customer called the broker and put in an order to sell. More likely, however, a market-maker sold you the stock. As their name implies, market-makers make markets. If you want to buy, they sell, and if you want to sell, they buy. In order to earn a living, market-makers sell for a high price and buy for a low price. If you deal with a market-maker, therefore, you buy for a high price and sell for a low price. This difference between the price at which you can buy and the price at which you can sell is called the bid-ask spread.11 In practice, the bidask spread on the stock you are buying may be $49.75 to $50. This means that you can buy for $50/share and sell for $49.75/share. If you were to buy immediately and then sell, you would pay the commission twice, and you would pay the bid-ask spread.
Example 1.1 Suppose XYZ is bid at $49.75 and offered at $50, and the commission is $15. If you buy 100 shares of the stock you pay ($50 × 100) + $15 = $5015. If you immediately sell them again, you receive ($49.75 × 100) − $15 = $4960. Your round-trip transaction cost—the difference between what you pay and what you receive from a sale, not counting changes in the bid and ask prices—is $5015 − $4960 = $55.
This discussion reveals where the terms “bid” and “ask” come from. The terminology seems backward, but rather than the price you pay, the bid price is what the market-maker pays; hence it is the price at which you sell. The offer price is what the market-maker will sell for; hence it is what you have to pay. The terminology reflects the perspective of the market-maker. What happens to your shares after you buy them? Unless you make other arrangements, shares are typically held in a central depository (in the U.S., at the DTCC) in the name of your broker. Such securities are said to be held in street name.

Ways to Buy or Sell
A buyer or seller of an asset can employ different strategies in trading the asset. You implement these different strategies by telling the broker (or the electronic trading system) what kind of order you are submitting.
A market order is an instruction to trade a specific quantity of the asset immediately, at the best price that is currently available. The advantage of a market order is that the trade

11. If you think a bid-ask spread is unreasonable, ask what a world without dealers would be like. Every buyer would have to find a seller, and vice versa. The search would be costly and take time. Dealers, because they maintain inventory, offer an immediate transaction, a service called immediacy.

15

16

Chapter 1. Introduction to Derivatives

is executed as soon as possible. The disadvantage of a market order is that you might have been able to get a better price had you been more patient.
A limit order is an instruction to trade a specific quantity of the asset at a specified or better price. A limit order to buy 100 shares at $47.50 can be filled at $47.50 or below. A limit order to sell at $50.25 can be filled at $50.25 or above. Having your limit order filled depends upon whether or not anyone else is willing to trade at that price. As time passes, your order may or may not be filled. Thus, the advantage of a limit order is obtaining a better price. The disadvantage is the possibility that the order is never filled.
There are other kinds of orders. For example, suppose you own 100 shares of XYZ.
If you enter a stop-loss order at $45, then your order becomes a market order to sell once the price falls to $45. Because your order is a market order, you may end up selling for less than $45.
In the earlier example we supposed that the bid and ask prices for XYZ were $49.75 and $50. You can think of those prices as limit orders—someone has offered to buy at $49.75 and (possibly someone different) to sell at $50.
The box on page 17 illustrates bid and offer prices for one prediction market.

Short-Selling
The sale of a stock you do not already own is called a short-sale. In order to understand short-sales, we first need to understand the terms “long” and “short.”
When we buy something, we are said to have a long position in that thing. For example, if we buy the stock of XYZ, we pay cash and receive the stock. Some time later, we sell the stock and receive cash. This transaction is a form of lending, in that we pay money today and receive money back in the future. For many assets the rate of return we receive is not known in advance (the return depends upon whether the stock price goes up or down), but it is a loan nonetheless.
The opposite of a long position is a short position. A short-sale of XYZ entails borrowing shares of XYZ from an owner, and then selling them, receiving the cash.12 Some time later, we buy back the XYZ stock, paying cash for it, and return it to the lender. A shortsale can be viewed as a way of borrowing money. With ordinary borrowing, you receive money today and repay it later, paying a rate of interest set in advance. This also happens with a short-sale, except that typically you don’t know in advance the rate you will pay.
There are at least three reasons to short-sell:
1. Speculation: A short-sale, considered by itself, makes money if the price of the stock goes down. The idea is to first sell high and then buy low. (With a long position, the idea is to first buy low and then sell high.)
2. Financing: A short-sale is a way to borrow money, and it is frequently used as a form of financing. This is very common in the bond market, for example.
3. Hedging: You can undertake a short-sale to offset the risk of owning the stock or a derivative on the stock. This is frequently done by market-makers and traders.

12. Most brokerage agreeements give your broker the right to lend your shares to another investor. The broker earns fees from doing this. You generally do not know if your shares have been loaned.

1.5 Buying and Short-Selling Financial Assets

BOX

17

1.2: Bid and Ask Prices in a Prediction Market

I

n the box on page 12 we discussed prediction markets. These markets have bid-ask spreads, as you would expect. Here is a table showing the bid-ask spreads in the U.S. presidential market from the Iowa Presidential Nomination Market on May 24, 2012. The “vote share” contract pays the percentage of the popular vote received by that party’s candidate, while the “winner take all” contract pays $1 if the candidate wins and zero otherwise. Share of Vote
Party

Bid

Winner Takes All
Ask

Party

Bid

Ask

Democratic

0.512

0.527

Democratic

0.592

0.600

Republican

0.470

0.484

Republican

0.400

0.409

Total

0.982

1.011

Total

0.992

1.009

If you wished to buy the Democratic winnertake-all contract, you would pay $0.600 per contract. An immediate sale would earn you
$0.592. If you bought both the Democratic and
Republican contracts, you would be guaranteed to earn $1 by the end of the convention, but the cost would be $1.009. Similarly, if you sold all four, you would have to pay $1 at the end of the convention but you would receive only $0.992.
The prices in the table are limit orders placed by other traders. What you cannot see in the table is how many contracts you can trade at those prices
(this is called market depth). You also cannot see any additional buy limit orders below the bid price and additional sell limit orders above the ask price.

These reasons are not mutually exclusive. For example, a market-maker might use a short-sale to simultaneously hedge and finance a position.
Example: Short-Selling Wine. Because short-sales can seem confusing, here is a detailed example that illustrates how short-sales work.
There are markets for many collectible items, such as fine wines. If you believe prices will rise, you would buy the wine on the market and plan to sell after the price rises.
However, suppose there is a wine from a particular vintage and producer that you believe to be overpriced and you expect the price to fall. How could you speculate based on this belief? The answer is that you can engage in a short-sale.
In order to short-sell, you must first obtain wine. You can do this by borrowing a case from a collector. The collector, of course, will want a promise that you will return the wine at some point; suppose you agree to return it 1 week later. Having reached agreement, you borrow the wine and then sell it at the market price. After 1 week, you acquire a replacement case on the market, then return it to the collector from whom you originally borrowed the wine. If the price has fallen, you will have bought the replacement wine for less than the price at which you sold the original, so you make money. If the price has risen, you have lost money. Either way, you have just completed a short-sale of wine. The act of buying replacement wine and returning it to the lender is said to be closing or covering the short position. Note that short-selling is a way to borrow money. Initially, you received money from selling the wine. A week later you paid the money back (you had to buy a replacement case to return to the lender). The rate of interest you paid was low if the price of the replacement case was low, and high if the price of the replacement case was high.
This example is obviously simplified. We have assumed several points:

18

Chapter 1. Introduction to Derivatives

TABLE 1.4

Cash flows associated with short-selling a share of IBM for
90 days. S0 and S90 are the share prices on days 0 and 90.
Note that the short-seller must pay the dividend, D, to the share-lender. Day 0
Action
Security
Cash

.

.

.

Dividend Ex-Day

Day 90

Borrow shares
Sell shares
+S0



−D

Return shares
Purchase shares
−S90

It is easy to find a lender of wine.
It is easy to buy, at a fair price, satisfactory wine to return to the lender: The wine you buy after 1 week is a perfect substitute for the wine you borrowed.
The collector from whom you borrowed is not concerned that you will fail to return the borrowed wine.

Example: Short-Selling Stock. Now consider a short-sale of stock. As with wine, when you short-sell stock you borrow the stock and sell it, receiving cash today. At some future date you buy the stock in the market and return it to the original owner.
Suppose you want to short-sell IBM stock for 90 days. Table 1.4 depicts the cash flows.
Observe in particular that if the share pays dividends, the short-seller must in turn make dividend payments to the share-lender. This issue did not arise with wine! This dividend payment is taxed to the recipient, just like an ordinary dividend payment, and it is taxdeductible to the short-seller.
Notice that the cash flows in Table 1.4 are exactly the opposite of the cash flows from purchasing the stock. Thus, short-selling is literally the opposite of buying.

The Lease Rate of an Asset
We have seen that when you borrow an asset it may be necessary to make payments to the lender. Dividends on short-sold stock are an example of this. We will refer to the payment required by the lender as the lease rate of the asset. This concept will arise frequently, and, as we will see, it provides a unifying concept for our later discussions of derivatives.
The wine example did not have a lease payment. But under some circumstances it might be necessary to make a payment to borrow wine. Wine does not pay an explicit dividend but does pay an implicit dividend if the owner enjoys seeing bottles in the cellar.
The owner might thus require a payment in order to lend a bottle. This would be a lease rate for wine.

Risk and Scarcity in Short-Selling
The preceding examples were simple illustrations of the mechanics and economics of shortselling, and they demonstrate the ideas you will need to understand our discussions of derivatives. It turns out, however, that some of the complexities we skipped over are easy

1.5 Buying and Short-Selling Financial Assets

to understand and are important in practice. In this section we use the wine example to illustrate some of these practical issues.
Credit Risk. As the short-seller, you have an obligation to the lender to return the wine.
The lender fears that you will renege on this obligation. This concern can be addressed with collateral: After you sell the wine, the lender can hold the money you received from selling the wine. You have an obligation to return the wine; the lender keeps the money in the event that you don’t.
Holding on to the money will help the lender feel more secure, but after thinking the matter over, the lender will likely want more from you than just the current value of the wine. Suppose you borrow $5000 worth of wine. What happens, the lender will think, if the price of that particular wine rises 1 week later to $6000? This is a $1000 loss on your short-sale. In order to return the wine, you will have to pay $6000 for wine you just sold for
$5000. Perhaps you cannot afford the extra $1000 and you will fail to return the borrowed wine. The lender, thinking ahead, will be worried at the outset about this possibility and will ask you to provide more than the $5000 the wine is worth—say, an extra $1000. This extra amount is called a haircut and serves to protect the lender against your failure to return the wine when the price rises.13 In practice, short-sellers must have funds—called capital—to be able to pay haircuts. The amount of capital places a limit on their ability to short-sell. Scarcity. As the short-seller, do you need to worry about the short-sale proceeds? The lender is going to have $6000 of your money. Most of this, however, simply reflects your obligation, and we could ask a trustworthy third party, such as a bank, to hold the money so the lender cannot abscond with it. However, when you return the wine, you are going to want your money back, plus interest. This raises the question: What rate of interest will the lender pay you? Over the course of the short-sale, the lender can invest your money, earning, say, 6%. The lender could offer to pay you 4% on the funds, thinking to keep as a fee the 2% difference between the 6% earned on the money and the 4% paid to you. What happens if the lender and borrower negotiate?
Here is the interesting point: The rate of interest the lender pays on the collateral is going to depend on how many people want to borrow wine from the particular vintage and producer, and how many are willing to lend it! As a practical matter, it may not be easy to find a lender; the wine may be “hard to locate.” If there is high demand for borrowed wine, the lender will offer a low rate of interest, essentially earning a fee for being willing to lend something that is scarce. However, if no one else wants to borrow the wine, the lender might conclude that a small fee is better than nothing and offer you a rate of interest close to the market rate.
The rate paid on collateral is called different things in different markets: the repo rate in bond markets and the short rebate in the stock market. Whatever it is called, the difference between this rate and the market rate of interest is another cost to your short-sale.

13. Note that the lender is not concerned about your failure to perform when the price goes down because the lender has the money!

19

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Chapter 1. Introduction to Derivatives

CHAPTER SUMMARY
Derivatives are financial instruments with a payoff determined by the price of something else. They can be used as a tool for risk management and speculation, and to reduce transaction costs or avoid taxes and regulation.
One important function of financial markets is to facilitate optimal risk-sharing. There are different ways to measure the size and activity of stock, bond, and derivatives markets, but these markets are by any measure large and growing. The growth of derivatives markets over the last 50 years has coincided with an increase in the risks evident in various markets.
Events such as the 1973 oil shock, the abandonment of fixed exchange rates, and the deregulation of energy markets have created a new role for derivatives.
There are costs to trading an asset, one of which is the bid-ask spread. The bid-ask spread is a key means by which those traders who make markets are compensated for doing so. In markets without explicit market-makers, limit orders create a bid-ask spread.
An important transaction is a short-sale, which entails borrowing a security, selling it, making dividend (or other cash) payments to the security lender, and then returning it. A short-sale is conceptually the opposite of a purchase. Short-sales can be used for speculation, as a form of financing, or as a way to hedge. Many of the details of short-selling in practice can be understood as a response to credit risk of the short-seller and scarcity of shares that can be borrowed. Short-sellers typically leave the short-sale proceeds on deposit with lenders, along with additional capital called a haircut. The rate paid on this collateral is called the short rebate, and is less than the interest rate.

FURTHER READING
The derivatives exchanges have websites that list their contracts and provide further details.
Because the web addresses can change (e.g., due to mergers), the easiest way to find them is with a web search.
Jorion (2006) discusses a number of derivatives disasters, and Jorion (1995) examines in detail one famous episode: Orange County in California. Bernstein (1992) presents a history of the development of financial markets, and Bernstein (1996) discusses the concept of risk measurement and how it evolved over the last 800 years. Miller (1986) discusses origins of past financial innovation, while Merton (1999) and Shiller (2003) provide a stimulating look at possible future developments in financial markets. Finally, Lewis (1989) is a classic, funny, insider’s account of investment banking, offering a different (to say the least) perspective on the mechanics of global risk-sharing.
The financial crisis spawned dozens of books. Lo (2012) reviews 21 of them. Informative and entertaining books aimed at a popular audience include Cohan (2009), Tett (2010),
Sorkin (2010), and Lewis (2011). Gorton (2010) provides a more detailed and technical perspective. PROBLEMS
1.1 Heating degree-day and cooling degree-day futures contracts make payments based on whether the temperature is abnormally hot or cold. Explain why the following businesses might be interested in such a contract:

Problems

a. Soft-drink manufacturers.
b. Ski-resort operators.
c. Electric utilities.
d. Amusement park operators.
1.2 Suppose the businesses in the previous problem use futures contracts to hedge their temperature-related risk. Who do you think might accept the opposite risk?
1.3 ABC stock has a bid price of $40.95 and an ask price of $41.05. Assume there is a
$20 brokerage commission.
a. What amount will you pay to buy 100 shares?
b. What amount will you receive for selling 100 shares?
c. Suppose you buy 100 shares, then immediately sell 100 shares with the bid and ask prices being the same in both cases. What is your round-trip transaction cost?
1.4 Repeat the previous problem supposing that the brokerage fee is quoted as 0.3% of the bid or ask price.
1.5 Suppose a security has a bid price of $100 and an ask price of $100.12. At what price can the market-maker purchase a security? At what price can a market-maker sell a security? What is the spread in dollar terms when 100 shares are traded?
1.6 Suppose you short-sell 300 shares of XYZ stock at $30.19 with a commission charge of 0.5%. Supposing you pay commission charges for purchasing the security to cover the short-sale, how much profit have you made if you close the short-sale at a price of $29.87?
1.7 Suppose you desire to short-sell 400 shares of JKI stock, which has a bid price of
$25.12 and an ask price of $25.31. You cover the short position 180 days later when the bid price is $22.87 and the ask price is $23.06.
a. Taking into account only the bid and ask prices (ignoring commissions and interest), what profit did you earn?
b. Suppose that there is a 0.3% commission to engage in the short-sale (this is the commission to sell the stock) and a 0.3% commission to close the short-sale
(this is the commission to buy the stock back). How do these commissions change the profit in the previous answer?
c. Suppose the 6-month interest rate is 3% and that you are paid nothing on the short-sale proceeds. How much interest do you lose during the 6 months in which you have the short position?
1.8 When you open a brokerage account, you typically sign an agreement giving the broker the right to lend your shares without notifying or compensating you. Why do brokers want you to sign this agreement?
1.9 Suppose a stock pays a quarterly dividend of $3. You plan to hold a short position in the stock across the dividend ex-date. What is your obligation on that date? If you are a taxable investor, what would you guess is the tax consequence of the payment?

21

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Chapter 1. Introduction to Derivatives

(In particular, would you expect the dividend to be tax deductible?) Suppose the company announces instead that the dividend is $5. Should you care that the dividend is different from what you expected?
1.10 Short interest is a measure of the aggregate short positions on a stock. Check an online brokerage or other financial service for the short interest on several stocks of your choice. Can you guess which stocks have high short interest and which have low? Is it theoretically possible for short interest to exceed 100% of shares outstanding? 1.11 Suppose that you go to a bank and borrow $100. You promise to repay the loan in
90 days for $102. Explain this transaction using the terminology of short-sales.
1.12 Suppose your bank’s loan officer tells you that if you take out a mortgage (i.e., you borrow money to buy a house), you will be permitted to borrow no more than 80% of the value of the house. Describe this transaction using the terminology of short-sales.
1.13 Pick a derivatives exchange such as CME Group, Eurex, or the Chicago Board
Options Exchange. Go to that exchange’s website and try to determine the following:
a. What products the exchange trades.
b. The trading volume in the various products.
c. The notional value traded.
What do you predict would happen to these measures if the notional value of a popular contract were cut in half? (For example, instead of an option being based on 100 shares of stock, suppose it were based on 50 shares of stock.)
1.14 Consider the widget exchange. Suppose that each widget contract has a market value of $0 and a notional value of $100. There are three traders, A, B, and C. Over one day, the following trades occur:
A long, B short, 5 contracts.
A long, C short, 15 contract.
B long, C short, 10 contracts.
C long, A short, 20 contracts.
a. What is each trader’s net position in the contract at the end of the day?
(Calculate long positions minus short positions.)
b. What are trading volume, open interest, and the notional values of trading volume and open interest? (Calculate open interest as the sum of the net long positions, from your previous answer.)
c. How would your answers have been different if there were an additional trade:
C long, B short, 5 contracts?
d. How would you expect the measures in part (b) to be different if each contract had a notional value of $20?

PART
Insurance, Hedging, and
Simple Strategies

I

n this part of the book, Chapters 2–4, we examine the basic derivatives contracts: forward contracts, futures contracts, call options, and put options. All of these are contracts between two parties, with a payoff at some future date based on the price of an underlying asset (this is why they are called derivatives).
There are a number of things we want to understand about these instruments.
What are they? How do they work and what do they cost? If you enter into a forward contract, futures contract, or option, what obligations or rights have you acquired?
Payoff and profit diagrams provide an important graphical tool to summarize the risk of these contracts.
Once we understand what the basic derivatives contracts are, what can we do with them? We will see that, among other things, they can be used to provide insurance, to convert a stock investment into a risk-free investment and vice versa,

and to speculate in a variety of ways. Derivatives can often be customized for a particular purpose. We will see how corporate risk managers can use derivatives, and some reasons for doing so.
In this part of the book we take the prices of derivatives as given; the underlying pricing models will be covered in much of the rest of the book. The main mathematical tool is present and future value calculations. We do, however, develop one key pricing idea: put-call parity. Put-call parity is important because it demonstrates a link among the different contracts we examine in these chapters, telling us how the prices of forward contracts, call options, and put options are related to one another.

2
T

An Introduction to
Forwards and Options

his chapter introduces the basic derivatives contracts: forward contracts, call options, and put options. These fundamental contracts are widely used, and serve as building blocks for more complicated derivatives that we discuss in later chapters. We explain here how the contracts work and how to think about their risk. We also introduce an extremely important tool for analyzing derivatives positions—namely, payoff and profit diagrams. The terminology and concepts introduced in this chapter are fundamental and will be used throughout this book.

2.1 FORWARD CONTRACTS
To understand a forward contract, it is helpful to first consider the process of buying or selling stock. Such a transaction entails at least three separate steps: (1) the buyer and seller agree to transact and set the price to be paid, (2) cash is transferred from the buyer to the seller, and (3) shares are transferred from the seller to the buyer. Typically, steps 2 and 3 occur shortly after the buyer and seller agree to transact.1 However, as a logical matter, a price could be set today and the transfer of shares and cash could then occur at a specified date in the future.
This is in fact the definition of a forward contract: It sets today the terms at which you buy or sell an asset or commodity at a specific time in the future. A forward contract does the following:
.

Specifies the quantity and exact type of the asset or commodity the seller must deliver.

.

Specifies delivery logistics, such as time, date, and place.

.

Specifies the price the buyer will pay at the time of delivery.

.

Obligates the seller to sell and the buyer to buy, subject to the above specifications.

The time at which the contract settles is called the expiration date. The asset or commodity on which the forward contract is based is called the underlying asset. Apart from commissions and bid-ask spreads (see Section 1.5), a forward contract requires no initial payment or premium. The contractual forward price simply represents the price at which

1. The current industry standard is for steps 2 and 3 to occur no later than three days after the agreement to transact. This is called T+3 settlement.

25

26

Chapter 2. An Introduction to Forwards and Options

FIGURE 2.1
Index futures price listings.

Data from the Wall Street Journal, April 15, 2010, p. C-7.

consenting adults agree today to transact in the future, at which time the buyer pays the seller the forward price and the seller delivers the asset.
Futures contracts are similar to forward contracts in that they create an obligation to buy or sell at a predetermined price at a future date. We will discuss the institutional and pricing differences between forwards and futures in Chapter 5. For the time being, think of them as interchangeable.
Figure 2.1 shows futures price listings from the Wall Street Journal for futures contracts on several stock indices, including the Dow Jones Industrial Average (DJ 30) and the Standard and Poor’s 500 (S&P 500). The indices are the underlying assets for the contracts. (A stock index is the average price of a group of stocks. In these examples we work with this group price rather than the price of just one stock.) The first column of the listing gives the expiration month. The columns that follow show the price at the beginning of the day (the open), the high and low during the day, and the settlement price, which reflects the last transactions of the day.
The listing also gives the price change from the previous day and open interest, which measures the number of contracts outstanding. (Since each trade of a contract has both a buyer and a seller, a buyer-seller pair counts as one contract.) Finally, the head of the listing tells us where the contracts trade (the Chicago Board of Trade [CBT], Chicago Mercantile
Exchange [CME] and the Intercontinental Exchange[ICE]), and the size of the contract, which for the S&P 500, for example, is $250 times the index value. Note that there is also an S&P 500 “mini” contract. This is the same as the S&P 500 contract except for scale:
The Mini S&P 500 futures contract is one-fifth the size of the regular S&P 500 futures contract. The mini contracts were introduced to make index futures more appealing to ordinary investors. We will discuss index futures contracts in more detail in Chapter 5. There are many more exchange-traded stock index futures contracts than those in Figure 2.1, both

2.1 Forward Contracts

TABLE 2.1

27

Some indexes on which futures contracts are traded.

Index

Exchange

Weights

Description

S&P 500 Index
DJ Industrial Average
NASDAQ 100

CME
CME
CME

Market
Price
Market

S&P Midcap 400
Russell 1000
Russell 2000
MSCI World
MSCI EAFE (Europe,
Australasia, Far East)
Euro Stoxx 50
Nikkei 225

Market
Market
Market
Market
Market

Hang Seng

CME
ICE
ICE
LIFFE
LIFFE,
CME
Eurex
SGX, OSE,
CME
HKEx

DAX

Eurex

Market

S&P Goldman Sachs
Commodity Index

CME

Production

500 large U.S. stocks
30 large U.S. stocks
100 large global non-financial firms listed on Nasdaq
400 mid-cap U.S. stocks
Largest 1000 U.S. companies
2000 small-cap U.S. companies
1500 stocks from 23 developed countries
Stocks from 21 developed countries, excluding Canada and the U.S.
50 blue-chip Eurozone stocks
225 stocks listed on the Tokyo Stock
Exchange
43 of the largest companies on the Hong
Kong Stock Exchange
30 large German companies listed on the
Frankfurt Stock Exchange
Wide range of commodities on which futures contracts are traded

Market
Price
Market

Abbreviations: CME = Chicago Mercantile Exchange, ICE = Intercontinental Exchange, LIFFE = London International Financial Futures
Exchange, SGX = Singapore Exchange, OSE = Osaka Stock Exchange, HKEx = Hong Kong Exchange and Clearing. For the weights, “market” means weights are proportional to market capitalization, “price” means weights are proportional to the stock price, “production” means weights are proporational to global production.

in the United States and around the world. Table 2.1 is a non-exhaustive list of global stock indexes. The price quotes in Figure 2.1 are from April. The prices are therefore set in April for purchase of the index in later months. For example, the December futures settlement price for the S&P 500 index is $1197.10.2 By contrast, the current S&P index price that day
(not in Figure 2.1) is $1210.65. This is the spot price for the index—the market price for immediate delivery of the index.
As we will see in Chapters 5, 6, and 7, there are also futures contracts on interest rates and commodities. Futures are widely used in risk management and as an alternative way to invest in the underlying asset. Agricultural futures (such as corn and soybeans) can be used

2. The use and nonuse of dollar signs for futures prices can be confusing. Many futures prices, in particular those for index futures, are in practice quoted without dollar signs, and multiplied by a dollar amount to determine the value of the contract. In this and the next several chapters, we will depart from this convention and use dollar signs for index futures prices. When we discuss the S&P 500 index futures contract in Chapter
5, however, we will follow practice and omit the dollar sign.

28

Chapter 2. An Introduction to Forwards and Options

by farmers and others to hedge crop prices. The box on page 28 discusses an unsuccessful proposal for a new futures contract that was in the news in 2003.
We will discuss in Chapter 5 how forward and futures prices are determined and more details about how futures contracts work. In this chapter we take prices as given and examine profit and loss on a forward contract. We will also see how a position in a forward contract is similar to and different from alternative investments, such as a direct investment in the underlying index.

BOX

2.1: Terrorism Futures?

Newspaper readers in July 2003 were undoubtedly startled to see the headline “Pentagon Prepares a Futures Market on Terror Attacks” (New
York Times, July 29, 2003, p. A1). The article continued: Traders bullish on a biological attack on
Israel or bearish on the chances of a North
Korean missile strike would have the opportunity to bet on the likelihood of such events on a new Internet site established by the Defense Advanced Research Projects
Agency.
The Pentagon called its latest idea a new way of predicting events and part of its search for the
“broadest possible set of new ways to prevent terrorist attacks.”
Critics immediately attacked the plan:
Two Democratic senators who reported the plan called it morally repugnant and grotesque. . . . One of the two senators, Byron L. Dorgan of North Dakota, said the idea seemed so preposterous that he had trouble persuading people it was not a hoax. “Can you imagine,” Mr. Dorgan asked, “if another country set up a betting parlor so that people could go in . . . and bet on the assassination of an
American political figure?”

The other critic, Senator Ron Wyden of Oregon, described the plan:
You may think early on that Prime Minister
X is going to be assassinated. So you buy the futures contracts for 5 cents each. As more people begin to think the person’s going to be assassinated, the cost of the contract could go up, to 50 cents. The payoff if he’s assassinated is
$1 per future. So if it comes to pass, and those who bought at 5 cents make 95 cents. Those who bought at 50 cents make 50 cents.
Later the same day (July 29), this headline appeared on the New York Times website: “Pentagon
Abandons Plan for Futures Market on Terror.”
Before dropping the plan, Defense officials defended it: “Research indicates that markets are extremely efficient, effective, and timely aggregators of dispersed and even hidden information.
Futures markets have proven themselves to be good at predicting such things as elections results; they are often better than expert opinions.”
A common concern about futures markets is the possibility that markets can be manipulated by better informed traders. The possibility of manipulation in this case was described as a
“technical challenge and uncertainty.” The natural worry was that terrorists would use the futures market to make money from attacks, or to mislead authorities about where they would attack.

2.1 Forward Contracts

TABLE 2.2

Payoff after 6 months from a long S&R forward contract and a short S&R forward contract at a forward price of
$1020. If the index price in 6 months is $1020, both the long and short have a 0 payoff. If the index price is greater than $1020, the long makes money and the short loses money. If the index price is less than $1020, the long loses money and the short makes money.

S&R Index in 6 Months
900
950
1000
1020
1050
1100

S&R Forward
Long
Short
−$120
−70
−20
0
30
80

$120
70
20
0
−30
−80

The Payoff on a Forward Contract
Every forward contract has both a party agreeing to buy and one agreeing to sell. The term long is used to describe the buyer and short is used to describe the seller. Generally, a long position is one that makes money when the price goes up and a short is one that makes money when the price goes down. Because the long has agreed to buy at the fixed forward price, a long position profits if prices rise.
The payoff to a contract is the value of the position at expiration. The payoff to a long forward contract is
Payoff to long forward = Spot price at expiration − forward price

(2.1)

Because the short has agreed to sell at the fixed forward price, the short profits if prices fall.
The payoff to a short forward contract is
Payoff to short forward = Forward price − spot price at expiration

(2.2)

To illustrate these calculations, consider a forward contract on a hypothetical stock index.
Suppose the non-dividend-paying S&R (“Special and Rich”) 500 index has a current price of $1000 and the 6-month forward price is $1020.3 The holder of a long position in the S&R forward contract is obligated to pay $1020 in 6 months for one unit of the index. The holder of the short position is obligated to sell one unit of the index for $1020. Table 2.2 lists the payoff on the position for various possible future values of the index.

3. We use a hypothetical stock index—the S&R—in order to avoid complications associated with dividends.
We discuss dividends—and real stock indices—in Chapter 5.

29

30

Chapter 2. An Introduction to Forwards and Options

FIGURE 2.2

Payoff ($)

Long and short forward positions on the S&R 500 index. 200

Long forward
Short forward

150
100
50
0
–50
–100
–150
–200
–250
800

1020
850

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

Example 2.1 Suppose the index price is $1050 in 6 months. A holder who entered a long position at a forward price of $1020 is obligated to pay $1020 to acquire the index, and hence earns $1050 − $1020 = $30 per unit of the index. The short is likewise obligated to sell for $1020, and thus loses $30.
This example illustrates the mechanics of a forward contract, showing why the long makes money when the price rises and the short makes money when the price falls.

Graphing the Payoff on a Forward Contract
We can graph the information in Table 2.2 to show the payoff in 6 months on the forward contract as a function of the index. Figure 2.2 graphs the long and short positions, with the index price at the expiration of the forward contract on the horizontal axis and payoff on the vertical axis. As you would expect, the two positions have a zero payoff when the index price in 6 months equals the forward price of $1020. The graph for the short forward is a mirror image (about the x-axis) of the graph for the long forward. For a given value of the index, the payoff to the short is exactly the opposite of the payoff to the long. In other words, the gain to one party is the loss to the other.
This kind of graph is widely used because it summarizes the risk of the position at a glance. Comparing a Forward and Outright Purchase
The S&R forward contract is a way to acquire the index by paying $1020 after 6 months.
An alternative way to acquire the index is to purchase it outright at time 0, paying $1000. Is

2.1 Forward Contracts

FIGURE 2.3
Comparison of payoff after
6 months of a long position in the S&R index versus a forward contract in the S&R index. Payoff ($)
2500
2000
1500
1000
500
0
–500

Long forward
Long physical index

–1000
–1500

–1020
0

500

1020
1000

1500

2000

2500

S&R Index Price ($)

there any advantage to using the forward contract to buy the index, as opposed to purchasing it outright?
If we buy the S&R index today, it costs us $1000. The value of the position in 6 months is the value of the S&R index. The payoff to a long position in the physical S&R index is graphed in Figure 2.3. For comparison, the payoff to the long forward position, is graphed as well. Note that the axes have different scales in Figures 2.3 and 2.2.
To see how Figure 2.3 is constructed, suppose the S&R index price is $0 after 6 months. If the index price is $0, the physical index will be worth $0; hence we plot a 0 on the y-axis against 0 on the x-axis. Similarly, for all other prices of the S&R index, the payoff equals the value of the S&R index. For example, if we own the index and the price in 6 months is $750, the value of the position is $750.
If the index price in 6 months is $0, the payoff to the forward contract, using equation (2.1), is
Payoff to long forward = 0 − $1020 = −$1020
If instead the index price is $1020, the long index position will be worth $1020 and the forward contract will be worth $0.
With both positions, we own the index after 6 months. What the figure does not reflect, however, is the different initial investments required for the two positions. With the cash index, we invest $1000 initially and then we own the index. With the forward contract, we invest $0 initially and $1020 after 6 months; then we own the index. The financing of the two positions is different. The payoff graph tells us how much money we end up with after
6 months, but does not account for the initial $1000 investment with the outright purchase.
We will refer to a position that has been paid in full as funded, and one for which payment is deferred as unfunded. In this example, the index position is funded and the forward contract

31

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Chapter 2. An Introduction to Forwards and Options

is unfunded. Figure 2.3 is accurate, but it does not answer our question—namely, whether there is an advantage to either a forward purchase or an outright purchase.
Both positions give us ownership of the S&R index after 6 months. We can compare them fairly if we equate the amounts initially invested and then account for interest earned over the 6 months. We can do this in either of two equivalent ways:
1. Invest $1000 in zero-coupon bonds (for example, Treasury bills) along with the forward contract, in which case each position initially costs $1000 at time 0. This creates a funded position in the forward contract.
2. Borrow to buy the physical S&R index, in which case each position initially costs $0 at time 0. This creates an unfunded position in the index.
Suppose the 6-month interest rate is 2%. With alternative 1, we pay $1000 today.
After 6 months the zero-coupon bond is worth $1000 × 1.02 = $1020. At that point, we use the bond proceeds to pay the forward price of $1020. We then own the index. The net effect is that we pay $1000 initially and own the index after 6 months, just as if we bought the index outright. Investing $1000 and at the same time entering a long forward contract mimics the effect of buying the index outright.
With alternative 2, we borrow $1000 to buy the index, which costs $1000. Hence we make no net cash payment at time 0. After 6 months we owe $1000 plus interest. At that time we repay $1000 × 1.02 = $1020 for the borrowed money. The net effect is that we invest nothing initially, and after six months pay $1020. We also own the index. Borrowing to buy the stock therefore mimics the effect of entering into a long forward contract.4
We conclude that when the index pays no dividends, the only difference between the forward contract and the cash index investment is the timing of a payment that will be made for certain. Therefore, we can compare the two positions by using the interest rate to shift the timing of payments. In the above example, we conclude that the forward contract and the cash index are equivalent investments, differing only in the timing of the cash flows.
Neither form of investing has an advantage over the other.
This analysis suggests a way to systematically compare positions that require different initial investments. We can assume that we borrow any required initial payment. At expiration, we receive the payoff from the contract, and repay any borrowed amounts. We will call this the net payoff or profit. Because this calculation accounts for differing initial investments in a simple fashion, we will primarily use profit rather than payoff diagrams throughout the book.5 Note that the payoff and profit diagrams are the same for a forward contract because it requires no initial investment.
To summarize, a payoff diagram graphs the cash value of a position at a point in time. A profit diagram subtracts from the payoff the future value of the investment in the position. This discussion raises a question: Given our assumptions, should we really expect the forward price to equal $1020, which is the future value of the index? The answer in this case is yes, but we defer a detailed explanation until Chapter 5.

4. If the index paid a dividend in this example, then we would receive the dividend by holding the physical index, but not when we entered into the forward contract. We will see in Chapter 5 how to take dividends into account in this comparison.
5. The term “profit” is defined variously by accountants and economists. All of our profit calculations are for the purpose of comparing one position with another, not computing profit in any absolute sense.

2.1 Forward Contracts

FIGURE 2.4

Payoff ($)

Payoff diagram for a long
S&R forward contract, together with a zero-coupon bond that pays $1020 at maturity. Summing the value of the long forward plus the bond at each S&R Index price gives the line labeled
“Forward + bond.”

Long forward
$1000 bond
Forward + bond

2500
2000
1500
$1020
1000
500
0
–500
–1000

1020

–1020
0

500

1000

1500

2000

2500

S&R Index Price ($)

Zero-Coupon Bonds in Payoff and Profit Diagrams
The preceding discussion showed that the long forward contract and outright purchase of the physical S&R index are essentially the same once we take time value of money into account.
Buying the physical index is like entering into the forward contract and simultaneously investing $1000 in a zero-coupon bond. We can see this same point graphically by using a payoff diagram where we include a zero-coupon bond.
Suppose we enter into a long S&R index forward position, and at the same time purchase a $1000 zero-coupon bond, which will pay $1020 after 6 months. (This was alternative 1 in the previous section.) Algebraically, the payoff to the forward plus the bond is
Forward + bond = Spot price at expiration − $1020 + $1020
Forward payoff

Bond payoff

= Spot price at expiration
This is the same as the payoff to investing in the physical index.
The payoff diagram for this position is an easy modification of Figure 2.3. We simply add a line representing the value of the bond after 6 months ($1000 × 1.02 = $1020), and then add the bond payoff to the forward payoff. This is graphed in Figure 2.4. The forward plus bond looks exactly like the physical index in Figure 2.3.
What is the profit diagram corresponding to this payoff diagram? For the forward contract, profit is the same as the payoff because there is no initial investment. Profit for the forward plus bond is obtained by subtracting the future value of the initial investment. The initial investment was the cost of the bond, $1000. Its future value is, by definition, $1020, the value of the bond after 6 months. Thus, the profit diagram for a forward contract plus a bond is obtained by ignoring the bond! Put differently, adding a bond to a position leaves a profit diagram unaffected.

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Chapter 2. An Introduction to Forwards and Options

Depending on the context, it can be helpful to draw either payoff or profit diagrams.
Bonds can be used to shift payoff diagrams vertically, but they do not change the profit calculation. Cash Settlement Versus Delivery
The foregoing discussion assumed that at expiration of the forward contract, the contract called for the seller (the party short the forward contract) to deliver the cash S&R index to the buyer (the party long the forward contract). However, a physical transaction in a broad stock index will likely have significant transaction costs. An alternative settlement procedure that is widely used is cash settlement. Instead of requiring delivery of the actual index, the forward contract settles financially. The two parties make a net cash payment, which yields the same cash flow as if delivery had occurred and both parties had then closed out their positions. We can illustrate this with an example.
Example 2.2 Suppose that the S&R index at expiration is $1040. Because the forward price is $1020, the long position has a payoff of $20. Similarly, the short position loses $20.
With cash settlement, the short simply pays $20 to the long, with no transfer of the physical asset, and hence no transaction costs. It is as if the long paid $1020, acquired the index worth $1040, and then immediately sold it with no transaction costs.
If the S&R index price at expiration had instead been $960, the long position would have a payoff of −$60 and the short would have a payoff of $60. Cash settlement in this case entails the long paying $60 to the short.
Cash settlement is feasible only when there is an accepted reference price upon which the settlement can be based. Cash settlement is not limited to forward contracts—virtually any financial contract can be settled using cash rather than delivery.

Credit Risk
Any forward or futures contract—indeed, any derivatives contract—has credit risk, which means there is a possibility that the counterparty who owes money fails to make a payment.
If you agree to sell the index in one year at a fixed price and the spot price turns out to be lower than the forward price, the counterparty is obligated to buy the index for more than it is worth. You face the risk that the counterparty will for some reason fail to pay the forward price for the index. Similarly, the counterparty faces the risk that you will not fulfill the contract if the spot price in 1 year turns out to be higher than the forward price.
With exchange-traded contracts, the exchange goes to great lengths to minimize this risk by requiring collateral of all participants and being the ultimate counterparty in all transactions. We will discuss credit risk and collateral in more detail when we discuss futures contracts in Chapter 5. With over-the-counter contracts, the fact that the contracts are transacted directly between two parties means that each counterparty bears the credit risk of the other.6

6. Of course, credit risk also exists in exchange-traded contracts. The specific details of how exchanges are structured to minimize credit risk is a complicated and fascinating subject (see Edwards and Ma (1992), ch. 3, for details). In practice, exchanges are regarded by participants as good credit risks.

2.2 Call Options

Credit risk is an important problem with all derivatives, but it is also quite complicated.
Credit checks of counterparties and credit protections such as collateral and bank letters of credit are commonly employed to guard against losses from counterparty default.

2.2 CALL OPTIONS
We have seen that a forward contract obligates the buyer (the holder of the long position) to pay the forward price at expiration, even if the value of the underlying asset at expiration is less than the forward price. Because losses are possible with a forward contract, it is natural to wonder: Could there be a contract where the buyer has the right to walk away from the deal? The answer is yes; a call option is a contract where the buyer has the right to buy, but not the obligation to buy. Here is an example illustrating how a call option works at expiration. Example 2.3 Suppose that the call buyer agrees to pay $1020 for the S&R index in 6 months but is not obligated to do so. (The buyer has purchased a call option.) If in 6 months the S&R price is $1100, the buyer will pay $1020 and receive the index. This is a payoff of
$80 per unit of the index. If the S&R price is $900, the buyer walks away.
Now think about this transaction from the seller’s point of view. The buyer is in control of the option, deciding when to buy the index by paying $1020. Thus, the rights of the option buyer are obligations for the option seller.
Example 2.4 If in 6 months the S&R price is $1100, the seller will receive $1020 and give up an index worth more, for a loss of $80 per unit of the index. If the S&R price is less than $1020, the buyer will not buy, so the seller has no obligation. Thus, at expiration, the seller will have a payoff that is zero (if the S&R price is less than $1020) or negative (if the
S&R price is greater than $1020).
Does it seem as if something is wrong here? Because the buyer can decide whether to buy, the seller cannot make money at expiration. This situation suggests that the seller must, in effect, be “bribed” to enter into the contract in the first place. At the time the buyer and seller agree to the contract, the buyer must pay the seller an initial price, the premium. This initial payment compensates the seller for being at a disadvantage at expiration. Contrast this with a forward contract, for which the initial premium is zero.

Option Terminology
Here are some key terms used to describe options:
Strike price. The strike price, or exercise price, of a call option is what the buyer pays for the asset. In the example above, the strike price was $1020. The strike price can be set at any value.
Exercise. The exercise of a call option is the act of paying the strike price to receive the asset. In Example 2.3, the buyer decided after 6 months whether to exercise the option—that is, whether to pay $1020 (the strike price) to receive the S&R index.

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Chapter 2. An Introduction to Forwards and Options

Expiration. The expiration of the option is the date by which the option must either be exercised or it becomes worthless. The option in Example 2.3 had an expiration of 6 months. Exercise style. The exercise style of the option governs the time at which exercise can occur. In the above example, exercise could occur only at expiration. Such an option is said to be a European-style option. If the buyer has the right to exercise at any time during the life of the option, it is an American-style option. If the buyer can only exercise during specified periods, but not for the entire life of the option, the option is a Bermudan-style option. (The terms “European” and “American,” by the way, have nothing to do with geography. European, American, and Bermudan options are bought and sold worldwide.)
To summarize, a European call option gives the owner of the call the right, but not the obligation, to buy the underlying asset on the expiration date by paying the strike price.
The option described in Examples 2.3 and 2.4 is a 6-month European-style S&R call with a strike price of $1020. The buyer of the call can also be described as having a long position in the call.
For the time being, we will discuss European-style options exclusively. We do this because European options are the simplest to discuss and are also quite common in practice.
Most exchange-traded options are American; later in the book we will discuss American options in more detail.

Payoff and Profit for a Purchased Call Option
We can graph call options as we did forward contracts. The buyer is not obligated to buy the index, and hence will only exercise the option if the payoff is greater than zero. The algebraic expression for the payoff to a purchased call is therefore
Purchased call payoff = max[0, spot price at expiration − strike price]

(2.3)

The expression max[a, b] means take the greater of the two values a and b. (Spreadsheets contain a max function, so it is easy to compute option payoffs in a spreadsheet.)
Example 2.5 Consider a call option on the S&R index with 6 months to expiration and a strike price of $1000. Suppose the index in 6 months is $1100. Clearly it is worthwhile to pay the $1000 strike price to acquire the index worth $1100. Using equation (2.3), the call payoff is max[0, $1100 − $1000] = $100
If the index is 900 at expiration, it is not worthwhile paying the $1000 strike price to buy the index worth $900. The payoff is then max[0, $900 − $1000] = $0
As discussed before, the payoff does not take account of the initial cost of acquiring the position. For a purchased option, the premium is paid at the time the option is acquired.
In computing profit at expiration, suppose we defer the premium payment; then by the time of expiration we accrue 6 months’ interest on the premium. The option profit is computed as
Purchased call profit = max[0, spot price at expiration − strike price]
− future value of option premium

(2.4)

2.2 Call Options

BOX

37

2.2: How Do You Buy an Option?

How would you actually buy an option? The quick answer is that buying an option is just like buying a stock. Option premiums are quoted just like stock prices. Figure 2.5 provides an example.
(For current quotes see, for example, http://www
.cboe.com; you can find bid and ask prices at the Chicago Board Options Exchange.) Using either an online or flesh-and-blood broker, you can enter an order to buy an option. As with stocks, in addition to the option premium, you pay a commission, and there is a bid-ask spread.
Options on numerous stocks are traded on exchanges, and for any given stock or index, there can be over a hundred options available, differing in strike price and expiration date. (In

July 2012, a quick count at the Chicago Board
Options Exchange website showed over 1200 options, with differing strikes and maturities, both puts and calls, with the S&P 500 index as the underlying asset.) Options may be either
American or European. If you buy an American option, you have to be aware that exercising the option prior to expiration may be optimal. Thus, you need to have some understanding of why and when exercise might make sense.
You can also sell, or write, options. In this case, you have to post collateral (called margin) to protect others against the possibility you will default. See Appendix 2.A for a discussion of this and other issues.

The following example illustrates the computation of the profit.
Example 2.6 Use the same option as in Example 2.5, and suppose that the risk-free rate is 2% over 6 months. Assume that the index spot price is $1000 and that the premium for this call is $93.81.7 Hence, the future value of the call premium is $93.81 × 1.02 = $95.68.
If the S&R index price at expiration is $1100, the owner will exercise the option. Using equation (2.4), the call profit is max[0, $1100 − $1000] − $95.68 = $4.32
If the index is 900 at expiration, the owner does not exercise the option. It is not worthwhile paying the $1000 strike price to buy the index worth $900. Profit is then max[0, $900 − $1000] − $95.68 = −$95.68 reflecting the loss of the premium.
We graph the call payoff by computing, for any index price at expiration, the payoff on the option position as a function of the price. We graph the call profit by subtracting from this the future value of the option premium. Table 2.3 computes the payoff and profit at different index values, computed as in Examples 2.5 and 2.6. Note that because the strike price is fixed, a higher market price at expiration of the S&R index benefits the call buyer.

7. It is not important at this point how we compute this price, but if you wish to replicate the option premiums, they are computed using the Black-Scholes formula, which we discuss in Chapter 12. Using the BSCall spreadsheet function accompanying this book, the call price is computed as BSCall(1000, 1000,
0.3, 2 × ln(1.02), 0.5, 0) = 93.81.

38

Chapter 2. An Introduction to Forwards and Options

TABLE 2.3

Payoff and profit after 6 months from a purchased 1000strike S&R call option with a future value of premium of $95.68. The option premium is assumed to be $93.81 and the effective interest rate is 2% over 6 months. The payoff is computed using equation (2.3), and the profit using equation (2.4).

S&R Index in 6 Months

Call
Payoff

Future Value of Premium

Call
Profit

800
850
900
950
1000
1050
1100
1150
1200

$0
0
0
0
0
50
100
150
200

−$95.68
−95.68
−95.68
−95.68
−95.68
−95.68
−95.68
−95.68
−95.68

−$95.68
−95.68
−95.68
−95.68
−95.68
−45.68
4.32
54.32
104.32

Figure 2.5 graphs the call payoff that is computed in Table 2.3. The graph clearly shows the “optionality” of the option: Below the strike price of $1000, the payoff is zero, while it is positive and increasing above $1000.
The last column in Table 2.3 computes the call profit at different index values. Because a purchased call and a forward contract are both ways to buy the index, it is interesting to contrast the two. Thus, Figure 2.6 plots the profit on both a purchased call and a long forward contract. Note that profit and payoff diagrams for an option differ by the future value of the premium, whereas for a forward contract they are the same.
If the index rises, the forward contract is more profitable than the option because it does not entail paying a premium. If the index falls sufficiently, however, the option is more profitable because the most the option buyer loses is the future value of the premium.
This difference suggests that we can think of the call option as an insured position in the index. Insurance protects against losses, and the call option does the same. Carrying the analogy a bit further, we can think of the option premium as, in part, reflecting the cost of that insurance. The forward, which is free, has no such insurance, and potentially has losses larger than those on the call.
This discussion highlights the important point that there are always trade-offs in selecting a position. The forward contract outperforms the call if the index rises and underperforms the call if the index falls sufficiently. When all contracts are fairly priced, you will not find a contract that has higher profits for all possible index market prices.

Payoff and Profit for a Written Call Option
Now let’s look at the option from the point of view of the seller. The seller is said to be the option writer, or to have a short position in a call option. The option writer is the

2.2 Call Options

FIGURE 2.5

Payoff ($)

The payoff at expiration of a purchased S&R call with a
$1000 strike price.

200

Purchased call

150
100
50
0
–50
–100
–150
–200
–250
800

FIGURE 2.6

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

Profit ($)

Profit at expiration for purchase of a 6-month S&R index call with a strike price of $1000 versus profit on a long S&R index forward position. 850

200
150
100
Index price = $1020

50
0
–50

Profit = –$95.68

–100

Index price = $1000

–150

Purchased call
Long forward

–200
–250
800

850

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

counterparty to the option buyer. The writer receives the premium for the option and then has an obligation to sell the underlying security in exchange for the strike price if the option buyer exercises the option.
The payoff and profit to a written call are just the opposite of those for a purchased call: 39

40

Chapter 2. An Introduction to Forwards and Options

Written call payoff = − max[0, spot price at expiration − strike price]
Written call profit = − max[0, spot price at expiration − strike price]
+ future value of option premium

(2.5)
(2.6)

This example illustrates the option writer’s payoff and profit. Just as a call buyer is long in the call, the call seller has a short position in the call.
Example 2.7 Consider a 1000-strike call option on the S&R index with 6 months to expiration. At the time the option is written, the option seller receives the premium of $93.81.
Suppose the index in 6 months is $1100. It is worthwhile for the option buyer to pay the $1000 strike price to acquire the index worth $1100. Thus, the option writer will have to sell the index, worth $1100, for the strike price of $1000. Using equation (2.5), the written call payoff is
− max[0, $1100 − $1000] = −$100
The premium has earned 2% interest for 6 months and is now worth $95.68. Profit for the written call is
−$100 + $95.68 = −$4.32
If the index is 900 at expiration, it is not worthwhile for the option buyer to pay the $1000 strike price to buy the index worth $900. The payoff is then
− max[0, $900 − $1000] = $0.
The option writer keeps the premium, for a profit after 6 months of $95.68.
Figure 2.7 depicts a graph of the option writer’s profit, graphed against a short forward contract. Note that it is the mirror image of the call buyer’s profit in Figure 2.6.

FIGURE 2.7

Profit ($)

Profit for the writer of a 6month S&R call with a strike of $1000 versus profit for a short S&R forward.

250

Written call
Short forward

200
150

Profit = $95.68

100

Index price = $1000

50
0
–50

Index price = $1020

–100
–150
–200
800

850

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

2.3 Put Options

2.3 PUT OPTIONS
We introduced a call option by comparing it to a forward contract in which the buyer need not buy the underlying asset if it is worth less than the agreed-to purchase price. Perhaps you wondered if there could also be a contract in which the seller could walk away if it is not in his or her interest to sell. The answer is yes. A put option is a contract where the seller has the right to sell, but not the obligation. Here is an example to illustrate how a put option works.
Example 2.8 Suppose that the seller agrees to sell the S&R index for $1020 in 6 months but is not obligated to do so. (The seller has purchased a put option.) If in 6 months the
S&R price is $1100, the seller will not sell for $1020 and will walk away. If the S&R price is $900, the seller will sell for $1020 and will earn $120 at that time.
A put must have a premium for the same reason a call has a premium. The buyer of the put controls exercise; hence the seller of the put will never have a positive payoff at expiration. A premium paid by the put buyer at the time the option is purchased compensates the put seller for this no-win position.
It is important to be crystal clear about the use of the terms “buyer” and “seller” in the above example, because there is potential for confusion. The buyer of the put owns a contract giving the right to sell the index at a set price. Thus, the buyer of the put is a seller of the index! Similarly, the seller of the put is obligated to buy the index, should the put buyer decide to sell. Thus, the buyer of the put is potentially a seller of the index, and the seller of the put is potentially a buyer of the index. (If thinking through these transactions isn’t automatic for you now, don’t worry. It will become second nature as you continue to think about options.)
Other terminology for a put option is the same as for a call option, with the obvious change that “buy” becomes “sell.” In particular, the strike price is the agreed-upon selling price ($1020 in Example 2.8), exercising the option means selling the underlying asset in exchange for the strike price, and the expiration date is that on which you must exercise the option or it is valueless. As with call options, there are European, American, and Bermudan put options.

Payoff and Profit for a Purchased Put Option
We now see how to compute payoff and profit for a purchased put option. The put option gives the put buyer the right to sell the underlying asset for the strike price. The buyer does this only if the asset is less valuable than the strike price. Thus, the payoff on the put option is
Put option payoff = max[0, strike price − spot price at expiration]

(2.7)

The put buyer has a long position in the put. Here is an example.
Example 2.9 Consider a put option on the S&R index with 6 months to expiration and a strike price of $1000.
Suppose the index in 6 months is $1100. It is not worthwhile to sell the index worth
$1100 for the $1000 strike price. Using equation (2.7), the put payoff is max[0, $1000 − $1100] = $0

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If the index were 900 at expiration, it is worthwhile selling the index for $1000. The payoff is then max[0, $1000 − $900] = $100
As with the call, the payoff does not take account of the initial cost of acquiring the position. At the time the option is acquired, the put buyer pays the option premium to the put seller; we need to account for this in computing profit. If we borrow the premium amount, we must pay 6 months’ interest. The option profit is computed as
Purchased put profit = max[0, strike price − spot price at expiration]
− future value of option premium

(2.8)

The following example illustrates the computation of profit on the put.
Example 2.10 Use the same option as in Example 2.9, and suppose that the risk-free rate is 2% over 6 months. Assume that the premium for this put is $74.20.8 The future value of the put premium is $74.20 × 1.02 = $75.68.
If the S&R index price at expiration is $1100, the put buyer will not exercise the option. Using equation (2.8), profit is max[0, $1000 − $1100] − $75.68 = −$75.68 reflecting the loss of the premium.
If the index is $900 at expiration, the put buyer exercises the put, selling the index for
$1000. Profit is then max[0, $1000 − $900] − $75.68 = $24.32
Table 2.4 computes the payoff and profit on a purchased put for a range of index values at expiration. Whereas call profit increases as the value of the underlying asset increases, put profit increases as the value of the underlying asset decreases.
Because a put is a way to sell an asset, we can compare it to a short forward position, which is a mandatory sale. Figure 2.8 graphs profit from the purchased put described in
Table 2.4 against the profit on a short forward.
We can see from the graph that if the S&R index goes down, the short forward, which has no premium, has a higher profit than the purchased put. If the index goes up sufficiently, the put outperforms the short forward. As with the call, the put is like an insured forward contract. With the put, losses are limited should the index go up. With the short forward, losses are potentially unlimited.

Payoff and Profit for a Written Put Option
Now we examine the put from the perspective of the put writer. The put writer is the counterparty to the buyer. Thus, when the contract is written, the put writer receives the premium. At expiration, if the put buyer elects to sell the underlying asset, the put writer must buy it.

8. This price is computed using the Black-Scholes formula for the price of a put: BSPut(1000, 1000, 0.3, 2× ln(1.02), 0.5, 0) = 74.20. We will discuss this formula in Chapter 12.

2.3 Put Options

TABLE 2.4

Profit after 6 months from a purchased 1000-strike S&R put option with a future value of premium of $75.68.

S&R Index in 6 Months

Put
Payoff

Future Value of Premium

Put
Profit

$800
850
900
950
1000
1050
1100
1150
1200

$200
150
100
50
0
0
0
0
0

−$75.68
−75.68
−75.68
−75.68
−75.68
−75.68
−75.68
−75.68
−75.68

$124.32
74.32
24.32
−25.68
−75.68
−75.68
−75.68
−75.68
−75.68

FIGURE 2.8

Profit ($)

Profit on a purchased S&R index put with a strike price of $1000 versus a short S&R index forward.

Purchased put
Short forward

250
200
150
100

Index price = $1020

50
0

Profit = –$75.68

–50
–100

Index price = $1000

–150
–200
800

850

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

The payoff and profit for a written put are the opposite of those for the purchased put:
Written put payoff = − max[0, strike price − spot price at expiration]
Written put profit = − max[0, strike price − spot price at expiration]
+ future value of option premium
The put seller has a short position in the put.

(2.9)
(2.10)

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Chapter 2. An Introduction to Forwards and Options

FIGURE 2.9
Written S&R index put option with a strike of $1000 versus a long S&R index forward contract.

Profit ($)
200
150
100

Index price = $1000

50

Profit = $75.68

0
–50

Index price = $1020

–100
–150

Written put
Long forward

–200
–250
800

850

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

Example 2.11 Consider a 1000-strike put option on the S&R index with 6 months to expiration. At the time the option is written, the put writer receives the premium of $74.20.
Suppose the index in 6 months is $1100. The put buyer will not exercise the put. Thus, the put writer keeps the premium, plus 6 months’ interest, for a payoff of 0 and profit of
$75.68.
If the index is $900 in 6 months, the put owner will exercise, selling the index for
$1000. Thus, the option writer will have to pay $1000 for an index worth $900. Using equation (2.9), the written put payoff is
− max[0, $1000 − $900] = −$100
The premium has earned 2% interest for 6 months and is now worth $75.68. Profit for the written put is therefore
−$100 + $75.68 = −$24.32
Figure 2.9 graphs the profit diagram for a written put. As you would expect, it is the mirror image of the purchased put.

The “Moneyness” of an Option
Options are often described by their degree of moneyness. This term describes whether the option payoff would be positive if the option were exercised immediately. (The term is used to describe both American and European options even though European options cannot be exercised until expiration.) An in-the-money option is one which would have a positive payoff (but not necessarily positive profit) if exercised immediately. A call with a strike price less than the asset price and a put with a strike price greater than the asset price are both in-the-money.

2.4 Summary of Forward and Option Positions

An out-of-the-money option is one that would have a negative payoff if exercised immediately. A call with a strike price greater than the asset price and a put with a strike price less than the asset price are both out-of-the-money.
An at-the-money option is one for which the strike price is approximately equal to the asset price.

2.4 SUMMARY OF FORWARD AND OPTION POSITIONS
We have now examined six different positions: short and long forwards, and purchased and written calls and puts. There are different ways to categorize these positions. One way is by their potential for gain and loss. Table 2.5 summarizes the maximum possible gain and loss at maturity for forwards and European options.
Another way to categorize the positions is by whether the positions represent buying or selling the underlying asset. Those that represent buying are fundamentally long with respect to the underlying asset, while those that represent selling are fundamentally short with respect to the underlying asset. If you find yourself confused by “long” and “short,” you can consult Box 2.3.

Positions Long with Respect to the Index
The following positions are long in the sense that there are circumstances in which they represent either a right or an obligation to buy the underlying asset:
Long forward: An obligation to buy at a fixed price.
Purchased call: The right to buy at a fixed price if it is advantageous to do so.
Written put: An obligation of the put writer to buy the underlying asset at a fixed price if it is advantageous to the option buyer to sell at that price. (Recall that the option buyer decides whether or not to exercise.)

TABLE 2.5

Maximum possible profit and loss at maturity for long and short forwards and purchased and written calls and puts. FV(premium) denotes the future value of the option premium. Position

Maximum Loss

Maximum Gain

Long forward
Short forward
Long call
Short call
Long put
Short put

−Forward price
Unlimited
−FV(premium)
Unlimited
−FV(premium)
FV(premium) − Strike price

Unlimited
Forward price
Unlimited
FV(premium)
Strike price − FV(premium)
FV(premium)

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Chapter 2. An Introduction to Forwards and Options

2.3: “Long” and “Short” Revisited

hen you purchase an option you are said to have a long position in that option. Thus, a call that you purchase is a “long call” and a put that you purchase is a “long put.” In both cases, you make money if the price of the option you purchased goes up. Similarly, any option that you sell is a short position in that option. A call that you sell is a “short call” and a put that you sell is a “short put.”
The terms “long” and “short” can be confusing, however, because they are often used more generally. In this more general usage, a position is long with respect to x if the value of the position goes up when x goes up, and it is short with respect to x if the value of the position goes down when x goes up.
A purchased call is therefore long with respect to the stock, because the call becomes more valuable when the stock price goes up. Similarly,

a purchased put is short with respect to the stock, because the put becomes more valuable when the stock price goes down. The case of a purchased put illustrates that a position can be simultaneously long with respect to one thing (its own price) and short with respect to something else (the price of the underlying asset).
Finally, a written put is a “short put” because you lose money if the put price goes up. However, the written put is long with respect to the stock, because the put price goes down, and hence the written put makes money, when the stock price goes up.
You may find that it takes a while to become comfortable with the long and short terminology.
A position can be simultaneously long with respect to one thing and short with respect to something else, so you must always be clear about what you are long or short with respect to.

Figure 2.10 compares these three positions. Note that the purchased call is long when the asset price is greater than the strike price, and the written put is long when the asset price is less than the strike price. All three of these positions benefit from a higher index price.

Positions Short with Respect to the Index
The following positions are short in the sense that there are circumstances in which they represent either a right or an obligation to sell the underlying asset:
Short forward: An obligation to sell at a fixed price.
Written call: An obligation of the call writer to sell the underlying asset at a fixed price if it is advantageous to the option holder to buy at that price (recall that the option buyer decides whether to exercise).
Purchased put: The right to sell at a fixed price if it is advantageous to do so.
Figure 2.11 compares these three positions. Note that the written call is short when the asset price is greater than the strike price, and the purchased put is short when the asset price is less than the strike price. All three of these positions benefit from a lower price of the index.

2.5 Options Are Insurance

FIGURE 2.10

Profit ($)

Profit diagrams for the three basic long positions: long forward, purchased call, and written put.

200
150
100
50
0
–50
–100
Purchased call
Written put
Long forward

–150
–200
–250
800

FIGURE 2.11

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

Profit ($)

Profit diagrams for the three basic short positions: short forward, written call, and purchased put.

850

250

Written call
Purchased put
Short forward

200
150
100
50
0
–50
–100
–150
–200
800

850

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

2.5 OPTIONS ARE INSURANCE
In many investment strategies using options, we will see that options serve as insurance against a loss. In what sense are options the same as insurance? In this section we answer this question by considering homeowner’s insurance. You will see that options are literally insurance, and insurance is an option.

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A homeowner’s insurance policy promises that in the event of damage to your house, the insurance company will compensate you for at least part of the damage. The greater the damage, the more the insurance company will pay. Your insurance policy thus derives its value from the value of your house: It is a derivative.

Homeowner’s Insurance Is a Put Option
To demonstrate how homeowner’s insurance acts as a put option, suppose that you own a house that costs $200,000 to build. To make this example as simple as possible, we assume that physical damage is the only thing that can affect the market value of the house.
Let’s say you buy a $15,000 insurance policy to compensate you for damage to the house. Like most policies, this has a deductible, meaning that there is an amount of damage for which you are obligated to pay before the insurance company pays anything. Suppose the deductible is $25,000. If the house suffers $4000 damage from a storm, you pay for all repairs yourself. If the house suffers $45,000 in damage from a storm, you pay $25,000 and the insurance company pays the remaining $20,000. Once damage occurs beyond the amount of the deductible, the insurance company pays for all further damage, up to
$175,000. (Why $175,000? Because the house can be rebuilt for $200,000, and you pay
$25,000 of that—the deductible—yourself.)
Let’s graph the profit to you for this insurance policy. On the vertical axis is the profit on the insurance policy—the payoff less the insurance premium—and on the horizontal axis, the value of the house. If the house is undamaged (the house value is $200,000) the payoff is zero, and profit is the loss from the unused insurance premium, $15,000. If the house suffers $50,000 damage, the insurance payoff is $50,000 less the $25,000 deductible, or
$25,000. The profit is $25,000 − $15,000 = $10,000. If the house is completely destroyed, the policy pays $175,000, and your profit is $160,000.
Figure 2.12 graphs the profit on the insurance policy. The insurance policy in Figure 2.12 has the same shape as the put option in Figure 2.8. An S&R put is insurance against a fall in the price of the S&R index, just as homeowner’s insurance insures against a fall in the price of the house. Insurance companies are in the business of writing put options! The
$15,000 insurance premium is like the premium of a put, and the $175,000 level at which insurance begins to make payments is like the strike price on a put.
The idea that a put option is insurance also helps us understand what makes a put option cheap or expensive. Two important factors are the riskiness of the underlying asset and the amount of the deductible. Just as with insurance, options will be more expensive when the underlying asset is riskier. Also, the option, like insurance, will be less expensive as the deductible gets larger (for the put option, this means lowering the strike price).
You have probably recognized that there are some practical differences between a financial put option and homeowner’s insurance. One important difference is that the S&R put pays off no matter why the index price declines. Homeowner’s insurance, on the other hand, pays off only if the house declines in value for specified reasons. In particular, a simple decline in real estate prices is not covered by typical homeowner’s insurance policies. We avoided this complication by assuming at the outset that only damage could affect the value of the house.

But I Thought Insurance Is Prudent and Put Options Are Risky . . .
If we accept that insurance and put options are the same thing, how do we reconcile this with the common idea that buying insurance is prudent and buying put options is risky?

2.5 Options Are Insurance

FIGURE 2.12
Profit from insurance policy on a $200,000 house.

Profit ($)

200,000
$160,000
150,000

Gain on insurance due to damage

100,000
Loss of $15,000 premium if house is undamaged
50,000
$175,000
Deductible
15,000

50,000

100,000 150,000 200,000 250,000
House Price ($)

The risk of a derivative or any other asset or security can only be evaluated in context.
Figure 2.12 depicts the risk of an insurance contract without considering the risk of the insured asset. This would be like owning insurance on your neighbor’s house. It would be “risky” because you would buy the insurance policy, and you would lose your entire investment if there were no insurance claim.9 We do not normally think of insurance like this, but it illustrates the point that an insurance policy is a put option on the insured asset.
In the same way, Figure 2.8 depicts the risk of a put option without considering the risk of any other positions an investor might be holding. In contrast to homeowner’s insurance, many investors do own put options without owning the underlying asset. This is why options have a reputation for being risky while homeowner’s insurance does not. With stock options it is possible to own the insurance without the asset. Of course, many investors who own put options also own the stock. For these investors, the risk is like that of insurance, which we normally think of as risk-reducing rather than risk-increasing.

Call Options Are Also Insurance
Call options can also be insurance. Whereas a put option is insurance for an asset we already own, a call option is insurance for an asset we plan to own in the future. Put differently, a put option is insurance for a long position while a call option is insurance for a short position.
Return to the earlier example of the S&R index. Suppose that the current price of the
S&R index is $1000 and that we plan to buy the index in the future. If we buy an S&R call

9. Of course, in real life no insurance company will sell you insurance on your neighbor’s house. The reason is that you will then be tempted to cause damage in order to make your policy valuable. Insurance companies call this “moral hazard.”

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option with a strike price of $1000, this gives us the right to buy S&R for a maximum cost of $1000/share. By buying a call, we have bought insurance against an increase in the price.

2.6 EXAMPLE: EQUITY-LINKED CDS
Although options and forwards are important in and of themselves, they are also commonly used as building blocks in the construction of new financial instruments. For example, banks and insurance companies offer investment products that allow investors to benefit from a rise in a stock index and that provide a guaranteed return if the market declines. We can
“reverse-engineer” such equity-linked CDs and notes using the tools we have developed thus far.10
1
A simple 5 2 year CD with a return linked to the S&P 500 might have the following structure: At maturity, the CD is guaranteed to repay the invested amount, plus 70% of the simple appreciation in the S&P 500 over that time.11
We can ask several questions about the CD:
.

Is the CD fairly priced?

.

How can we decompose the product in terms of options and bonds?

.

How does the issuing bank hedge the risk associated with issuing the product?

.

How does the issuing bank make a profit?

To understand this product, suppose the S&P index is 1300 initially and an investor invests $10,000. If the index is below 1300 after 5.5 years, the CD returns to the investor the original $10,000 investment. If the index is above 1300 after 5.5 years, the investor receives
$10,000 plus 70% of the percentage gain on the index. For example, if the index is 2200, the investor receives
$10,000 × [1 + (2200/1300 − 1) × 70%] = $14,846
At first glance this product appears to permit gains but no losses. However, by now you are probably skeptical of a phrase like “gains but no losses”; the investor must pay something for an investment like this.

Graphing the Payoff on the CD
As a first step in analyzing the CD, we will draw a payoff diagram. If we invest $10,000, we receive at least $10,000. If the index rises to Sfinal > 1300, we also receive on our investment
70% of the rate of return
Sfinal
−1
1300

10. A CD (certificate of deposit) is a kind of interest-bearing bank account. You can think of a CD as being the same as a note or a bond.
11. This is the structure of a CD issued in 1999 by First Union National Bank.

2.6 Example: Equity-Linked CDs

FIGURE 2.13
Payoff at expiration to
$10,000 investment in an equity-linked CD that repays the initial investment at expiration plus 70% of the rate of appreciation of the market above 1300.

Payoff (thousands of $)
Payoff of CD
13
12
11
10
$1300

9
8
600

TABLE 2.6

800

1000 1200 1400 1600
S&P 500 at Expiration

1800

2000

Payoff of equity-linked CD at expiration.

S&P Index
After 5.5 Years

CD
Payoff

500
1000
1500
2000
2500
3000

$10,000.00
10,000.00
11,076.92
13,769.23
16,461.54
19,153.85

Thus, the CD pays
$10,000 × 1 + 0.7 × max 0,

Sfinal
−1
1300

(2.11)

Figure 2.13 graphs the payoff at expiration to this investment in the CD.
Recall the discussion in Section 2.1 of incorporating a zero-coupon bond into a payoff diagram. Per unit of the index (there are 10,000/1300 = 7.69 units of the index in a $10,000 investment), the CD buyer receives 0.7 of an index call option, plus a zero-coupon bond paying $1300 at expiration.
Table 2.6 computes the payoff to the equity-linked CD for different values of the index. 51

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Chapter 2. An Introduction to Forwards and Options

Economics of the CD
Now we are in a position to understand the economics of this product. Think about what happens if the index is below 1300 at expiration. We pay $10,000 and we receive $10,000 back, plus an option. Thus, we have forgone interest on $10,000 in exchange for the possibility of receiving 70% of the gains on the S&P. Suppose that the effective annual
1
interest rate is 6%; after 5 2 years, the buyer has lost interest with a present value of
$10,000 × (1.06)−5.5 − $10,000 = −$2,742
Essentially, the buyer forgoes interest in exchange for a call option on the index.
With this description we have reverse-engineered the CD, decomposing it in terms of an option and a bond. The question of whether the CD is fairly priced turns on whether the
$2742 is a fair price for the index option implicit in the CD. Given information about the interest rate, the volatility of the index, and the dividend yield on the index, it is possible to price the option to determine whether the CD is fairly priced. We perform that analysis for this example in Chapter 15.

Why Equity-Linked CDs?
With reverse-engineering, we see that an investor could create the equivalent of an equitylinked CD by buying a zero-coupon bond and 0.7 call options. Why, then, do products like this exist?
Consider what must be done to replicate the payoff. If a retail investor were to insure an index investment using options, the investor would have to learn about options, decide what maturity, strike price, and quantity to buy, and pay transaction costs. Exchange-traded options have at most 3 years to maturity, so obtaining longer-term protection requires rolling over the position at some point.
An equity-linked CD provides a prepackaged solution. It may provide a pattern of market exposure that many investors could not otherwise obtain at such low transaction costs. The idea that a prepackaged deal may be attractive should be familiar to you. Supermarkets sell whole heads of lettuce—salad building blocks, as it were—and they also sell, at a premium price, lettuce already washed, torn into bite-sized pieces, and mixed as a salad.
The transaction cost of salad preparation leads some consumers to prefer the prepackaged salads. What does the financial institution get out of this? Just as the supermarket earns profit on prepackaged salads, the issuing bank wants to earn a transaction fee on the CD. When it sells a CD, the issuing bank borrows money (the zero-coupon bond portion of the CD) and receives the premium for writing a call option. The cost of the CD to the bank is the cost of the zero-coupon bond plus the cost of the call option. Obviously the bank would not issue the equity-linked CD in the first place unless it was less expensive than alternative ways to attract deposits, such as standard CDs. The equity-linked CD is risky because the bank has written a call, but the bank can manage this risk in several ways, one of which is to purchase call options from a dealer to offset the risk of having written calls. Using data from the early 1990s, Baubonis et al. (1993) estimated that issuers of equity-linked CDs

Chapter Summary

TABLE 2.7

Forwards, calls, and puts at a glance: A summary of forward and option positions.

Derivative
Position

Position with Respect to Underlying Asset

Asset Price
Contingency

Long forward
Short forward
Long call
Short call

Long (buy)
Short (sell)
Long (buy)
Short (sell)

Always
Always
> Strike
> Strike

Long put
Short put

Short (sell)
Long (buy)

< Strike
< Strike

Strategy
Guaranteed purchase price
Guaranteed sale price
Insures against high price
Sells insurance against high price
Insures against low price
Sells insurance against low price earned about 3.5% of the value of the CD as a fee, with about 1% as the transaction cost of hedging the written call.12
In this discussion we have viewed the equity-linked CD from several perspectives.
The end-user is interested in the product and whether it meets a financial need at a fair cost.
The market-maker (the bank in this case) is interested in making a profit without bearing risk from having issued the CD. And the economic observer is interested in knowing why equity-linked CDs exist. The three perspectives overlap, and a full explanation of the product touches on all of them.

CHAPTER SUMMARY
Forward contracts and put and call options are the basic derivative instruments that can be used directly and that serve as building blocks for other instruments. A long forward contract represents an obligation to buy the underlying asset at a fixed price, a call option gives its owner the right (but not the obligation) to buy the underlying asset at a fixed price, and a put option gives its owner the right (but not the obligation) to sell the underlying asset at a fixed price. Payoff and profit diagrams are commonly used tools for evaluating the risk of these contracts. Payoff diagrams show the gross value of a position at expiration, and profit diagrams subtract from the payoff the future value of the cost of the position.
Table 2.7 summarizes the characteristics of forwards, calls, and puts, showing which are long or short with respect to the underlying asset. The table describes the strategy associated with each: Forward contracts guarantee a price, purchased options are insurance, and written options are selling insurance. Figure 2.14 provides a graphical summary of these positions. 12. A back-of-the-envelope calculation in Chapter 15 suggests the issuer fees for this product are in the neighborhood of 4% to 5%.

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FIGURE 2.14

Profit

Profit
Long forward

Short forward

Stock Price

Stock Price

The basic profit diagrams: long and short forward, long and short call, and long and short put.
Profit

Profit
Long call

Short call

Stock Price

Stock Price

Profit

Profit
Long put

Short put

Stock Price

Stock Price

Options can also be viewed as insurance. A put option gives the owner the right to sell if the price declines, just as insurance gives the insured the right to sell (put) a damaged asset to the insurance company.

FURTHER READING
We use the concepts introduced in this chapter throughout the rest of this book. Chapter 3 presents a number of basic option strategies which are widely used in practice, including caps, collars, and floors. Chapter 4 presents the use of options in risk management.
A more general question raised implicitly in this chapter is how the prices of forwards and options are determined. Chapter 5 covers financial forwards and futures in detail, and
Chapter 10 introduces the basic ideas underlying option pricing.
Brokerages routinely supply options customers with an introductory pamphlet about options entitled Characteristics and Risks of Standardized Options. This is available online from http://www.optionsclearing.com. You can also obtain current option prices from websites such as the CBOE’s and various brokerage sites.
The notion that options are insurance has been applied in practice. Sharpe (1976), for example, analyzed optimal pension funding policy taking into account pension insurance provided by the Pension Benefit Guaranty Corporation. Merton (1977a) observed that bank deposit insurance and in fact any loan guarantee can be modeled as a put option. Baubonis et al. (1993) discuss equity-linked CDs.

Problems

PROBLEMS
In the following problems, if the “effective annual interest rate” is r, a $1 investment yields
1 + r after 1 year.
2.1 Suppose XYZ stock has a price of $50 and pays no dividends. The effective annual interest rate is 10%. Draw payoff and profit diagrams for a long position in the stock.
Verify that profit is 0 at a price in 1 year of $55.
2.2 Using the same information as the previous question, draw payoff and profit diagrams for a short position in the stock. Verify that profit is 0 at a price in 1 year of
$55.
2.3 What position is the opposite of a purchased call? The opposite of a purchased put?
2.4

a. Suppose you enter into a long 6-month forward position at a forward price of $50. What is the payoff in 6 months for prices of $40, $45, $50, $55, and
$60?
b. Suppose you buy a 6-month call option with a strike price of $50. What is the payoff in 6 months at the same prices for the underlying asset?
c. Comparing the payoffs of parts (a) and (b), which contract should be more expensive (i.e., the long call or long forward)? Why?

2.5

a. Suppose you enter into a short 6-month forward position at a forward price of $50. What is the payoff in 6 months for prices of $40, $45, $50, $55, and
$60?
b. Suppose you buy a 6-month put option with a strike price of $50. What is the payoff in 6 months at the same prices for the underlying asset?
c. Comparing the payoffs of parts (a) and (b), which contract should be more expensive (i.e., the long put or short forward)? Why?

2.6 A default-free zero-coupon bond costs $91 and will pay $100 at maturity in 1 year.
What is the effective annual interest rate? What is the payoff diagram for the bond?
The profit diagram?
2.7 Suppose XYZ stock pays no dividends and has a current price of $50. The forward price for delivery in 1 year is $55. Suppose the 1-year effective annual interest rate is 10%.
a. Graph the payoff and profit diagrams for a forward contract on XYZ stock with a forward price of $55.
b. Is there any advantage to investing in the stock or the forward contract? Why?
c. Suppose XYZ paid a dividend of $2 per year and everything else stayed the same. Now is there any advantage to investing in the stock or the forward contract? Why?
2.8 Suppose XYZ stock pays no dividends and has a current price of $50. The forward price for delivery in one year is $53. If there is no advantage to buying either the stock or the forward contract, what is the 1-year effective interest rate?

55

56

Chapter 2. An Introduction to Forwards and Options

2.9 An off-market forward contract is a forward where either you have to pay a premium or you receive a premium for entering into the contract. (With a standard forward contract, the premium is zero.) Suppose the effective annual interest rate is 10% and the S&R index is 1000. Consider 1-year forward contracts.
a. Verify that if the forward price is $1100, the profit diagrams for the index and the 1-year forward are the same.
b. Suppose you are offered a long forward contract at a forward price of $1200.
How much would you need to be paid to enter into this contract?
c. Suppose you are offered a long forward contract at $1000. What would you be willing to pay to enter into this forward contract?
2.10 For Figure 2.6, verify the following:
a. The S&R index price at which the call option diagram intersects the x-axis is $1095.68.
b. The S&R index price at which the call option and forward contract have the same profit is $924.32.
2.11 For Figure 2.8, verify the following:
a. The S&R index price at which the put option diagram intersects the x-axis is
$924.32.
b. The S&R index price at which the put option and forward contract have the same profit is $1095.68.
2.12 For each entry in Table 2.5, explain the circumstances in which the maximum gain or loss occurs.
2.13 Suppose the stock price is $40 and the effective annual interest rate is 8%.
a. Draw on a single graph payoff and profit diagrams for the following options:
(i) 35-strike call with a premium of $9.12.
(ii) 40-strike call with a premium of $6.22.
(iii) 45-strike call with a premium of $4.08.
b. Consider your payoff diagram with all three options graphed together. Intuitively, why should the option premium decrease with the strike price?
2.14 Suppose the stock price is $40 and the effective annual interest rate is 8%. Draw payoff and profit diagrams for the following options:
a. 35-strike put with a premium of $1.53.
b. 40-strike put with a premium of $3.26.
c. 45-strike put with a premium of $5.75.
Consider your payoff diagram with all three options graphed together. Intuitively, why should the option premium increase with the strike price?
2.15 The profit calculation in the chapter assumes that you borrow at a fixed interest rate to finance investments. An alternative way to borrow is to short-sell stock. What

2.A More on Buying a Stock Option

complications would arise in calculating profit if you financed a $1000 S&R index investment by shorting IBM stock, rather than by borrowing $1000?
2.16 Construct a spreadsheet that permits you to compute payoff and profit for a short and long stock, a short and long forward, and purchased and written puts and calls.
The spreadsheet should let you specify the stock price, forward price, interest rate, option strikes, and option premiums. Use the spreadsheet’s max function to compute option payoffs.

Appendix 2.A MORE ON BUYING A STOCK OPTION
The box on page 37 discusses buying options. There are at least four practical issues that an option buyer should be aware of: dividends, exercise, margins, and taxes. In this section we will focus on retail investors and exchange-traded stock options. Be aware that specific rules regarding margins and taxes change frequently. This section is intended to help you identify issues and is not intended as a substitute for professional brokerage, accounting, or legal advice.

Dividends
The owner of a standard call option has the right to buy a fixed number of shares of stock at a fixed price, but has no right to receive dividends paid on the underlying stock over the life of the option. When a stock pays a dividend, the stock price declines by approximately the amount of the dividend. This decline in the price lowers the return to the owner of a call option. For exchange-traded options in the United States, there is typically no adjustment in the terms of the option if the stock pays an “ordinary” dividend (one that is typical for the stock). However, if the stock pays an unusual dividend, then officials at the Options Clearing
Corporation (OCC) decide whether or not to make an adjustment.
In June 2003, Iomega Corporation declared a $5 dividend, payable on October 1.
At the time of the declaration, Iomega’s share price was $11.40. Since the dividend was
44% of the share price, the OCC reduced all Iomega option strike prices by $5, effective
October 2.13
When we discuss option pricing, we will see that it is necessary to take dividends into account when pricing an option.

Exercise
Some options, for example those that are cash-settled, are automatically exercised at maturity; the option owner need not take any action. Suppose you own a traded option that is not cash-settled and not automatically exercised. In this case you must provide exercise

13. Reducing the strike price by the amount of the dividend leaves call holders worse off, albeit better off than if no adjustment had been made. If S is the cum-dividend stock price and S − D the ex-dividend stock price, Merton (1973b, p. 152) shows that to leave the value of a call position unchanged, it is necessary to reduce the strike price by the factor (S − D)/S, and give the option holder S/(S − D) − 1 additional options. An option with a value protected against dividends is said to be payout-protected.

57

58

Chapter 2. An Introduction to Forwards and Options

instructions prior to the broker’s deadline. If you fail to do so, the option will expire worthless. When you exercise the option, you generally pay a commission. If you do not wish to own the stock, exercising the option would require that you pay a commission to exercise and then a commission to sell the shares. It might be preferable to sell the option instead of exercising it. If you do wish to own the underlying asset, you can exercise the option.
The option writer who is obligated to fulfill the option exercise (delivering the shares for a call or buying the shares for a put) is said to have been assigned. Assignment can involve paying a commission.
American-style options can be exercised prior to expiration. If you own an option and fail to exercise when you should, you will lose money relative to following the optimal exercise strategy. If you write the option, and it is exercised (you are assigned), you will be required to sell the stock (if you sold a call) or buy the stock (if you sold a put). Therefore, if you buy or sell an American option, you need to understand the circumstances under which exercise might be optimal. Dividends are one factor that can affect the exercise decision.
We discuss early exercise in Chapters 9 and 11.

Margins for Written Options
Purchased options for which you fully pay require no margin, as there is no counterparty risk. With written option positions, however, you can incur a large loss if the stock moves against you. When you write an option, therefore, you are required to post collateral to insure against the possibility that you will default. This collateral is called margin.
Margin rules are beyond the scope of this book and change over time. Moreover, different option positions have different margin rules. Both brokers and exchanges can provide information about current margin requirements.

Taxes
Tax rules for derivatives in general can be complicated, and they change frequently as the tax law changes. The taxation of simple option transactions is straightforward.
If you purchase a call option or stock and then sell it, gain or loss on the position is treated like gain or loss on a stock, and accorded long-term or short-term capital gains treatment depending on the length of time for which the position has been held. If you purchase a call option and then exercise it, the cost basis of the resulting stock position is the exercise price plus the option premium plus commissions. The holding period for the resulting stock position begins the day after the option is exercised. The time the option is held does not contribute to the holding period.
The rules become more intricate when forwards and options are held in tandem with the underlying asset. The reasons for this complexity are not hard to understand. Tax laws in the United States accord different tax treatment to different kinds of income. The tax code views interest income, dividend income, and capital gains income as distinct and subject to different tax rules. Futures also have special rules. However, using derivatives, one kind of income can be turned into another. We saw in this chapter, for example, that buying zero-coupon bonds and a forward contract mimics a stock investment.
One category of special rules governs a constructive sale. If you own a stock, entering into certain option or forward positions can trigger a constructive sale, meaning that even if you continue to own the stock, for tax purposes you are deemed to have sold it at the time you enter into the forward or option positions. By shorting a forward against the stock,

2.A More on Buying a Stock Option

for example, the stock position is transformed into a bond position. When you have no risk stemming from stock ownership, tax law deems you to no longer be an owner.
The so-called straddle rules are tax rules intended to control the recognition of losses for tax purposes when there are offsetting risks as with constructive sales. Such positions often arise when investors are undertaking tax arbitrage, which is why the positions are accorded special treatment. A stock owned together with a put is a tax straddle.14 Generally, the straddle rules prevent loss recognition on only a part of the entire position. A straddle for tax purposes is not the same thing as an option straddle, discussed in Chapter 3.
It is probably obvious to you that if you are taxable and transact in options, and especially if you have both stock and offsetting option positions, you should be prepared to seek professional tax advice.

14. For an illustration of the complexity, in this particular case, an exception to the straddle rules occurs if the stock and put are a “married put,” meaning that the two are purchased together and the stock is delivered to settle the put.

59

3
I

Insurance, Collars, and
Other Strategies

n this chapter we continue to discuss forwards, calls, and puts. We elaborate on the idea, introduced in the last chapter, that options are insurance. We also continue to examine the link between forward prices and option prices, including the important concept of put-call parity. Finally, we look at some common option strategies, such as spreads, straddles, and collars. Among your goals in this chapter should be to become familiar with drawing and interpreting profit and loss diagrams for option positions.

3.1 BASIC INSURANCE STRATEGIES
There are an infinity of ways to combine options to create different payoffs. In this section we examine two important kinds of strategies in which the option is combined with a position in the underlying asset. First, options can be used to insure long or short asset positions.
Second, options can be written against an asset position, in which case the option writer is selling insurance. We consider four positions: being long the asset coupled with a purchased put or written call, and being short the asset coupled with a purchased call or written put.
In this section we continue to use the S&R index examples presented in Sections
2.2 and 2.3. We assumed an index price of $1000, a 2% effective 6-month interest rate, and premiums of $93.809 for the 1000-strike 6-month call and $74.201 for the 1000-strike
6-month put.

Insuring a Long Position: Floors
The analysis in Section 2.5 demonstrated that put options are insurance against a fall in the price of an asset. Thus, if we own the S&R index, we can insure the position by buying an S&R put option. The purchase of a put option is also called a floor, because we are guaranteeing a minimum sale price for the value of the index.
To examine this strategy, we want to look at the combined payoff of the index position and put. In the last chapter we graphed them separately; now we add them together to see the net effect of holding both positions at the same time.
Table 3.1 summarizes the result of buying a 1000-strike put with 6 months to expiration, in conjunction with holding an index position with a current value of $1000. The table computes the payoff for each position and sums them to obtain the total payoff. The final

61

62

Chapter 3. Insurance, Collars, and Other Strategies

TABLE 3.1

Payoff and profit at expiration from purchasing the S&R index and a 1000-strike put option. Payoff is the sum of the first two columns. Cost plus interest for the position is
($1000 + $74.201) × 1.02 = $1095.68. Profit is payoff less $1095.68.

Payoff at Expiration
S&R Index
S&R Put

Payoff

−(Cost + Interest)

Profit

$900
950
1000
1050
1100
1150
1200

$1000
1000
1000
1050
1100
1150
1200

−$1095.68
−1095.68
−1095.68
−1095.68
−1095.68
−1095.68
−1095.68

−$95.68
−95.68
−95.68
−45.68
4.32
54.32
104.32

$100
50
0
0
0
0
0

column takes account of financing cost by subtracting cost plus interest from the payoff to obtain profit. “Cost” here means the initial cash required to establish the position. This is positive when payment is required, and negative when cash is received. We could also have computed profit separately for the put and index. For example, if the index is $900 at expiration, we have
$900 − ($1000 × 1.02) + $100 − ($74.201 × 1.02) = −$95.68
Profit on S&R Index

Profit on Put

This gives the same result as the calculation performed in Table 3.1. The level of the floor is −$95.68, which is the lowest possible profit.
Figure 3.1 graphs the components of Table 3.1. Panels (c) and (d) show the payoff and profit for the combined index and put positions. The combined payoff graph in panel
(c) is created by adding at each index price the value of the index and put positions; this is equivalent to summing columns 1 and 2 in Table 3.1.
Notice in Figure 3.1 that the combined position created by adding the index and the put looks like a call. Intuitively this equivalence makes sense. A call has a limited loss—the premium—and benefits from gains in the index above the strike price. Similarly, when we own the index and buy a put, the put limits losses, but it permits us to benefit from gains in the index. Thus, the call on the one hand and the insured index position on the other have similar characteristics.
Panel (c), however, illustrates that the payoff to the combined position (index plus put) is not identical to the payoff from buying a call (compare panel (c) to Figure 2.5). The difference stems from the fact that buying a call entails paying only the option premium, while buying the index and put entails paying for both the index and the put option, which together are more expensive than buying a call. The profit diagram in panel (d) of Figure
3.1, however, does look like a call. In fact, it is identical to the profit diagram for buying

3.1 Basic Insurance Strategies

FIGURE 3.1
Panel (a) shows the payoff diagram for a long position in the index (column 1 in
Table 3.1). Panel (b) shows the payoff diagram for a purchased index put with a strike price of $1000 (column 2 in Table 3.1). Panel
(c) shows the combined payoff diagram for the index and put (column 3 in Table 3.1).
Panel (d) shows the combined profit diagram for the index and put, obtained by subtracting $1095.68 from the payoff diagram in panel
(c) (column 5 in Table 3.1).

Payoff ($)

63

Payoff ($)
Long S&R Index

Long S&R Put

2000

2000

1000

1000

0

0

–1000

–1000

–2000

0

500 1000 1500 2000

–2000

0

500 1000 1500 2000

S&R Index at Expiration

S&R Index at Expiration

(a)

(b)

Payoff ($)

Profit ($)
Combined Payoff

2000
1000

1000
$1000

0
–1000
–2000

Combined Profit
2000

0
–$95.68

–1000
0

500 1000 1500 2000
S&R Index at Expiration
(c)

–2000

0

500 1000 1500 2000
S&R Index at Expiration
(d)

an S&R index call with a strike price of $1000, graphed in Figure 2.6. We can see this by comparing Table 2.3 with Table 3.1. The profit of −$95.68 for prices below $1000 is exactly the future value of the 1000-strike 6-month to expiration call premium above.
We discussed in Section 2.1 that adding a bond to a payoff diagram shifts it vertically but leaves the corresponding profit diagram unaffected. The combined position of index plus put in panel (c) is therefore equivalent to buying a 1000-strike call—depicted by itself in panel (d)—and buying a zero-coupon bond that pays $1000 at expiration of the option. The point that buying an asset and a put generates a position that looks like a call can also be seen using the homeowner’s insurance example from Section 2.5. There, we examined the insurance policy in isolation. However, in practice, a buyer of homeowner’s insurance also owns the insured asset (the house). Owning a home is analogous to owning the stock index, and insuring the house is like owning a put. Thus, owning a home plus insurance is like owning the index and owning a put. Figure 3.2 depicts the insurance policy from Figure 2.12, together with the uninsured house and the combined position. Interpreting the house as the S&R index and insurance as the put, Figure 3.2 looks exactly like Figure 3.1.
An insured house has a profit diagram that looks like a call option.

64

Chapter 3. Insurance, Collars, and Other Strategies

Profit ($)

FIGURE 3.2
Payoff to owning a house and owning insurance. We assume a $25,000 deductible and a $200,000 house, with the policy costing $15,000.

200,000
150,000
Insurance
100,000
50,000
0
Insured house
–50,000
–100,000

Uninsured house

–150,000
–200,000
50,000

150,000
House Price ($)

250,000

Insuring a Short Position: Caps
If we have a short position in the S&R index, we experience a loss when the index rises.
We can insure a short position by purchasing a call option to protect against a higher price of repurchasing the index.1 Buying a call option is also called a cap.
Table 3.2 presents the payoff and profit for a short position in the index coupled with a purchased call option. Because we short the index, we earn interest on the short proceeds less the cost of the call option, giving −$924.32 as the future value of the cost.
Figure 3.3 graphs the columns of Table 3.2. The payoff and profit diagrams resemble those of a purchased put. As with the insured index position in Figure 3.1, we have to be careful in dealing with cash flows. The payoff in panel (c) of Figure 3.3 is like that of a purchased put coupled with borrowing. In this case, the payoff diagram for shorting the index and buying a call is equivalent to that from buying a put and borrowing the present value of $1000 ($980.39). Since profit diagrams are unaffected by borrowing, however, the profit diagram in panel (d) is exactly the same as that for a purchased S&R index put. You can see this by comparing panel (d) with Figure 2.8. Not only does the insured short position look like a put, it has the same loss as a purchased put if the price is above $1000: $75.68, which is the future value of the $74.201 put premium.

1. Keep in mind that if you have an obligation to buy the index in the future but the price is not fixed, then you have an implicit short position (if the price goes up, you will have to pay more). A call is insurance for both explicit and implicit short-sellers.

3.1 Basic Insurance Strategies

TABLE 3.2

65

Payoff and profit at expiration from short-selling the S&R index and buying a 1000-strike call option at a premium of
$93.809. The payoff is the sum of the first two columns.
Cost plus interest for the position is (−$1000 + $93.809)
× 1.02 = −$924.32. Profit is payoff plus $924.32.

Payoff at Expiration
Short S&R Index
S&R Call

Payoff

−(Cost + Interest)

Profit

−$900
−950
−1000
−1050
−1100
−1150
−1200

−$900
−950
−1000
−1000
−1000
−1000
−1000

$924.32
924.32
924.32
924.32
924.32
924.32
924.32

$24.32
−25.68
−75.68
−75.68
−75.68
−75.68
−75.68

FIGURE 3.3
Panel (a) shows the payoff diagram for a short position in the index (column 1 in
Table 3.2). Panel (b) shows the payoff diagram for a purchased index call with a strike price of $1000 (column 2 in Table 3.2). Panel
(c) shows the combined payoff diagram for the short index and long call (column
3 in Table 3.2). Panel (d) shows the combined profit diagram for the short index and long call, obtained by adding $924.32 to the payoff diagram in panel (c) (column
5 in Table 3.2).

$0
0
0
50
100
150
200

Payoff ($)

Payoff ($)
Short S&R Index

Long S&R Call

2000

2000

1000

1000

0

0

–1000

–1000

–2000

0

500 1000 1500 2000
S&R Index at Expiration

–2000

0

500 1000 1500 2000
S&R Index at Expiration

(a)
Payoff ($)

(b)
Profit ($)

Combined Payoff
2000

Combined Profit
2000

1000

–$1000

1000

0

0

–1000

–1000

–2000

0

500 1000 1500 2000
S&R Index at Expiration
(c)

–2000

–$75.68

0

500 1000 1500 2000
S&R Index at Expiration
(d)

66

Chapter 3. Insurance, Collars, and Other Strategies

Selling Insurance
We can expect that some investors want to purchase insurance. However, for every insurance buyer there must be an insurance seller. In this section we examine strategies in which investors sell insurance.
It is possible, of course, for an investor to simply sell calls and puts. Often, however, investors also have a position in the asset when they sell insurance. Writing an option when there is a corresponding long position in the underlying asset is called covered writing, option overwriting, or selling a covered call. All three terms mean essentially the same thing.2 In contrast, naked writing occurs when the writer of an option does not have a position in the asset.
In addition to the covered writing strategies we will discuss here, there are other insurance-selling strategies, such as delta-hedging, which are less risky than naked writing and are used in practice by market-makers. We will discuss these other strategies later in the book, particularly in Chapter 13.
Covered Call Writing. If we own the S&R index and simultaneously sell a call option, we have written a covered call. A covered call will have limited profitability if the index increases, because an option writer is obligated to sell the index for the strike price. Should the index decrease, the loss on the index is offset by the premium earned from selling the call. A payoff with limited profit for price increases and potentially large losses for price decreases sounds like a written put.
The covered call looks exactly like a written put, and the maximum profit will be the same as with a written put. Suppose the index is $1100 at expiration. The profit is
$1100 − ($1000 × 1.02) + ($93.809 × 1.02) − $100 = $75.68
Profit on S&R Index

Profit on written call

which is the future value of the premium received from writing a 1000-strike put.
The profit from owning the index and writing the 1000-strike call is computed in
Table 3.3 and graphed in Figure 3.4. If the index falls, we lose money on the index but the option premium partially offsets the loss. If the index rises above the strike price, the written option loses money, negating gains on the index.
Comparing Table 3.3 with Table 2.4, we can see that writing the covered call generates exactly the negative of the profit from buying a put.
Why would anyone write a covered call? Suppose you have the view that the index is unlikely to move either up or down. (This is sometimes called a “neutral” market view.)
If in fact the index does not move and you have written a call, then you keep the premium.
If you are wrong and the stock appreciates, you forgo gains you would have had if you did not write the call.
Covered Puts. A covered put is achieved by writing a put against a short position on the index. The written put obligates you to buy the index—for a loss—if it goes down in price.
Thus, for index prices below the strike price, the loss on the written put offsets the short stock. For index prices above the strike price, you lose on the short stock.

2. Technically, “option overwriting” refers to selling a call on stock you already own, while a “covered write” entails simultaneously buying the stock and selling a call. The distinction is irrelevant for our purposes. 3.1 Basic Insurance Strategies

TABLE 3.3

Payoff and profit at expiration from purchasing the S&R index and selling a 1000-strike call option. The payoff column is the sum of the first two columns. Cost plus interest for the position is ($1000 − $93.809) × 1.02 =
$924.32. Profit is payoff less $924.32.

Payoff at Expiration
S&R Index
Short S&R Call
$900
950
1000
1050
1100
1150
1200

FIGURE 3.4
Payoff and profit diagrams for writing a covered S&R call. Panel (a) is the payoff to a long S&R position.
Panel (b) is the payoff to a short S&R call with strike price of 1000. Panel (c) is the combined payoff for the S&R index and written call. Panel (d) is the combined profit, obtained by subtracting ($1000 −
$93.809) × 1.02 = $924.32 from the payoff in panel (c).

67

−(Cost + Interest)

Profit

$900
950
1000
1000
1000
1000
1000

$0
0
0
−50
−100
−150
−200

Payoff

−$924.32
−924.32
−924.32
−924.32
−924.32
−924.32
−924.32

−$24.32
25.68
75.68
75.68
75.68
75.68
75.68

Payoff ($)

Payoff ($)
Long S&R Index

Short S&R Call

2000

2000

1000

1000

0

0

–1000

–1000

–2000

0

500 1000 1500 2000

–2000

0

500 1000 1500 2000

S&R Index at Expiration

S&R Index at Expiration

(a)

(b)

Payoff ($)

Profit ($)
Combined Payoff

Combined Profit

2000

2000

1000

1000

0

0
$1000

–1000
–2000

0

500 1000 1500 2000
S&R Index at Expiration
(c)

$75.68

–1000
–2000

0

500 1000 1500 2000
S&R Index at Expiration
(d)

68

Chapter 3. Insurance, Collars, and Other Strategies

Payoff ($)

FIGURE 3.5
Payoff and profit diagrams for writing a covered S&R put. Panel (a) is the payoff to a short S&R position.
Panel (b) is the payoff to a short S&R put with a strike price of $1000. Panel (c) is the combined payoff for the short S&R index and written put. Panel (d) is the combined profit, obtained by adding ($1000 + $74.201)
× 1.02 = $1095.68 to the payoff in panel (c).

Payoff ($)
Short S&R Index

Short S&R Put

2000

2000

1000

1000

0

0

–1000

–1000

–2000

0

500 1000 1500 2000
S&R Index at Expiration

–2000

0

500 1000 1500 2000
S&R Index at Expiration

(a)
Payoff ($)

(b)
Profit ($)

Combined Payoff
2000

Combined Profit
2000

1000

–$1000

$95.68

1000

0

0

–1000

–1000

–2000

0

500 1000 1500 2000
S&R Index at Expiration
(c)

–2000

0

500 1000 1500 2000
S&R Index at Expiration
(d)

A position where you have a constant payoff below the strike and increasing losses above the strike sounds like a written call. In fact, shorting the index and writing a put produces a profit diagram that is exactly the same as for a written call. Figure 3.5 shows this graphically, and Problem 3.2 asks you to verify this by constructing a payoff table.

3.2 PUT-CALL PARITY
We now discuss put-call parity, which is one of the most important relationships in the theory of options. The parity equation tells us the difference in the premiums of puts and calls, when the two options have the same strike price and time to expiration. In this section we will first discuss the use of options to create synthetic forward contracts. We then develop the put-call parity equation. You should be aware that we will discuss parity at much greater length in Chapter 9, and that parity will arise frequently throughout the rest of the book.

Synthetic Forwards
It is possible to mimic a long forward position on an asset by buying a call and selling a put, with each option having the same strike price and time to expiration. For example, we could buy the 6-month 1000-strike S&R call for $93.81 and sell the 6-month 1000-strike
S&R put for $74.20. This position resembles a long forward contract: In 6 months we will

3.2 Put-Call Parity

FIGURE 3.6
Purchase of a 1000-strike
S&R call, sale of a 1000strike S&R put, and the combined position. The combined position resembles the profit on a long forward contract.

Profit ($)
200
150
100
50

$1020

0
–50
–100
Purchased call
Written put
Combined position

–150
–200
–250
800

850

900

950 1000 1050 1100 1150 1200
S&R Index Price ($)

pay $1000 to buy the index. An important difference between the forward and the option position is that the forward contract has no premium, but the options have a net cost of
$93.81 − $74.20 = $19.61.
To better understand this position, suppose that the index in 6 months is at 900. We will not exercise the call, but we have written a put. The put buyer will exercise the right to sell the index for $1000; therefore we are obligated to buy the index at $1000. If instead the index is at $1100, the put is not exercised, but we exercise the call, buying the index for
$1000. Thus, whether the index rises or falls, when the options expire we buy the index for the strike price of the options, $1000.
The purchased call, written put, and combined positions are shown in Figure 3.6. The purchase of a call and sale of a put creates a synthetic long forward contract, which has two minor differences from the actual forward:
1. The forward contract has a zero premium, while the synthetic forward requires that we pay the net option premium.
2. With the forward contract we pay the forward price, while with the synthetic forward we pay the strike price.
If you think about it, these two considerations must be related. If we set the strike price low, we are obligated to buy the index at a discount relative to the forward price. Buying at a lower price than the forward price is a benefit. In order to obtain this benefit we have to pay the positive net option premium, which stems from the call being more expensive than the put. In fact, in Figure 3.6, the implicit cost of the synthetic forward—the price at which the profit on the combined call-put position is zero—is $1020, which is the S&R forward price. Similarly, if we set the strike price high, we are obligated to buy the index at a high price relative to the forward price. To offset the extra cost of acquiring the index using the high strike options, it makes sense that we would receive payment initially. This would occur if the put that we sell is more expensive than the call we buy.

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Chapter 3. Insurance, Collars, and Other Strategies

Finally, if we set the strike price equal to the forward price, then to mimic the forward the initial premium must equal zero. In this case, put and call premiums must be equal.

The Put-Call Parity Equation
We can summarize this argument by saying that the net cost of buying the index at a future date using options must equal the net cost of buying the index using a forward contract. If at time 0 we enter into a long forward position expiring at time T , we obligate ourselves to buy the index at the forward price, F0, T . The present value of buying the index in the future is just the present value of the forward price, PV(F0, T ).
If instead we buy a call and sell a put today to guarantee the purchase price for the index in the future, the present value of the cost is the net option premium for buying the call and selling the put, Call(K , T ) − Put(K , T ), plus the present value of the strike price,
PV(K). (The notations “Call(K , T )” and “Put(K , T )” denote the premiums of options with strike price K and with T periods until expiration.)
Equating the costs of the alternative ways to buy the index at time t gives us
PV(F0, T ) = [Call(K , T ) − Put(K , T )] + PV(K)
We can rewrite this as
Call(K , T ) − Put(K , T ) = PV(F0, T − K)

(3.1)

In words, the present value of the bargain element from buying the index at the strike price
[the right-hand side of equation (3.1)] must equal the initial net option premium [the lefthand side of equation (3.1)]. Equation (3.1) is known as put-call parity.
Example 3.1 As an example of equation (3.1), consider buying the 6-month 1000-strike
S&R call for a premium of $93.809 and selling the 6-month 1000-strike put for a premium of $74.201. These transactions create a synthetic forward permitting us to buy the index in 6 months for $1000. Because the actual forward price is $1020, this synthetic forward permits us to buy the index at a bargain of $20, the present value of which is $20/1.02 = $19.61. The difference in option premiums is also $19.61 ($93.809 − $74.201 = $19.61). This result is exactly what we get with equation (3.1):
$93.809 − $74.201 = PV($1020 − $1000)
A forward contract for which the premium is not zero is sometimes called an offmarket forward. This terminology arises since a true forward by definition has a zero premium. Therefore, a forward contract with a nonzero premium must have a forward price that is “off the market (forward) price.” Unless the strike price equals the forward price, buying a call and selling a put creates an off-market forward.
Equivalence of Different Positions. We have seen earlier that buying the index and buying a put generates the same profit as buying a call. Similarly, selling a covered call (buying the index and selling a call) generates the same profit as selling a put. Equation (3.1) explains why this happens.
Consider buying the index and buying a put, as in Section 3.1. Recall that, in this example, we have the forward price equal to $1020 and the index price equal to $1000.

3.3 Spreads and Collars

Thus, the present value of the forward price equals the index price. Rewriting equation
(3.1) gives
PV(F0, T ) + Put(K , T ) = Call(K , T ) + PV(K)
$1000 + $74.201 = $93.809 + $980.39
That is, buying the index and buying the put cost the same, and generate the same payoff, as buying the call and buying a zero-coupon bond costing PV(K). (Recall from Section 2.1 that a bond does not affect profit.)
Similarly, in the case of writing a covered call, we have
PV(F0, T ) − Call(K , T ) = PV(K) − Put(K , T )
That is, writing a covered call has the same profit as lending PV(K) and selling a put.
Equation (3.1) provides a tool for constructing equivalent positions.
No Arbitrage. In deriving equation (3.1), and in some earlier discussions, we relied on the idea that if two different investments generate the same payoff, they must have the same cost. This commonsensical idea is one of the most important in the book. If equation
(3.1) did not hold, there would be both low-cost and high-cost ways to acquire the index at time T . We could simultaneously buy the index at low cost and sell the index at high cost.
This transaction has no risk (since we both buy and sell the index) and generates a positive cash flow (because of the difference in costs). Taking advantage of such an opportunity is called arbitrage, and the idea that prices should not permit arbitrage is called “no-arbitrage pricing.” We implicitly illustrated this idea earlier in showing how owning the index and buying a put has the same profit as a call, etc. No-arbitrage pricing will be a major theme in Chapter 5 and beyond.3

3.3 SPREADS AND COLLARS
There are many well-known, commonly used strategies that combine two or more options.
In this section we discuss some of these strategies and explain the motivation for using them. The underlying theme in this section is that there are always trade-offs in designing a position: It is always possible to lower the cost of a position by reducing its payoff. Thus there are many variations on each particular strategy.
All the examples in this section will use the set of option prices in Table 3.4. We will assume the continuously compounded interest rate is 8%.

Bull and Bear Spreads
An option spread is a position consisting of only calls or only puts, in which some options are purchased and some written. Spreads are a common strategy. In this section we define some typical spread strategies and explain why you might use a spread.
Suppose you believe a stock will appreciate. Let’s compare two ways to speculate on this belief: entering into a long forward contract or buying a call option with the strike

3. Another way to express the principle of no arbitrage is using profit diagrams. Given two profit diagrams, there is an arbitrage opportunity if one diagram is nowhere below and somewhere above the other.

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TABLE 3.4

Black-Scholes option prices assuming stock price = $40, volatility = 30%, effective annual risk-free rate = 8.33%
(8%, continuously compounded), dividend yield = $0, and
91 days to expiration.

Strike

Call

Put

35
40
45

6.13
2.78
0.97

0.44
1.99
5.08

price equal to the forward price. The forward contract has a zero premium and the call has a positive premium. A difference in payoffs explains the difference in premiums. If the stock price at expiration is greater than the forward price, the forward contract and call have the same payoff. If the stock price is less than the forward price, however, the forward contract has a loss and the call is worth zero. Put-call parity tells us that the call is equivalent to the forward contract plus a put option. Thus, the call premium equals the cost of the put, which is insurance against the stock price being less than the forward price.
You might ask: Is there a lower-cost way to speculate that the stock price will rise, that still has the insurance implicit in the call? The answer is yes: You can lower the cost of your strategy if you are willing to reduce your profit should the stock appreciate. You can do this by selling a call at a higher strike price. The owner of this second call buys appreciation above the higher strike price and pays you a premium. You achieve a lower cost by giving up some portion of profit. A position in which you buy a call and sell an otherwise identical call with a higher strike price is an example of a bull spread.
Bull spreads can also be constructed using puts. Perhaps surprisingly, you can achieve the same result either by buying a low-strike call and selling a high-strike call, or by buying a low-strike put and selling a high-strike put.
Spreads constructed with either calls or puts are sometimes called vertical spreads.
The terminology stems from the way option prices are typically presented, with strikes arrayed vertically (as in Table 3.4).
Example 3.2 To see how a bull spread arises, suppose we want to speculate on the stock price increasing. Consider buying a 40-strike call with 3 months to expiration. From
Table 3.4, the premium for this call is $2.78. We can reduce the cost of the position—and also the potential profit—by selling the 45-strike call.
An easy way to construct the graph for this position is to emulate a spreadsheet: For each price, compute the profit of each option position and add up the profits for the individual positions. It is worth working through one example in detail to see how this is done.
The initial net cost of the two options is $2.78 − $0.97 = $1.81. With 3 months interest, the total cost at expiration is $1.81 × (1.0833)0.25 = $1.85. Table 3.5 computes the cash flow at expiration for both options and computes profit on the position by subtracting the future value of the net premium.
Figure 3.7 graphs the position in Table 3.5. You should verify that if you buy the
40-strike put and sell the 45-strike put, you obtain exactly the same graph.

3.3 Spreads and Collars

TABLE 3.5

Profit at expiration from purchase of 40-strike call and sale of 45-strike call.

Stock Price at Expiration

Purchased
40-Call

Written
45-Call

Premium
Plus Interest

Total

$35.0
37.5
40.0
42.5
45.0
47.5
50.0

$0.0
0.0
0.0
2.5
5.0
7.5
10.0

$0.0
0.0
0.0
0.0
0.0
−2.5
−5.0

−$1.85
−1.85
−1.85
−1.85
−1.85
−1.85
−1.85

−$1.85
−1.85
−1.85
0.65
3.15
3.15
3.15

FIGURE 3.7
Profit diagram for a 40–
45 bull spread: buying a
40-strike call and selling a
45-strike call.

Profit ($)
4

Bull spread

3
2
1
0
–1
–2
20

25

30

35

40

45

50

55

60

XYZ Stock Price ($)

The opposite of a bull spread is a bear spread. Using the options from the above example, we could create a bear spread by selling the 40-strike call and buying the 45-strike call. The profit diagram would be exactly the opposite of Figure 3.7.

Box Spreads
A box spread is accomplished by using options to create a synthetic long forward at one price and a synthetic short forward at a different price. This strategy guarantees a cash flow in the future. Hence, it is an option spread that is purely a means of borrowing or lending

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Chapter 3. Insurance, Collars, and Other Strategies

money: It is costly but has no stock price risk. The reasons for using a box spread are discussed in the box on page 75.
Example 3.3 Suppose we simultaneously enter into the following two transactions:
1. Buy a 40-strike call and sell a 40-strike put.
2. Sell a 45-strike call and buy a 45-strike put.
The first transaction is a synthetic forward purchase of a stock for $40, while the second transaction is the synthetic forward sale of the stock for $45. Clearly the payoff at expiration will be $5; hence, the transaction has no stock price risk. Using the assumptions in Table 3.4, the cost of the strategy should be
5 × (1.0833)−0.25 = $4.90
In fact, using the premiums in Table 3.4, the initial cash flow is
$1.99 − $2.78 + ($0.97 − $5.08) = −$4.90
Another way to view this transaction is that we have bought a 40–45 bull spread using calls
(buy 40 call, sell 45 call), and bought a 40–45 bear spread using puts (sell 40 put, buy 45 put). Ratio Spreads
A ratio spread is constructed by buying m options at one strike and selling n options at a different strike, with all options having the same type (call or put), same time to maturity, and same underlying asset. You are asked to construct ratio spreads in Problem 3.15. Also, a ratio spread constructed by buying a low-strike call and selling two higher-strike calls is one of the positions depicted in the chapter summary in Figure 3.16.
Since ratio spreads involve buying and selling unequal numbers of options, it is possible to construct ratio spreads with zero premium. The significance of this may not be obvious to you now, but we will see in Chapter 4 that by using ratio spreads we can construct paylater strategies: insurance that costs nothing if the insurance is not needed.
The trade-off to this, as you might guess, is that the insurance is more costly if it is needed.

Collars
A collar is the purchase of a put and the sale of a call with a higher strike price, with both options having the same underlying asset and the same expiration date. If the position is reversed (sale of a put and purchase of a call), the collar is written. The collar width is the difference between the call and put strikes.4
Example 3.4 Suppose we sell a 45-strike call with a $0.97 premium and buy a 40-strike put with a $1.99 premium. This collar is shown in Figure 3.8. Because the purchased put has a higher premium than the written call, the position requires investment of $1.02.

4. Collars in which call and put strikes are equidistant from the forward price are sometimes called risk reversals. This term also has a different but related meaning that we will discuss in Chapter 12.

3.3 Spreads and Collars

BOX

A

75

3.1: The Use of Box Spreads

box spread is an alternative to buying a bond. Option market-makers in particular have low transaction costs and can sell box spreads, which is equivalent to borrowing. Box spreads can therefore be a source of funds. In the past, box spreads also provided a tax benefit for some investors. Although a change in the tax law in 1993 ostensibly eliminated this use of box spreads, the issue provides an illustration of why derivatives create problems for the tax authorities.
Consider a taxable investor who has sold stock investments at a loss. This loss is classified for tax purposes as a capital loss. In the United States, capital gains are always taxed, but capital losses are only deductible against capital gains. (The exception to this is that individual investors are allowed to deduct a limited amount of capital losses against ordinary income.) Thus, a taxable investor with large capital losses would like to find a mechanism to generate income which can be labeled as capital gains. This is not as easy as it sounds. A risk-free zero-coupon bond— which is certain to appreciate over its life— generates interest income, which cannot be used to offset capital losses. A stock held to generate gains could instead go down in price, generating additional losses.
A box spread sounds as if it should enable investors to generate capital gains as needed: It

is a synthetic bond, guaranteed to appreciate in value just like a bond. Moreover, the gain or loss on an option is a capital gain or loss. If the change in value of a box spread were taxed as a capital gain, box spreads could be used to create riskfree capital gains income, against which capital losses could be offset.
Lawmakers in the United States have anticipated strategies like this. Section 1258 of the
U.S. Tax Code, enacted in 1993, explicitly states that capital income should be taxed as ordinary income if all expected return is due to time value of money on the investment (in other words, if the investment is equivalent to a bond). This would seem to eliminate the tax motive for entering into box spreads. The problem for the tax authorities, however, is how to identify taxpayers using box spreads for this purpose. There is nothing wrong with entering into a box spread; the law is only violated if the taxpayer reports the resulting income as a capital gain. This is difficult to detect. Tax rules may also differ internationally. In Griffin v.
Citibank Investments Ltd. (2000), for example,
British courts ruled that a box spread was not necessarily equivalent to a loan.
The fundamental problem is that the tax code calls for different taxation of bonds and options, but options can be used to create bonds. There are many similar illustrations of this problem.

If you hold this book at a distance and squint at Figure 3.8, the collar resembles a short forward contract. Economically, it is like a short forward contract in that it is fundamentally a short position: The position benefits from price decreases in the underlying asset and suffers losses from price increases. A collar differs from a short forward contract in having a range between the strikes in which the expiration payoff is unaffected by changes in the value of the underlying asset.
In practice, collars are frequently used to implement insurance strategies—for example, by buying a collar when we own the stock. This position, which we will call a collared stock, entails buying the stock, buying a put, and selling a call. It is an insured position because we own the asset and buy a put. The sale of a call helps to pay for the purchase of the put. The collared stock looks like a bull spread; however, it arises from a different set

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Chapter 3. Insurance, Collars, and Other Strategies

FIGURE 3.8
Profit diagram of a purchased collar constructed by selling a 45-strike call and buying a 40-strike put.

Profit ($)
Purchased collar

20
15
10
5
0
–5
–10
–15
–20
20

25

30

35
40
45
50
XYZ Stock Price ($)

55

60

of transactions. The bull spread is created by buying one option and selling another. The collared stock begins with a position in the underlying asset that is coupled with a collar.
Example 3.5 Suppose that you own shares of XYZ for which the current price is $40, and you wish to buy insurance. You do this by purchasing a put option. A way to reduce the cost of the insurance is to sell an out-of-the-money call. The profit calculations for this set of transactions—buy the stock, buy a 40-strike put, sell a 45-strike call—are shown in
Table 3.6. Comparing this table to Table 3.5 demonstrates that profit on the collared stock position is identical to profit on the bull spread. Note that it is essential to account for interest as a cost of holding the stock.
If you have a short position in the stock, you can collar the position by buying a call for insurance and selling an out-of-the-money put to partially fund the call purchase. The result looks like a bear spread.
Zero-Cost Collars. The collar depicted in Table 3.6 entails paying a net premium of $1.02:
$1.99 for the purchased put, against $0.97 for the written call. It is possible to find strike prices for the put and call such that the two premiums exactly offset one another. This position is called a zero-cost collar.
To illustrate a zero-cost collar, suppose you buy the stock and buy the 40-strike put that has a premium of $1.99. Trial and error with the Black-Scholes formula reveals that a call with a strike of $41.72 also has a premium of $1.99. Thus, you can buy a 40-strike put and sell a 41.72-strike call without paying any premium. The result is depicted in Figure 3.9.
At expiration, the collar exposes you to stock price movements between $40 and $41.72, coupled with downside protection below $40. You pay for this protection by giving up gains should the stock move above $41.72.

3.3 Spreads and Collars

TABLE 3.6

Profit at expiration from purchase of 40-strike put and sale of 45-strike call.

Stock Price at Expiration

Purchased
40-Put

Written
45-Call

Premium
Plus Interest

Profit on Stock

Total

$35.00
37.50
40.00
42.50
45.00
47.50
50.00

$5.00
2.50
0.00
0.00
0.00
0.00
0.00

$0.00
0.00
0.00
0.00
0.00
−2.50
−5.00

−$1.04
−1.04
−1.04
−1.04
−1.04
−1.04
−1.04

−$5.81
−3.31
−0.81
1.69
4.19
6.69
9.19

−$1.85
−1.85
−1.85
0.65
3.15
3.15
3.15

FIGURE 3.9
Zero-cost collar on XYZ, created by buying XYZ at
$40, buying a 40-strike put with a premium of $1.99, and selling a 41.72-strike call with a premium of $1.99.

Profit ($)
20
15
10
5
0
–5
–10
–15
Long XYZ stock
Stock + Zero-cost collar

–20
–25
20

25

30

35

40

45

50

55

60

XYZ Stock Price ($)

For any given stock there is an infinite number of zero-cost collars. One way to see this is to first pick the desired put strike below the forward price. It is then possible to find a strike above the forward price such that a call has the same premium.
Understanding Collars. One aspect of the zero-cost collar that may seem puzzling is that you can finance the purchase of an at-the-money put by selling an out-of-the-money call.
In the above example, with the stock at $40, you were able to costlessly buy a 40-strike put by also selling a 41.72-strike call. This makes it seem as if you have free insurance with some possibility of gain. Even if you are puzzled by this, you probably realize that “free”

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I

Chapter 3. Insurance, Collars, and Other Strategies

3.2: Bernard Madoff: A Collar by Any Other Name

n 2009, Bernard L. Madoff confessed to having run a Ponzi scheme in his investment fund,
Bernard L. Madoff Investment Securities, LLC.
A Ponzi scheme (named after Charles Ponzi) is a fraudulent investment vehicle in which funds paid in by late investors are used to pay off early investors. The sheer scale of Madoff’s fraud was astonishing: There was an estimated $65 billion missing from investor accounts. The story dominated the news for weeks, with numerous accounts of individual investors who had lost huge sums in the swindle.
Madoff had claimed that his investment fund had returns that were both high on average and with low volatility. Investors were attracted, though there were reports that many professional investors were skeptical. In fact, in 2005, Harry
Markopolos sent to the SEC a now-famous 17page memo in which he alleged that Madoff was running a fraudulent investment fund. Among his

assertions were that Madoff’s returns reported to clients were too great to be real, and that the correlation of his returns to the S&P 500 index was too low.
Madoff claimed to be using collars to achieve his stable high returns. According to Markopolos,
Madoff claimed to investors that he achieved his returns using a “split-strike conversion” strategy on a basket of large stocks. A conversion consists of buying a stock, buying a put, and selling a call, with the options having the same strike. In a
“split-strike” conversion, the put strike is below the call strike—in other words, it’s a collar.
Stutzer (2010) and Bernard and Boyle (2009) have both examined Madoff’s investment strategy and evaluated Markopolos’s claims. Both papers conclude that Madoff’s reported returns are statistically implausible. In particular, the strategy should have exhibited a greater return volatility and a higher correlation with the index.

insurance is not possible, and something must be wrong with this way of thinking about the position. This puzzle is resolved by taking into account financing cost. Recall that if you pay
$40 for stock and sell it for $40 in 91 days, you have not broken even. You have lost money, because you have forgone $40 × (1.08330.25 − 1) = $0.808 in interest. Thus, the true breakeven stock price in this example is $40.808, about halfway between $40 and $41.72.
To illustrate the use and pricing of collars, consider an executive who owns a large position in company stock. Such executives frequently hedge their stock positions, using zero-cost collars with several years to maturity. Suppose, for example, that a firm has a price of $30/share and an executive wishes to hedge 1 million shares. If the executive buys a 30strike put with 3 years to maturity, what 3-year call will have the same premium? Assuming an effective annual risk-free rate of 6%, a zero dividend yield, and a 40% volatility, the
Black-Scholes price is $5.298 for a 30-strike put with 3 years to maturity. Using trial and error (or a numerical solver), a call option with a strike of $47.39 has the same premium.
Once again, the zero-cost collar seems highly asymmetric. However, this comparison does not take into account financing cost. The executive selling stock in 3 years for $30/share will in fact have lost 3 years’ worth of interest: $30 × [(1.06)3 − 1] = $5.73.
Box 3.2 discusses another example of how collars were (or were not) used.
The Cost of the Collar and the Forward Price. Suppose you try to construct a zero-cost collar in which you set the strike of the put option at the stock price plus financing cost—

3.4 Speculating on Volatility

i.e., the future value of the stock price. In the 91-day example above, this would require that you set the put strike equal to $40.808, which gives a premium of $2.39. The call premium at this strike is also $2.39! If you try to insure against all losses on the stock, including interest, then a zero-cost collar has zero width.
This is an implication of put-call parity, equation (3.1). It turns out that $40.808 is also the theoretical forward price. If we set the strike equal to the forward price, the call premium equals the put premium.

3.4 SPECULATING ON VOLATILITY
The positions we have just considered are all directional: A bull spread or a collared stock is a bet that the price of the underlying asset will increase. Options can also be used to create positions that are nondirectional with respect to the underlying asset. With a nondirectional position, the holder does not care whether the stock goes up or down, but only how much it moves. We now examine straddles, strangles, and butterfly spreads, which are examples of nondirectional speculations.

Straddles
Consider the strategy of buying a call and a put with the same strike price and time to expiration: This strategy is called a straddle. The general idea of a straddle is simple: If the stock price rises, there will be a profit on the purchased call, and if the stock price declines, there will be a profit on the purchased put. Thus, the advantage of a straddle is that it can profit from stock price moves in both directions. The disadvantage to a straddle is that it has a high premium because it requires purchasing two options. If the stock price at expiration is near the strike price, the two premiums are lost. The profit diagram for a 40-strike straddle is graphed in Figure 3.10. The initial cost of the straddle at a stock price of $40 is $4.77:
$2.78 for the call and $1.99 for the put.
Figure 3.10 demonstrates that a straddle is a bet that volatility will be high: The buyer of an at-the-money straddle is hoping that the stock price will move but does not care about the direction of the move. Because option prices reflect the market’s estimate of volatility, the cost of a straddle will be greater when the market’s perception is that volatility is greater.
If at a given set of option prices all investors found it desirable to buy straddles, then option prices would increase. Thus, purchasing a straddle is really a bet that volatility is greater than the market’s assessment of volatility, as reflected in option prices.
Strangle. The disadvantage of a straddle is the high premium cost. To reduce the premium, you can buy out-of-the-money options rather than at-the-money options. Such a position is called a strangle. For example, consider buying a 35-strike put and a 45-strike call, for a total premium of $1.41, with a future value of $1.44. These transactions reduce your maximum loss if the options expire with the stock near $40, but they also increase the stock-price move required for a profit.
Figure 3.11 shows the 40-strike straddle graphed against the 35–45 strangle. This comparison illustrates a key point: In comparing any two fairly priced option positions, there will always be a region where each outperforms the other. Indeed, this is necessary to have a fairly priced position.

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FIGURE 3.10
Combined profit diagram for a purchased 40-strike straddle—i.e., purchase of one 40-strike call option and one 40-strike put option.

Profit ($)
Straddle

20

15

10

5

0

–5
20

25

30

35
40
45
50
XYZ Stock Price ($)

55

60

In Figure 3.11, the strangle outperforms the straddle roughly when the stock price at expiration is between $36.57 and $43.43. Obviously, there is a much broader range in which the straddle outperforms the strangle. How can you decide which is the better investment?
The answer is that unless you have a particular view on the stock’s performance, you cannot say that one position is preferable to the other. An option pricing model implicitly evaluates the likelihood that one strategy will outperform the other, and it computes option prices so that the two strategies are equivalently fair deals. An investor might have a preference for one strategy over the other due to subjective probabilities that differ from the market’s.
Written Straddle. What if an investor believes that volatility is lower than the market’s assessment? Because a purchased straddle is a bet that volatility is high (relative to the market’s assessment), a written straddle—selling a call and put with the same strike price and time to expiration—is a bet that volatility is low (relative to the market’s assessment).
Figure 3.12 depicts a written straddle, which is exactly the opposite of Figure 3.10, the purchased straddle. The written straddle is most profitable if the stock price is $40 at expiration, and in this sense it is a bet on low volatility. What is striking about Figure 3.12, however, is the potential for loss. A large change in the stock price in either direction leads to a large, potentially unlimited, loss.
It might occur to you that an investor wishing to bet that volatility will be low could write a straddle and acquire insurance against extreme negative outcomes. That intuition is correct and leads to our next strategy.

Butterfly Spreads
The straddle writer can insure against large losses on the straddle by buying options to protect against losses on both the upside and downside. Buying an out-of-the-money put provides insurance on the downside, protecting against losses on the at-the-money written

3.4 Speculating on Volatility

FIGURE 3.11
40-strike straddle and strangle composed of 35strike put and 45-strike call.

Profit ($)
Straddle
Strangle

20

15

10

5

0

–5
20

FIGURE 3.12
Profit at expiration from a written straddle—i.e., selling a 40-strike call and a
40-strike put.

25

30

35
40
45
50
XYZ Stock Price ($)

55

60

Profit ($)
Written straddle

15
10
5
0
–5
–10
–15
–20
20

25

30

35
40
45
50
XYZ Stock Price ($)

55

60

put. Buying an out-of-the-money call provides insurance on the upside, protecting against losses on the written at-the-money call.
Figure 3.13 displays the straddle written at a strike price of $40, along with the options to safeguard the position: a 35-strike put and a 45-strike call. The net result of combining these three strategies is an insured written straddle, which is called a butterfly spread,

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FIGURE 3.13
Written 40-strike straddle, purchased 45-strike call, and purchased 35-strike put.
These positions combined generate the butterfly spread graphed in Figure 3.14.

Profit ($)
Written straddle
Purchased call
Purchased put

15
10
5
0
–5
–10
–15
–20
20

25

30

35
40
45
50
XYZ Stock Price ($)

55

60

graphed in Figure 3.14. It can be thought of as a written straddle for the timid (or for the prudent!). Comparing the butterfly spread to the written straddle (Figure 3.14), we see that the butterfly spread has a lower maximum profit (due to the cost of insurance) if the stock at expiration is close to $40, and a higher profit if there is a large move in the stock price, in which case the insurance becomes valuable.
We will see in Chapter 9 that by understanding the butterfly spread we gain important insights into option prices. Also, the butterfly spread can be created in a variety of ways: solely with calls, solely with puts, or by using the stock and a combination of calls and puts.5 You are asked to verify this in Problem 3.18. The spread in Figure 3.14 can also be created by simultaneously buying a 35–40 bull spread and a 40–45 bear spread.

Asymmetric Butterfly Spreads
Examine Figure 3.15. It looks like a butterfly spread except that it is asymmetric: The peak is closer to the high strike than to the low strike. This picture was created by buying two 35strike calls, selling ten 43-strike calls (with a premium of $1.525, using the assumptions in
Table 3.4), and buying eight 45-strike calls. The position is like a butterfly in that it earns a profit if the stock stays within a small range, and the loss is the same for high and low stock prices. However, the profit diagram is now tilted to the right, rather than being symmetric.
Suppose you knew that you wanted a position that looks like Figure 3.15. How would you know how many options to buy and sell to construct this position? In order to obtain

5. Technically, a true butterfly spread is created solely with calls or solely with puts. A butterfly spread created by selling a straddle and buying a strangle is called an “iron butterfly.”

3.4 Speculating on Volatility

FIGURE 3.14
Comparison of the 35–
40–45 butterfly spread, obtained by adding the profit diagrams in Figure 3.13, with the written 40-strike straddle. Profit ($)
Written straddle
Butterfly

15
10
5
0
–5
–10
–15
–20
20

25

30

35

40

45

50

55

60

XYZ Stock Price ($)

FIGURE 3.15
Asymmetric butterfly, created by buying two 35-strike calls and eight 45-strike calls and selling ten 43-strike calls. Profit ($)
Asymmetric butterfly
10
8
6
4
2
0
–2
–4
–6
–8
–10
20

25

30

35
40
45
50
XYZ Stock Price ($)

55

60

this position, the strikes clearly have to be at 35, 43, and 45. The total distance between 35 and 45 is 10. The number 43 is 80% (= 43−35 ) of the way from 35 to 45. In fact, we can
10
write 43 as
43 = (0.2 × 35) + (0.8 × 45)

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This way of writing 43 tells us our call position: For every written 43-strike call, we want to buy 0.2 35 calls and 0.8 45 calls. Thus if we sell ten 43-strike calls, we buy two 35 calls and eight 45-strike calls.
In general, consider the strike prices K1, K2, and K3, where K1 < K2 < K3. Define λ so that λ= K3 − K2
K3 − K1

or
K2 = λK1 + (1 − λ)K3
For example, if K1 = 35, K2 = 43, and K3 = 45, then λ = 0.2, as in the above example. In order to construct an asymmetric butterfly, for every K2 call we write, we buy λ K1 calls and 1 − λ K3 calls.
You should verify that if you buy two 35-strike puts, sell ten 43-strike puts, and buy eight 45-strike puts, you duplicate the profit diagram in Figure 3.15.

CHAPTER SUMMARY
Puts are insurance against a price decline and calls are insurance against a price increase.
Combining a long or short position in the asset with an offsetting position in options (for example, a long position in the asset is coupled either with a purchased put or written call) leads to the various possible positions and their equivalents in Table 3.7.
Buying a call and selling a put with the same strike price and time to expiration creates an obligation to buy the asset at expiration by paying the strike price. This is a synthetic forward. A synthetic forward must have the same cost in present value terms as a true forward. This observation leads to equation (3.1):
Call(K , T ) − Put(K , T ) = PV(F0, T − K)

(3.1)

This relationship, called put-call parity, explains the difference in call and put premiums for otherwise identical options. It is one of the most important relationships in derivatives. TABLE 3.7
Position
Index + Put
Index − Call
−Index + Call
−Index − Put

Summary of equivalent positions from Section 3.1.

Is Equivalent To

And Is Called

Zero-Coupon Bond + Call
Zero-Coupon Bond − Put
−Zero-Coupon Bond + Put
−Zero-Coupon Bond − Call

Insured Asset (floor)
Covered Written Call
Insured Short (cap)
Covered Written Put

Chapter Summary

Profit

FIGURE 3.16

Profit
Bull Spread

Collar

Stock Price

Stock Price

Profit diagrams for positions discussed in the chapter: bull spread, collar, straddle, strangle, butterfly, and 2:1 ratio spread.
Profit

Profit
Straddle

Strangle

Stock Price

Stock Price

Profit

Profit
Butterfly

Stock Price

TABLE 3.8

Ratio Spread

Stock Price

Positions consistent with different views on the stock price and volatility direction. Volatility Will Increase
Price will fall
No price view
Price will increase

No Volatility View

Volatility Will Fall

Buy puts
Buy straddle
Buy calls

Sell underlying
Do nothing
Buy underlying

Sell calls
Sell straddle
Sell puts

There are numerous strategies that permit speculating on the direction of the stock or on the size of stock price moves (volatility). Some of these positions are summarized graphically in Figure 3.16. We also categorize in Table 3.8 various strategies according to whether they reflect bullish or bearish views on the stock price direction or volatility.6

6. Table 3.8 was suggested by David Shimko.

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FURTHER READING
In Chapter 4 we will see how firms can use these strategies to manage risk. We will further explore put-call parity in Chapter 9, in which we also will use bull, bear, and butterfly spreads to say more about what it means for an option to be fairly priced.
Put-call parity was first demonstrated in Stoll (1969). Merton (1973a) corrected the original analysis for the case of American options, for which, because of early exercise, parity need not hold. Ronn and Ronn (1989) provide a detailed examination of price bounds and returns on box spreads.
There are numerous practitioner books on option trading strategies. A classic practitioner reference is McMillan (1992).

PROBLEMS
3.1 Suppose that you buy the S&R index for $1000, buy a 1000-strike put, and borrow
$980.39. Perform a payoff and profit calculation mimicking Table 3.1. Graph the resulting payoff and profit diagrams for the combined position.
3.2 Suppose that you short the S&R index for $1000 and sell a 1000-strike put. Construct a table mimicking Table 3.1 that summarizes the payoff and profit of this position.
Verify that your table matches Figure 3.5.
For the following problems assume the effective 6-month interest rate is 2%, the S&R 6month forward price is $1020, and use these premiums for S&R options with 6 months to expiration: Strike

Call

Put

$950
1000

$120.405
93.809

$51.777
74.201

1020
1050

84.470
71.802

84.470
101.214

1107

51.873

137.167

3.3 Suppose you buy the S&R index for $1000 and buy a 950-strike put. Construct payoff and profit diagrams for this position. Verify that you obtain the same payoff and profit diagram by investing $931.37 in zero-coupon bonds and buying a 950-strike call.
3.4 Suppose you short the S&R index for $1000 and buy a 950-strike call. Construct payoff and profit diagrams for this position. Verify that you obtain the same payoff and profit diagram by borrowing $931.37 and buying a 950-strike put.
3.5 Suppose you short the S&R index for $1000 and buy a 1050-strike call. Construct payoff and profit diagrams for this position. Verify that you obtain the same payoff and profit diagram by borrowing $1029.41 and buying a 1050-strike put.
3.6 Verify that you earn the same profit and payoff by (a) buying the S&R index for
$1000 and (b) buying a 950-strike S&R call, selling a 950-strike S&R put, and lending $931.37.

Problems

3.7 Verify that you earn the same profit and payoff by (a) shorting the S&R index for $1000 and (b) selling a 1050-strike S&R call, buying a 1050-strike put, and borrowing $1029.41.
3.8 Suppose the premium on a 6-month S&R call is $109.20 and the premium on a put with the same strike price is $60.18. What is the strike price?
3.9 Construct payoff and profit diagrams for the purchase of a 950-strike S&R call and sale of a 1000-strike S&R call. Verify that you obtain exactly the same profit diagram for the purchase of a 950-strike S&R put and sale of a 1000-strike S&R put. What is the difference in the payoff diagrams for the call and put spreads? Why is there a difference? 3.10 Construct payoff and profit diagrams for the purchase of a 1050-strike S&R call and sale of a 950-strike S&R call. Verify that you obtain exactly the same profit diagram for the purchase of a 1050-strike S&R put and sale of a 950-strike S&R put. What is the difference in the initial cost of these positions?
3.11 Suppose you invest in the S&R index for $1000, buy a 950-strike put, and sell a 1050strike call. Draw a profit diagram for this position. What is the net option premium?
If you wanted to construct a zero-cost collar keeping the put strike equal to $950, in what direction would you have to change the call strike?
3.12 Suppose you invest in the S&R index for $1000, buy a 950-strike put, and sell a 1107strike call. Draw a profit diagram for this position. How close is this to a zero-cost collar? 3.13 Draw profit diagrams for the following positions:
a. 1050-strike S&R straddle.
b. Written 950-strike S&R straddle.
c. Simultaneous purchase of a 1050-strike straddle and sale of a 950-strike S&R straddle. 3.14 Suppose you buy a 950-strike S&R call, sell a 1000-strike S&R call, sell a 950-strike
S&R put, and buy a 1000-strike S&R put.
a. Verify that there is no S&R price risk in this transaction.
b. What is the initial cost of the position?
c. What is the value of the position after 6 months?
d. Verify that the implicit interest rate in these cash flows is 2% over 6 months.
3.15 Compute profit diagrams for the following ratio spreads:
a. Buy 950-strike call, sell two 1050-strike calls.
b. Buy two 950-strike calls, sell three 1050-strike calls.
c. Consider buying n 950-strike calls and selling m 1050-strike calls so that the premium of the position is zero. Considering your analysis in (a) and (b), what can you say about n/m? What exact ratio gives you a zero premium?

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3.16 In the previous problem we saw that a ratio spread can have zero initial premium.
Can a bull spread or bear spread have zero initial premium? A butterfly spread? Why or why not?
3.17 Construct an asymmetric butterfly using the 950-, 1020-, and 1050-strike options.
How many of each option do you hold? Draw a profit diagram for the position.
3.18 Verify that the butterfly spread in Figure 3.14 can be duplicated by the following transactions (use the option prices in Table 3.4):
a. Buy 35 call, sell two 40 calls, buy 45 call.
b. Buy 35 put, sell two 40 puts, buy 45 put.
c. Buy stock, buy 35 put, sell two 40 calls, buy 45 call.
3.19 Here is a quote from an investment website about an investment strategy using options: One strategy investors are applying to the XYZ options is using “synthetic stock.” A synthetic stock is created when an investor simultaneously purchases a call option and sells a put option on the same stock. The end result is that the synthetic stock has the same value, in terms of capital gain potential, as the underlying stock itself. Provided the premiums on the options are the same, they cancel each other out so the transaction fees are a wash.
Suppose, to be concrete, that the premium on the call you buy is the same as the premium on the put you sell, and both have the same strikes and times to expiration.
a. What can you say about the strike price?
b. What term best describes the position you have created?
c. Suppose the options have a bid-ask spread. If you are creating a synthetic purchased stock and the net premium is zero inclusive of the bid-ask spread, where will the strike price be relative to the forward price?
d. If you create a synthetic short stock with zero premium inclusive of the bid-ask spread, where will the strike price be relative to the forward price?
e. Do you consider the “transaction fees” to really be “a wash”? Why or why not? 3.20 Construct a spreadsheet for which you can input up to five strike prices and quantities of put and call options bought or sold at those strikes, and which will automatically construct the total expiration payoff diagram for that position. Modify the spreadsheet to permit you to choose whether to graph a payoff or profit function.

4
B

Introduction to Risk
Management

usiness, like life, is inherently risky. Firms convert inputs such as labor, raw materials, and machines into goods and services. A firm is profitable if the cost of what it produces exceeds the cost of the inputs. Prices can change, however, and what appears to be a profitable activity today may not be profitable tomorrow. Many instruments are available that permit firms to hedge various risks, ranging from commodity prices to weather. A firm that actively uses derivatives and other techniques to alter its risk and protect its profitability is engaging in risk management. In this chapter we take a look at how derivatives—such as forwards, calls, and puts—are used in practice to manage risk.
We begin by examining two hypothetical firms—Golddiggers, a gold-mining firm, and Auric Enterprises, a manufacturer using gold as an input—to see what risks they face and to demonstrate the use of derivatives strategies to manage those risks. After looking at these examples, we will explore some reasons firms seek to manage risk in the first place.

4.1 BASIC RISK MANAGEMENT: THE PRODUCER’S
PERSPECTIVE
Golddiggers is a gold-mining firm planning to mine and sell 100,000 ounces of gold over the next year. For simplicity, we will assume that they sell all of the next year’s production precisely 1 year from today, receiving whatever the gold price is that day. The price of gold today is $405/oz. We will ignore production beyond the next year.
Obviously Golddiggers—like any producer—hopes that the gold price will rise over the next year. However, Golddiggers’s management computes estimated net income for a range of possible prices of gold in 1 year (Table 4.1). The net income calculation shows that
Golddiggers’s profit is affected by gold prices.
Should Golddiggers simply shut the mine if gold prices fall enough to make net income negative? The answer depends on the extent to which costs are fixed. The firm incurs the fixed cost whether or not it produces gold. Variable costs are incurred only if the mine operates. Thus, for any gold price above the variable cost of $50/oz, it will make sense to produce gold.1

1. Suppose that the gold price is $350/oz. If Golddiggers produces no gold, the firm loses its fixed cost, which we assume to be $33,000,000. With production of 100,000 ounces, this is an average cost of $330/oz.

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TABLE 4.1

Golddiggers’s estimated net income 1 year from today, unhedged. Gold Price in One Year

Fixed
Cost

Variable
Cost

Net
Income

$350
$400
$450
$500

−$330
−$330
−$330
−$330

−$50
−$50
−$50
−$50

−$30
$20
$70
$120

TABLE 4.2

Golddiggers’s net income 1 year from today, hedged with a forward sale of gold.

Gold Price in One Year

Fixed
Cost

Variable
Cost

Profit on
Short Forward

Net Income on Hedged Position

$350
$400
$450
$500

−$330
−$330
−$330
−$330

−$50
−$50
−$50
−$50

$70
$20
−$30
−$80

$40
$40
$40
$40

Hedging with a Forward Contract
Golddiggers can lock in a price for gold in 1 year by entering into a short forward contract, agreeing today to sell its gold for delivery in 1 year. Suppose that gold to be delivered in 1 year can be sold today for $420/oz and that Golddiggers agrees to sell forward all of its gold production. We will assume in all examples that the forward contract settles financially. As noted earlier, the payoff to a forward is the same with physical or financial settlement.
Profit calculations when Golddiggers is hedged are summarized in Table 4.2. This table adds the profit on the forward contract to net income from Table 4.1. Figure 4.1 contains three curves showing the following:
.

Unhedged profit: Since cost is $380/oz, the line labeled “unhedged seller” shows zero profit at $380, a loss at lower prices, and profit at higher prices. For example, at

If Golddiggers produces gold, the firm has a variable cost of $50/oz in addition to the fixed cost. If the price is $350/oz, the firm loses $350 − ($330 + $50) = −$30/oz. It is better to lose $30/oz instead of $330/oz, so
Golddiggers will produce even when they have negative net income. If the gold price were to fall below the variable cost of $50/oz, then it would make sense for Golddiggers to stop producing. For convenience we will refer to $330/oz as the fixed cost, but note that this presumes a production level, in this case, 100,000 ounces. 4.1 Basic Risk Management: The Producer’s Perspective

FIGURE 4.1

Profit ($)

Producer profit in 1 year, assuming hedging with a short forward contract at a forward price of $420/oz.

200

Unhedged seller
Short gold forward
Hedged seller

150
100
50

$40

0
–50
–100
–150
250

$380
300

$420

350
400
450
500
Gold Price in 1 Year ($)

550

$420, profit is $40/oz. Since it has gold in the ground, Golddiggers has a long position in gold.
.

.

Profit on the short forward position: The “short gold forward” line represents the profit from going short the gold forward contract at a forward price of $420/oz. We profit from locking in the price if prices are lower than $420, and we lose if prices are higher. Hedged profit: The line labeled “hedged seller” is the sum of the other two lines, adding them vertically at every gold price. It is flat at $40/oz, as we would expect from Table 4.2. A quick way to add the lines together is to notice that the “unhedged seller” graph has a positive slope of 1, and the “short gold forward” graph has a slope of −1. Added together vertically, the two graphs will have a slope of 0, so the only question is the height of the line. A profit calculation at a single point tells us that it must be at $40/oz.

Insurance: Guaranteeing a Minimum Price with a Put Option
A possible objection to hedging with a forward contract is that if gold prices do rise,
Golddiggers will still receive only $420/oz; there is no prospect for greater profit. Gold insurance—i.e., a put option—provides a way to have higher profits at high gold prices while still being protected against low prices. Suppose that the market price for a 420-strike put is $8.77/oz.2 This put provides a floor on the price.

2. This uses the Black-Scholes formula for the put price with inputs S = 420, K = 420, r = 4.879%, σ = 5.5%, δ = 4.879%, and t = 1 (year).

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TABLE 4.3

Golddiggers’s net income 1 year from today, hedged with a 420-strike put option.

Gold Price in One Year

Fixed
Cost

Variable
Cost

Profit on
Put Option

Net
Income

$350
$400
$450
$500

−$330
−$330
−$330
−$330

−$50
−$50
−$50
−$50

$60.79
$10.79
−$9.21
−$9.21

$30.79
$30.79
$60.79
$110.79

FIGURE 4.2

Profit ($)

Comparison of unhedged position, 420-strike put option, and unhedged position plus 420-strike put.

200

Unhedged seller
Purchased put
Hedged seller

150
100
50

$30.79

0
–50
–100
–150
250

$380
300

350

$420
400

450

500

550

Gold Price in 1 Year ($)

Since the put premium is paid 1 year prior to the option payoff, we must take into account interest cost when we compute profit in 1 year. The future value of the premium is $8.77 × 1.05 = $9.21. As with the forward contract, we assume financial settlement, although physical settlement would yield the same net income.
Table 4.3 shows the result of buying this put. If the price is less than $420, the put is exercised and Golddiggers sells gold for $420/oz. less the cost of the put. This gives net income of $30.79. If the price is greater than $420, Golddiggers sells gold at the market price. The insurance strategy—buying the put—performs better than shorting the forward if the price of gold in 1 year is more than $429.21. Otherwise the short forward outperforms insurance. Figure 4.2 shows the unhedged position, profit from the put by itself, and the result of hedging with the put.

4.1 Basic Risk Management: The Producer’s Perspective

FIGURE 4.3

Profit ($)

Comparison of payoffs for
Golddiggers hedged with a forward contract and hedged with a put option.

200

Seller hedged with forward
Seller hedged with put

150
100
50
0

$30.79

–50
–100
–150

$420
300

350
400
450
Gold Price in 1 Year ($)

500

What this analysis does not address is the probability that the gold price in 1 year will be in different regions; that is, how likely is it that the gold price will exceed $429.21? The price of the put option implicitly contains information about the likelihood that the gold price will exceed $420, and by how much. The probability distribution of the gold price is a key factor determining the pricing of the put. We will see in later chapters how the distribution affects the put price and how to use information about the probability distribution to help us assess risk.
Figure 4.3 compares the profit from the two protective strategies we have examined:
Selling a forward contract and buying a put. As you would expect, neither strategy is clearly preferable; rather, there are trade-offs, with each contract outperforming the other for some range of prices.
The fact that no hedging strategy always outperforms the other will be true of all fairly priced strategies. In practice, considerations such as transaction costs and market views are likely to govern the choice of a strategy. The box on page 94 discusses oil hedging by
Mexico, and the attendant political risks of such a strategy.

Insuring by Selling a Call
With the sale of a call, Golddiggers receives a premium, which reduces losses, but the written call limits possible profits. One can debate whether this really constitutes insurance, but our goal is to see how the sale of a call affects the potential profit and loss for Golddiggers.
Suppose that instead of buying a put, Golddiggers sells a 420-strike call and receives an $8.77 premium. Golddiggers in this case would be said to have sold a cap.
Figure 4.4 shows the profit to this strategy. If we compute the actual profit 1 year from today, we see that if the gold price in 1 year exceeds $420, Golddiggers will show profits of
$420 + $9.21 − $380 = $49.21

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BOX

4.1: Mexico’s Oil Hedge

M

exico is one of the world’s largest oil producers and its economic well-being is affected by the price of oil. Mexico has been unusual among governments in hedging commodity prices (Blas and Thomson, 2008):
Mexico is taking steps to protect itself from the oil price remaining below $70 a barrel in the clearest sign yet of the concerns of producer countries at the impact of the global economic slowdown on their revenues.
The world’s sixth biggest oil producer hedged almost all of next year’s oil exports at prices ranging from $70 to $100 at a cost of about
$1.5bn through derivatives contracts, according to bankers familiar with the deal.
The cover is far higher than the country— which relies on oil for up to 40 per cent of government revenue—usually seeks. Last year,
Mexico hedged 20–30 per cent of its exports.
Mexico’s finance ministry declined to comment but said in its latest quarterly report that its oil income stabilisation fund spent about
$1.5bn on “financial investments, as part of the measures taken for risk management.”
Oil prices hit an all-time high of $147.27 a barrel in July but have since fallen to less than
$65 as the global economy cools.
Global investment banks were counterparties for Mexico’s trades, which earned it $5 billion. In
2009 Mexico also hedged its 2010 net exposure

of 230 million barrels at an average price of
$57/barrel (Williams, 2009).
Governments that hedge commodity price exposure face political risk (Blas, 2009):
“Very few government officials of commodity-exporting countries are secure enough in their jobs to place a bet on the future level of commodity prices,” wrote the oil economist Philip K. Verleger . . .
Such political fears were plainly exposed when the state of Alaska discussed whether to hedge its oil income in 2002. The state’s department of revenues warned that policymakers would “be reluctant to take the political risks” of a hedging programme.
“If a programme succeeded, it is unlikely the policymakers who took the initiative to create the programme would be rewarded with public congratulations,” the report said. “On the other hand, if the state lost significant sums . . . the conventional wisdom is that public criticism would be harsh,” it concluded.
The political risk also explains why the few countries or states that have launched sovereign oil hedging including Mexico and, in the past, Ecuador, Colombia, Algeria, Texas and
Louisiana, have made use of put options—an insurance policy that sets a price floor without giving away any potential upside—rather than futures that fix a price.
Based on Blas & Thomas 2008, Williams 2009

That is, Golddiggers sells gold for $420 (since the written call is exercised by the holder), receives the future value of the premium, and has a cost of $380. If the price of gold is less than $420, Golddiggers will make
Pg + $9.21 − $380
On the downside, Golddiggers has exposure to gold but keeps the option premium.
By writing the call, Golddiggers keeps the $8.77 call premium and 1 year later makes
$9.21 more than an unhedged gold seller. On the other hand, if the gold price exceeds $420, the call is exercised and the price Golddiggers receives is capped at $420. Thus, for gold prices above $429.21, an unhedged strategy has a higher payoff than that of writing a 420-

4.1 Basic Risk Management: The Producer’s Perspective

FIGURE 4.4

Profit ($)

Comparison of Golddiggers hedging with sale of 420strike call versus unhedged.

200
150

Unhedged seller
Written call
Seller hedged with written call

100
$49.21
50
$9.21
0
–50
–100
–150
250

$420
300

350

400

450

500

550

Gold Price in 1 Year ($)

strike call. Also, for prices below $410.79, being fully hedged is preferable to having sold the call.

Adjusting the Amount of Insurance
Consider again Golddiggers’s strategy of obtaining insurance against a price decline by purchasing a put option. A common objection to the purchase of insurance is that it is expensive. Insurance has a premium because it eliminates the risk of a large loss, while allowing a profit if prices increase. The cost of insurance reflects this asymmetry.
There are at least two ways to reduce the cost of insurance:
.

Reduce the insured amount by lowering the strike price of the put option.

.

Sell some of the gain.

Both of these strategies reduce the asymmetry between gains and losses, and hence lower the cost of insurance. The first strategy, lowering the strike price, permits some additional loss, while the second, selling some of the gain, puts a cap on the potential gain.
Reducing the strike price lowers the amount of insurance; therefore, the put option will have a lower premium. Figure 4.5 compares profit diagrams for Golddiggers’s hedging using put options with strikes of $400 (premium = $2.21), $420 (premium = $8.77), and
$440 (premium = $21.54). The 400-strike, low-premium option yields the highest profit if insurance is not needed (the price is high) and the lowest profit if insurance is needed (the price is low). The 440-strike, high-premium option yields the lowest profit if insurance is not needed, and the highest profit if insurance is needed.
The manager’s view of the market and willingness to absorb risk will undoubtedly influence the choice among these alternatives. Managers optimistic about the price of gold will opt for low-strike-price puts, whereas pessimistic managers will more likely choose high-strike puts. While corporations per se may not be risk-averse, managers may be. Also,

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FIGURE 4.5

Profit ($)

Comparison of profit for
Golddiggers using three different put strikes.

160
140

Hedged with 420-strike put
Hedged with 400-strike put
Hedged with 440-strike put

120
100
80
60
40
20
0
250

300

350

400

450

500

550

Gold Price in 1 Year ($)

some managers may perceive losses to be costly in terms of the public’s perception of the firm or the boss’s perception of them.
This problem of choosing the appropriate strike price is not unique to corporate risk management. Safe drivers and more careful homeowners often reduce premiums by purchasing auto and homeowner’s insurance with larger deductibles. This reflects their proprietary view of the likelihood that the insurance will be used. One important difference between gold insurance and property insurance, however, is that poor drivers would like smaller deductibles for their auto insurance; this differential demand by the quality of the insured is called adverse selection and is reflected in the premiums for different deductibles.
A driver known to be good would face a lower premium for any deductible than a driver known to be bad. With gold, however, the price of the put is independent of who is doing the buying.3

4.2 BASIC RISK MANAGEMENT: THE BUYER’S PERSPECTIVE
Auric Enterprises is a manufacturer of widgets, a product that uses gold as an input. We will suppose for simplicity that the price of gold is the only uncertainty Auric faces. In particular, we assume that
.

Auric sells each widget for a fixed price of $800, a price known in advance.

.

The fixed cost per widget is $340.

3. You might think that a dealer would charge a higher price for a purchased option if the dealer knew that an option buyer had superior information about the market for gold. However, in general the dealer will quickly hedge the risk from the option and therefore has less concern than an ordinary investor about future movements in the price of gold.

4.2 Basic Risk Management: The Buyer’s Perspective

TABLE 4.4
Revenue
per Widget
$800
$800
$800
$800

Auric estimated net income, unhedged, 1 year from today.

Gold Price in 1 Year

Fixed
Cost

Variable
Cost

Net
Income

$350
$400
$450
$500

$340
$340
$340
$340

$0
$0
$0
$0

$110
$60
$10
−$40

.

The manufacture of each widget requires 1 oz of gold as an input.

.

The nongold variable cost per widget is zero.

.

The quantity of widgets to be sold is known in advance.

Because Auric makes a greater profit if the price of gold falls, Auric’s gold position is implicitly short. As with Golddiggers, we will examine various risk-management strategies for Auric. The pro forma net income calculation for Auric is shown in Table 4.4.

Hedging with a Forward Contract
The forward price is $420 as before. Auric can lock in a profit by entering into a long forward contract. Auric thereby guarantees a profit of
Profit = $800 − $340 − $420 = $40
Note that whereas Golddiggers was selling in the forward market, Auric is buying in the forward market. Thus, Golddiggers and Auric are natural counterparties in an economic sense. In practice, they need not be direct counterparties since they can enter into forward contracts through dealers or on exchanges. But in an economic sense, one firm’s desire to sell forward has a counterpart in the other’s desire to buy forward.
Figure 4.6 compares the profit diagrams for the unhedged buyer and a long forward position in gold. It also shows the profit for the hedged buyer, which is generated by summing up the forward position and the unhedged payoff. We see graphically that the buyer can lock in a profit of $40/oz.

Insurance: Guaranteeing a Maximum Price with a Call Option
Rather than lock in a price unconditionally, Auric might like to pay $420/oz if the gold price is greater than $420/oz but pay the market price if it is less. Auric can accomplish this by buying a call option. As a future buyer, Auric is naturally short; hence, a call is insurance.
Suppose the call has a premium of $8.77/oz (recall that this is the same as the premium on the put with the same strike price). The future value of the premium is $8.77 × 1.05 = $9.21.
If Auric buys the insurance contract, net income on the hedged position will be as shown in Table 4.5. If the price is less than $420, the call is worthless at expiration and

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FIGURE 4.6

Profit ($)

Profit diagrams for unhedged buyer, long forward, and buyer hedged with long forward. Unhedged buyer
Long gold forward
Hedged buyer

200
150
100
50

$40

0
–50
–100
–150

TABLE 4.5
Gold Price in 1 Year
$350
$400
$450
$500

$420
300

$460

350
400
450
Gold Price in 1 Year ($)

500

Auric net income 1 year from today, hedged with 420-strike call option.

Unhedged Net Income from Table 4.4

Profit on
Call Option

Net
Income

$110
$60
$10
−$40

−$9.21
−$9.21
$20.79
$70.79

$100.79
$50.79
$30.79
$30.79

Auric buys gold at the market price. If the price is greater than $420, the call is exercised and Auric buys gold for $420/oz, less the cost of the call. This gives a profit of $30.79.
If the price of gold in 1 year is less than $410.79, insuring the price by buying the call performs better than locking in a price of $420. At low prices, the option permits us to take advantage of lower gold prices. If the price of gold in 1 year is greater than $410.79, insuring the price by buying the call performs worse than locking in a price of $420 since we have paid the call premium.
Figure 4.7 shows the profit from the call by itself, along with the results of hedging with the call. As before, the graph does not show the probability that the gold price in 1 year will be in different regions; hence, we cannot evaluate the likelihood of different outcomes.

4.3 Why Do Firms Manage Risk?

FIGURE 4.7
Comparison of profit for unhedged gold buyer, gold buyer hedged with call, and stand-alone call.

Profit ($)
Unhedged buyer
Purchased call
Buyer hedged with call

200
150
100
50

$30.79

0
–$9.21
–50
$420 $460
–100
250

300

350

400

450

500

550

Gold Price in 1 Year ($)

4.3 WHY DO FIRMS MANAGE RISK?
The Golddiggers and Auric examples illustrate how the two companies can use forwards, calls, and puts to reduce losses in case of an adverse gold price move, essentially insuring their future cash flows. Why would a firm use these strategies?
In Chapter 1 we listed four reasons that firms might use derivatives: to hedge, to speculate, to reduce transaction costs, and to effect regulatory arbitrage. In practice, more than one of these considerations may be important. We have already discussed the fact that market views—for example, opinions about the future price of gold—can affect the choice of a hedging strategy. Thus, the choice of a hedging strategy can have a speculative component. Managers often cite the accounting treatment of a transaction as important, and transaction costs are obviously a consideration.
In this section we discuss why firms might hedge, ignoring speculation, transaction costs, and regulation (but we do consider taxes). It seems obvious that managers would want to reduce risk. However, in a world with fairly priced derivatives, no transaction costs, and no other market imperfections such as taxes, derivatives change the distribution of cash flows but do not increase the value of cash flows. Moreover, large publicly held firms are owned by diverse shareholders. These shareholders can, in theory, configure their own portfolios to bear risk optimally, suiting their own taste. In order to hedge, the firm must pay commissions and bid-ask spreads, and bear counterparty credit risk. Why incur these costs? There are several reasons that firms might seek to manage risk. Before discussing them, let’s think about what derivatives accomplish. To be concrete, suppose that Golddiggers sells gold forward at $420/oz. As we saw, this guarantees a net income of $40/oz.
When hedged with the forward, Golddiggers will have a profit of $40 whatever the price in 1 year. In effect, the value of the reduced profits, should the gold price rise, subsidizes

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TABLE 4.6

Calculation of after-tax net income in states where the output price is $9.00 and $11.20. Expected after-tax income is (0.5 ×−$1) + (0.5 × $0.72) = −$0.14.

Price = $9
(1)
(2)
(3)

Pre-tax operating income
Taxable income
Tax @ 40% [0.4 × (2)]
After-tax income [(2) − (3)]

Price = $11.20

−$1
$0
0
−$1

$1.20
$1.20
$0.48
$0.72

the payment to Golddiggers should the gold price fall. If we use the term “state” to denote a particular gold price in 1 year, we can describe the hedging strategy as shifting dollars from more profitable states (when gold prices are high) to less profitable states (when gold prices are low).
This shifting of dollars from high gold price states to low gold price states will have value for the firm if the firm values the dollar more in a low gold price state than in a high gold price state. Why might a firm value a dollar differently in different states?

An Example Where Hedging Adds Value
Consider a firm that produces one unit per year of a good costing $10. Immediately after production, the firm receives a payment of either $11.20 or $9, with 50% probability. Thus, the firm has either a $1.20 profit or a $1 loss. On a pre-tax basis, the firm has an expected profit of
[0.5 × ($9 − $10)] + [0.5 × ($11.20 − $10)] = $0.10
However, on an after-tax basis, the firm could have an expected loss.
For example, suppose that when the firm reports a profit, 40% of the profit is taxed, but when the firm reports a loss, it pays no taxes and receives no tax refund. Table 4.6 computes expected after-tax profit under these circumstances. The taxation of profits converts an expected $0.10 pre-tax gain into an after-tax $0.14 loss.4 Because of taxes, the firm values a dollar of profit at $0.60 ($0.40 goes to the government), but values a dollar of loss at $1.
In this situation, it is desirable for the firm to trade pre-tax profits for pre-tax losses.
Suppose that there is a forward market for the firm’s output, and that the forward price is $10.10. If the firm sells forward, profit is computed as in Table 4.7. Instead of an expected loss of $0.14, we obtain a certain profit of $0.06. Hedging with a forward transfers net income from a less-valued to a more highly valued state, raising the expected value of cash flows.
Figure 4.8 depicts how the nondeductibility of losses affects after-tax cash flows.
First, observe that after-tax profit (line ACB) is a concave function of the output price. (A

4. Problem 4.15 asks you to compute profit when losses are deductible.

4.3 Why Do Firms Manage Risk?

TABLE 4.7

Calculation of hedged after-tax net income in states where the output price is $9.00 and $11.20. Expected after-tax income is $0.06.

Price = $9
(1)
(2)
(3)
(4)

−$1.00
$1.10
$0.10
$0.04
$0.06

Pre-tax operating income
Income from short forward
Taxable income [(1) + (2)]
Tax @ 40% [0.4 × (3)]
After-tax income [(3) − (4)]

FIGURE 4.8
After-tax profit as a function of pre-tax profit. Point A is profit at a price of $9
(−$1.00), point B is profit at $11.20 ($0.72), point C is profit at $10.10 ($0.06), and point D is expected profit if price is $9 or $11.20 with one-half probability,
−$0.14.

Price = $11.20
$1.20
−$1.10
$0.10
$0.04
$0.06

After-Tax Profit

Slope = 1 – Tax rate
B

$0.72
$0.06
0

–$1

$9

A

C
$11.20
$10.10
Output price
– $0.14

D

Slope = 1

concave function is one shaped like the cross section of an upside-down bowl.) When profits are concave, the expected value of profits is increased by reducing uncertainty. We can see this in the graph. If the price is certain to be $10.10, then profit will be given by point C.
However, if price can be either $9 or $11.20, expected profit is at point D, on the line ADB at the expected price of $10.10. Because ACB is concave, point D lies below point C, and hedging increases expected profits.5

5. This is an illustration of Jensen’s inequality, which is discussed in Appendix C; we will encounter it often in this book.

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Some of the hedging rationales we discuss hinge on there being concave profits, so that value is increased by reducing uncertainty.

Reasons to Hedge
There are in fact a number of reasons why losses might be more harmful than profits are beneficial. We now discuss some of those reasons.6
Taxes. The previous example illustrating the effect of taxes was oversimplified in assuming that losses are completely untaxed, but it is typically the case that governments tax profits but do not give full credits for losses. Tax systems usually permit a loss to be offset against a profit from a different year. However, in present value terms, the loss will have a lower effective tax rate than that applied to profits, which still generates a motive to hedge.
There are other aspects of the tax code that can encourage firms to shift income using derivatives; such uses may or may not appear to be hedging and may or may not be approved of by tax authorities. Tax rules that may entice firms to use derivatives include the separate taxation of capital and ordinary income (derivatives can be used to convert one form of income to another), capital gains taxation (derivatives can be used to defer taxation of capital gains income, as with collars), and differential taxation across countries (derivatives can be used to shift income from one country to another).
Bankruptcy and Distress Costs. An unusually large loss can threaten the survival of a firm, which may then be unable to meet fixed obligations, such as debt payments and wages.
If a firm appears to be in distress, customers may be less willing to purchase its goods. This can be a problem for companies that promise future goods or services, such as airline seats or warranties.
Actual or threatened bankruptcy can be costly; a dollar of loss can cost the company more than a dollar. As with taxes, this is a reason for firms to enter derivatives contracts that transfer income from profit states to loss states, thereby reducing the probability of bankruptcy or distress.
Costly External Financing. Even if a loss is not large enough to threaten the survival of a firm, the firm must pay for the loss, either by using cash reserves or by raising funds externally (for example, by borrowing or issuing new securities).
Raising funds externally can be costly. There are explicit costs, such as bank and underwriting fees. There can also be implicit costs. If you borrow money, the lender may worry that you need to borrow because you are in decline, which increases the probability that you will not repay the loan. The lender will therefore raise the interest rate on the loan.
For the same reason, you may be able to issue equity only at a reduced price.
At the same time, cash reserves are valuable because they reduce a firm’s need to raise funds externally in the future. So if the firm uses cash to pay for a loss, the reduction in cash increases the probability that the firm will need costly external financing in the future.
The fact that external financing is costly can even lead the firm to forgo investment projects it would have taken had cash been available to use for financing.

6. The following are discussed in Smith and Stulz (1985) and Froot et al. (1994).

4.3 Why Do Firms Manage Risk?

Thus, however the firm pays for the loss, a dollar of loss may actually cost the firm more than a dollar. Hedging can safeguard cash reserves and reduce the probability of costly external financing.
Increase Debt Capacity. Because of the deductibility of interest expense for tax purposes, firms may find debt to be a tax-advantaged way to raise funds.7 However, lenders, fearful of bankruptcy, may be unwilling to lend to firms with risky cash flows. The amount that a firm can borrow is its debt capacity.
A firm that credibly reduces the riskiness of its cash flows should be able to borrow more, since for any given level of debt, bankruptcy is less likely. Such a firm is said to have raised its debt capacity. To the extent debt has a tax advantage, such a firm will also be more valuable. Managerial Risk Aversion. While large, public firms are owned by well-diversified investors, firm managers are typically not well-diversified. Salary, bonus, and compensation options are all tied to the performance of the firm.
An individual who is unwilling to take a fair bet (i.e., one with an expected payoff equal to the money at stake) is said to be risk-averse. Risk-averse persons are harmed by a dollar of loss more than they are helped by a dollar of gain. Thus, they benefit from reducing uncertainty. The effect is analogous to that shown in Figure 4.8.
If managers are risk-averse and have wealth that is tied to the company, we might expect that they will try to reduce uncertainty. However, matters are not this simple:
Managers are often compensated in ways that encourage them to take more risk. For example, options given to managers as compensation, which we discuss in Chapter 16, are more valuable, other things equal, when the firm’s stock price is riskier. Thus, a manager’s risk aversion may be offset by compensation that is more valuable if the firm is riskier.
Nonfinancial Risk Management. Firms make risk-management decisions when they organize and design a business. For example, suppose you plan to sell widgets in Europe.
You can construct a plant in the United States and export to Europe, or you can construct the plant in Europe, in which case costs of construction, labor, interest rates, and other inputs will be denominated in the same currency as the widgets you sell. Exchange rate hedging, to take one example, would be unnecessary.
Of course, if you build in a foreign country, you will encounter the costs of doing business abroad, including dealing with different tax codes and regulatory regimes.
Risk can also be affected by such decisions as leasing versus buying equipment, which determines the extent to which costs are fixed. Firms can choose flexible production technologies that may be more expensive at the outset but which can be reconfigured at low cost. Risk is also affected by the decision to enter a particular line of business in the first place. Firms making computer mice and keyboards, for example, have to consider the possibility of lawsuits for repetitive stress injuries.
The point is that risk management is not a simple matter of hedging or not hedging using financial derivatives, but rather a series of decisions that start when the business is first conceived.

7. For a discussion of this issue, see Brealey et al. (2011, ch. 17).

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Reasons Not to Hedge
There are also reasons why firms might elect not to hedge:
.

.

.

.

Transacting in derivatives entails paying transaction costs, such as commissions and the bid-ask spread.
The firm must assess costs and benefits of a given strategy; this can require costly expertise. The firm must monitor transactions and have managerial controls in place to prevent unauthorized trading.
The firm must be prepared for tax and accounting consequences of their transactions.
In particular, this may complicate reporting.

Thus, while there are reasons to hedge, there are also costs. When thinking about costs and benefits, keep in mind that some of what firms do could be called risk management but may not obviously involve derivatives. For example, suppose Auric enters into a 2-year agreement with a supplier to buy gold at a fixed price. Will management think of this as a derivative? In fact this is a derivative under current accounting standards (it is a swap, which we discuss in Chapter 8), but it is exempt from derivatives accounting.8 Finally, firms can face collateral requirements (the need to post extra cash with their counterparty) if their derivatives position loses money.
Periodically, firms appears in the news for their hedging successes or, more commonly, failures. The boxes on pages 105 and 106 illustrate the challenges of managing corporate hedging programs.

Empirical Evidence on Hedging
We know surprisingly little about the risk-management practice and derivatives use of firms in real life. It is difficult to tell, from publicly available information, the extent to which firms use derivatives. Beginning in 2000, Statement of Financial Accounting Standards (SFAS)
133 required firms to recognize derivatives as assets or liabilities on the balance sheet, to measure them at fair value, and to report changes in their market value.9 This reporting does not necessarily reveal a firm’s hedging position (forward contracts have zero value, for example). Prior to 2000, firms had to report notional exposure; hence, much existing evidence relies on data from the 1990s.
Research tries to address two questions: How much do firms use derivatives and why?
Financial firms—commercial banks, investment banks, broker-dealers, and other financial institutions—transact in derivatives frequently. Risks are identifiable, and regulators encourage risk management. The more open question is the extent to which nonfinancial firms use derivatives. We can summarize research findings as follows:
.

Roughly half of nonfinancial firms report using derivatives, with usage greater among large firms (Bodnar et al., 1998; Bartram et al., 2004).

8. Current derivatives accounting rules contain a “normal purchases and sales” exemption. Firms need not use derivatives accounting for forward contracts with physical delivery, for quantities likely to be used or sold over a reasonable period in the normal course of business.
9. See Gastineau et al. (2001) for a discussion of SFAS 133 and previous accounting rules.

4.3 Why Do Firms Manage Risk?

BOX

105

4.2: Ford: A Hedge Too Far

F

ord Motor Co. stunned investors in January
2002 when it announced a $1 billion write-off on stockpiles of palladium, a precious metal Ford used in catalytic converters (devices that reduce polluting emissions from cars and trucks). Ironically, Ford sustained the loss while attempting to actively manage palladium risk.
According to the Wall Street Journal (see Gregory L. White, “A Mismanaged Palladium Stockpile Was Catalyst for Ford’s Write-Off,” February
6, 2002, p. A1), Ford in the late 1980s had begun to use palladium as a replacement for platinum.
Palladium prices were steady until 1997, when
Russia, a major supplier with a large stockpile of palladium, withheld supply from the market. Prices more than doubled to $350/oz at a time when Ford was planning to increase its use of the metal. By early 2000, prices had doubled again, to $700. While GM had begun work several years earlier to reduce reliance on palladium, Ford continued to rely heavily on the metal. .

.

.

In 2000, Ford management agreed to allow the purchasing staff to stockpile palladium. The purchasing staff evidently did not communicate with Ford’s treasury department, which had hedging experience. Thus, for example, Ford did not buy puts to protect against a drop in palladium prices. The purchasing staff also did not communicate with Ford’s research department, which was working to reduce reliance on palladium.
Ford continued to buy palladium in 2001 as prices exceeded $1000. However, by the middle of the year, palladium prices had fallen to $350.
By the end of 2001, Ford had developed technology that would eventually reduce the need for palladium by 50%. The year-end price of palladium was $440/oz.
As a result of this experience, “Ford has instituted new procedures to ensure that treasurydepartment staffers with experience in hedging are involved in any major commodities purchases in the future, [Ford Chief Financial Officer Martin] Inglis says.”

Among firms that do use derivatives, less than 25% of perceived risk is hedged, with firms likelier to hedge short-term risks (Bodnar et al., 1998).
Firms with more investment opportunities are likelier to hedge (G´ czy et al., 1997). e Firms that use derivatives may have a higher market value (Allayannis and Weston,
2001; Allayannis et al., 2004; Bartram et al., 2004; Carter et al., 2006; MacKay and
Moeller, 2007), more leverage (Graham and Rogers, 2002; Haushalter, 2000), and lower interest costs (Campello et al., 2010).10 Jin and Jorion (2006), however, find that hedging firms do not have a higher market value.

Guay and Kothari (2003) obtain many similar results but conclude that for most firms, derivatives use is of minor economic significance. In their sample of large firms, slightly more than half report derivatives usage. Among derivatives users, the authors estimate that the median firm hedges only about 3% to 6% of exposure to interest rates and exchange rates.
Bartram et al. (2009) examine annual reports from over 7000 international companies to link

10. Graham and Smith (1999) find that after-tax profits are concave for a majority of firms, as in Figure 4.8.
However, Graham and Rogers (2002) are unable to find a link between hedging and tax-induced concavity.

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BOX

4.3: Hedging Jet Fuel: Southwest Airlines

Southwest Airlines is well known for systematically hedging the cost of jet fuel. In the 1990s, fuel on average accounted for 10–15% of Southwest’s operating costs (Carter et al., 2004). In recent years, fuel costs have been as much as a quarter of operating expenses. Since fuel costs have risen over the last decade, Southwest has benefited from hedging. In its 2005 quarterly financial reports, for example, Southwest reported savings from its hedge program of $155, $196, $295, and
$258 million, against net income of $75, $159,
$227, and $86 million. Clearly, fuel hedging was important for Southwest’s profitability.
Southwest uses cross-hedges to hedge jet fuel, and hedges a significant portion of its future projected fuel needs. Here is what Southwest said about its hedging program in its 3rd quarter 2006 financial statement:
The Company endeavors to acquire jet fuel at the lowest possible cost. Because jet fuel is not traded on an organized futures exchange, liquidity for hedging is limited. However, the
Company has found commodities for hedging of jet fuel costs, primarily crude oil, and refined products such as heating oil and unleaded gasoline. . . . The Company currently has a mixture of purchased call options, collar structures, and fixed price swap agreements in place to protect

against over 85 percent of its remaining 2006 total anticipated jet fuel requirements . . .
Southwest also stated that it had hedged 85% of expected fuel purchases for 2007, over 43% for
2008, with smaller positions as far out as 2012.
A company that hedges the price of inputs can lose if prices fall. From the New York Times,
October 17, 2008:
Southwest Airlines said Thursday that it lost $120 million in the third quarter, its first quarterly loss in more than 17 years, because of a noncash charge to write down the declining value of its hedging contracts.
The airline took a one-time charge of $247 million, reflecting the decline in oil prices, which hit record high levels in July. The last time Southwest lost money was in the first quarter of 1991.
Without the charge, Southwest said it would have earned $69 million. . . .
“It’s like you can’t win,” said Betsy J. Snyder, an industry analyst with Standard & Poor’s
Ratings Services. “People bother you when you don’t hedge, and when you do, and prices go down, you get hit.”
Excerpts from New York Times Oct 17, 2008 and Southwest’s 2006 Financial Statement.

firm characteristics and derivatives usage. They argue that it is hard to test specific hedging theories because derivatives use is only one of many firm decisions that affect financial risk. Because data are hard to obtain, some studies have focused on particular industries and even firms. Tufano (1996), Petersen and Thiagarajan (2000), and Brown et al. (2003) have examined hedging behavior by gold-mining firms. Using a uniquely detailed data set,
Tufano found that most gold firms use some derivatives, with the median firm in his sample
(North American firms) selling forward about 25% of 3-year production. Fifteen percent of firms used no derivatives. Brown et al. found substantial variation over time in the amount of hedging by gold firms. Firms tended to increase hedging as the price rose, and managers reported that they adjusted hedges based on their views about gold prices.
The currency-hedging operations of a U.S.-based manufacturing firm are examined in detail by Brown (2001), who finds that foreign exchange hedging is an integral part of

4.4 Golddiggers Revisited

TABLE 4.8

Call and put premiums for gold options.

Strike
Price

Put
Premium

Call
Premium

21.54
8.77
2.21

2.49
8.77
21.26

440
420
400

Note: These prices are computed using the Black formula for options on futures, with a futures price of $420, effective annual interest rate of 5%, volatility of 5.5%, and 1 year to expiration.

firm operations, but the company has no clear rationale for hedging. For example, Brown reports one manager saying, “We do not take speculative positions, but the extent we are hedged depends on our views.” Bartram (2008) investigates in depth one large German firm and also finds that hedging reduces currency exposure. Faulkender (2005) finds consistent evidence for interest-rate hedging in the chemical industry. These firms increase exposure to short-term interest rates as the yield curve becomes more upward-sloping,11 but correlations between cash flows and interest rates do not affect behavior.
The varied evidence suggests that some use of derivatives is common, especially at large firms, but the evidence is weak that economic theories explain hedging.

4.4 GOLDDIGGERS REVISITED
We have looked at simple hedging and insurance strategies for buyers and sellers. We now examine some additional strategies that permit tailoring the amount and cost of insurance.
For simplicity we will focus primarily on Golddiggers; however, in every case there are analogous strategies for Auric.
Table 4.8 lists premiums for three calls and puts on gold with 1 year to expiration and three different strikes. The examples use these values.

Selling the Gain: Collars
As discussed earlier, we can reduce the cost of insurance by reducing potential profit, i.e., by selling our right to profit from high gold prices. We can do this by selling a call. If the gold price is above the strike on the call, we are contractually obligated to sell at the strike.
This caps our profits, in exchange for an initial premium payment.
A 420–440 Collar. Suppose that Golddiggers buys a 420-strike put option for $8.77 and sells a 440-strike call option for a premium of $2.49. If the price of gold in 1 year is $450/oz, the call owner will exercise and Golddiggers is obligated to sell gold at the strike price

11. An upward-sloping yield curve means that long-term bond yields are greater than short-term bond yields. This appears to make short-term financing less expensive. However, we will see in Chapters 7 and
8 that if a company hedges all of its future short-term financing costs, long-term and short-term financing will cost the same.

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FIGURE 4.9

Profit ($)

Net profit at expiration resulting from buying a 420strike put with premium of
$8.77 and selling a 440strike call with premium of
$2.49. The profit for gold prices between $420 and
$440 is ($2.49 − $8.77) ×
1.05 = −$6.60.

200

420–440 Gold collar

150
100
50
– $6.60
0
–50
–100
–150
250

$420
300

$440

350
400
450
500
Gold Price in 1 Year ($)

550

of $440, rather than the market price of $450. The $2.49 premium Golddiggers received initially compensates them for the possibility that this will happen.
Figure 4.9 depicts the combination of the purchased put and written call, while Figure
4.10 shows the two profit diagrams for Golddiggers hedged with a 420-strike put, as opposed to hedged with a 420-strike put plus writing a 440-strike call.
Note that the 420–440 collar still entails paying a premium. The 420 put costs $8.77, and the 440 call yields a premium of only $2.49. Thus, there is a net expenditure of $6.28. It is probably apparent, though, that we can tinker with the strike prices and pay a still lower net premium, including zero premium, if we wish. The trade-off is that the payoff on the collar becomes less attractive as we lower the required premium.
A Zero-Cost Collar. To construct a zero-cost collar, we could argue as follows: A 400strike put and a 440-strike call are equally distant from the forward price of $420. This equivalence suggests that the options should have approximately the same premium. As we can see from the table of premiums for different strike options, the 400-strike put has a premium of $2.21, while the 440-strike call has a premium of $2.49. The net premium we would receive from buying this collar is thus $0.28. We can construct a true zero-cost collar by slightly changing the strike prices, making the put more expensive (raising the strike) and the call less expensive (also raising the strike). With strikes of $400.78 for the put and
$440.78 for the call, we obtain a premium of $2.355 for both options.
In reality this zero-cost collar of width 40 would be sold at lower strike prices than
$400.78 and $440.78. The reason is that there is a bid-ask spread: Dealers are willing to buy a given option at a low price and sell it at a high price.
The purchased put will be bought at the dealer’s offer price and the call will be sold at the bid. The dealer can earn this spread in either of two ways: selling the 400.78–440.78 collar and charging an explicit transaction fee, or lowering the strike prices appropriately and charging a zero transaction fee. Either way, the dealer earns the fee. One of the tricky

4.4 Golddiggers Revisited

FIGURE 4.10

Profit ($)

Comparison of Golddiggers hedged with 420-strike put versus hedged with
420-strike put and written
440-strike call (420–440 collar). 150

Seller hedged with 420–440 collar
Seller hedged with 420-strike put

100
50
0

$33.40
$30.79

–50
–100
$420
–150

300

350

400

$440
450

500

Gold Price in 1 Year ($)

aspects of the more complicated derivatives is that it is relatively easy for dealers to embed fees that are invisible to the buyer. Of course a buyer can mitigate this problem by always seeking quotes from different dealers.
We can examine the payoffs by considering separately the three interesting regions of gold prices:
Price of gold < $400.78: In this region, Golddiggers can sell gold for $400.78 by exercising the put option.
Price of gold between $400.78 and $440.78: In this region, Golddiggers can sell gold at the market price.
Price of gold > $440.78: In this region, Golddiggers sells gold at $440.78. It has sold a call, so the owner of the call will exercise. This forces Golddiggers to sell gold to the call owner for the strike price of $440.78.
Figure 4.11 graphs the zero-cost collar against the unhedged position. Notice that between $400.78 and $440.78, the zero-cost collar graph is coincident with the unhedged profit. Above the 440.78-strike the collar provides profit of $60.78, and below the 400.78strike, the collar provides profit of $20.78.
The Forward Contract as a Zero-Cost Collar. Because the put and call with strike prices of $420 have the same premiums, we could also construct a zero-cost collar by buying the
$420-strike put and selling the $420-strike call. If we do this, here is what happens:
Price of gold < $420: Golddiggers will exercise the put option, selling gold at the price of $420.
Price of gold > $420: Golddiggers has sold a 420-strike call. The owner of that call will exercise, obligating Golddiggers to sell gold for $420.

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FIGURE 4.11

Profit ($)

Comparison of unhedged profit for Golddiggers versus zero-cost collar obtained by buying 400.78-strike put and selling 440.78-strike call.

200

Unhedged seller
Seller hedged with 400.78–440.78 collar

150
100
50

$60.78
$20.78

0
–50
–100
–150
250

$400.78
300

$440.78

350
400
450
500
Gold Price in 1 Year ($)

550

In either case, Golddiggers sells gold at $420. Thus, the “420–420 collar” is exactly like a forward contract. By buying the put and selling the call at the same strike price,
Golddiggers has synthetically created a short position in a forward contract. Since a short forward and 420–420 collar have the same payoff, they must cost the same. This is why the premiums on the 420-strike options are the same. This example is really just an illustration of equation (3.1).
Synthetic Forwards at Prices Other Than $420. We can easily extend this example to understand the relationship between option premiums at other strike prices. In the previous example, Golddiggers created a synthetic forward sale at $420. You might think that you could benefit by creating a synthetic forward contract at a higher price such as $440. Other things being equal, you would rather sell at $440 than $420. To accomplish this you buy the
440 put and sell the 440 call. However, there is a catch: The 440-strike put is in-the-money and the 440-strike call is out-of-the-money. Since we would be buying the expensive option and selling the inexpensive option, we have to pay a premium.
How much is it worth to Golddiggers to be able to lock in a selling price of $440 instead of $420? Obviously, it is worth $20 1 year from today, or $20 ÷ (1.05) = $19.05 in present value terms. Since locking in a $420 price is free, it should therefore be the case that we pay $19.05 in net premium in order to lock in a $440 price. In fact, looking at the prices of the 440-strike put and call in Table 4.8, we have premiums of $21.54 for the put and $2.49 for the call. This gives us
Net premium = $21.54 − $2.49 = $19.05
Similarly, suppose Golddiggers explored the possibility of locking in a $400 price for gold in 1 year. Obviously, Golddiggers would require compensation to accept a lower price. In fact, they would need to be paid $19.05—the present value of $20—today.

4.4 Golddiggers Revisited

Again we compute the option premiums and see that the 400-strike call sells for $21.26 while the 400-strike put sells for $2.21. Again we have
Net premium = $2.21 − $21.26 = −$19.05
Golddiggers in this case receives the net premium for accepting a lower price.

Other Collar Strategies
Collar-type strategies are quite flexible. We have focused on the case where the firm buys one put and sells one call. However, it is also possible to deal with fractional options. For example, consider the 400.78–440.78 collar above. We could buy one put to obtain full downside protection, and we could vary the strike price of the call by selling fractional calls at strike prices other than $440.78. For example, we could lower the call strike price below
$440.78, in which case we would obtain a higher premium per call. To offset the higher premium, we could sell less than one call. The trade-off is that we cap the gold price on part of production at a lower level, but we maintain some participation at any price above the strike.
Alternatively, we could raise the cap level (the strike price on the call) and sell more than one call. This would increase participation in gold price increases up to the cap level, but also have the effect of generating a net short position in gold if prices rose above the cap.

Paylater Strategies
A disadvantage to buying a put option is that Golddiggers pays the premium even when the gold price is high and insurance was, after the fact, unnecessary. One strategy to avoid this problem is a paylater strategy, where the premium is paid only when the insurance is needed. While it is possible to construct exotic options in which the premium is paid only at expiration and only if the option is in the money, the strategy we discuss here is a ratio spread using ordinary put options. The goal is to find a strategy where if the gold price is high, there is no net option premium. If the gold price is low, there is insurance, but the effective premium is greater than with an ordinary insurance strategy.
If there is no premium when the gold price is high, we must have no initial premium.
This means that we must sell at least one option. Consider the following strategy for
Golddiggers: Sell a 434.6-strike put and buy two 420-strike puts. Using our assumptions, the premium on the 434.6-strike put is $17.55, while the premium on the 420-strike put is
$8.77. Thus, the net option premium from this strategy is $17.55 − (2 × $8.775) = 0.
Figure 4.12 depicts the result of Golddiggers’s hedging with a paylater strategy. When the price of gold is greater than $434.60, neither put is exercised, and Golddiggers’s profit is the same as if it were unhedged. When the price of gold is between $420 and $434.60, because of the written $434.60 put, the firm loses $2 of profit for every $1 decline in the price of gold. Below $420 the purchased 420-strike puts are exercised, and profit becomes constant. The net result is an insurance policy that is not paid for unless it is needed.
Also depicted in Figure 4.12 is the familiar result from a conventional insurance strategy of hedging by purchasing a single 420-strike put. When the gold price is high, the paylater strategy with a zero premium outperforms the single put. When the gold price is low, the paylater strategy does worse because it offers less insurance. Thus, the premium is paid later, if insurance is needed.

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FIGURE 4.12

Profit ($)

Depiction of “paylater” strategy, in which Golddiggers sells a 434.6-strike put and buys two 420-strike puts, compared to the conventional insurance strategy of buying a 420-strike put.

200

Unhedged seller
Seller hedged with 420-strike put
Paylater
Seller hedged with paylater

150
100
50
0

$30.79
$25.40

–50
–100
–150
250

$420
300

$434.60

350
400
450
500
Gold Price in 1 Year ($)

550

4.5 SELECTING THE HEDGE RATIO
In the Golddiggers and Auric examples, we performed all calculations in terms of one unit of gold and made two important assumptions. First, we assumed perfect correlation between the price of gold and the price of what each company wants to hedge. Second, we assumed that the companies knew for certain the quantity of gold they would sell and buy. As a result of these assumptions, we effectively assumed that the hedge ratio is 1, where the hedge ratio is defined as the ratio of the forward position to the underlying asset.
In practice, neither assumption may be valid. We first examine the effect of widget price uncertainty on hedging, and discuss cross-hedging. We then examine quantity uncertainty using an agricultural example.

Cross-Hedging
In the Auric example we assumed that widget prices are fixed. However, since gold is used to produce widgets, widget prices might vary with gold prices. If widget and gold prices vary one-for-one, Auric’s profits would be independent of the price of gold and Auric would have no need to hedge.12
More realistically, the price of widgets could change with the price of gold, but not one-for-one; other factors could affect widget prices as well. In this case, Auric might find it helpful to use gold derivatives to hedge the price of the widgets it sells as well as the price of the gold it buys. Using gold to hedge widgets would be an example of cross-hedging: the use of a derivative on one asset to hedge another asset. Cross-hedging arises in many different contexts.

12. “One-for-one” in this context means that if the price of gold rises by $1, the price of a widget rises by
$1 times the amount of gold used to make the widget.

4.5 Selecting the Hedge Ratio

The hedging problem for Auric is to hedge the difference in the price of widgets and gold. Conceptually, we can think of hedging widgets and gold separately, and then combining those separate hedges into one net hedge.
We now generalize the Auric example. Suppose that we can produce one widget with λ ounces of gold. If we produce Nw widgets at a price of Pw and do not hedge, profit is
Profit = Nw Pw − λNw Pg

(4.1)

where Pg is the future spot price of gold.
Now suppose that we hedge by going long H gold futures contracts, with each contract covering q oz of gold. If F is the gold forward price, the profit on the hedged position is
Hedged profit = Nw Pw − λNw Pg + H q(Pg − F )
= Nw Pw + (H q − λNw )Pg − H qF
The variance of hedged profit is
2
2 2
2
σhedged = Nw σw + (H q − λNw )2σg + 2Nw (H q − λNw )ρσw σg

(4.2)

where σw is the standard deviation of the widget price, σg is the standard deviation of the gold price, and ρ is the price correlation between widgets and gold. The covariance between widget and gold prices is ρσw σg .
We can select H in equation (4.2) so as to minimize the variance of hedged profit.
The variance-minimizing hedge position, H ∗, satisfies13 qH ∗ = λNw − Nw

ρσw σg (4.3)

Equation (4.3) has a straightforward interpretation. The first term on the right-hand side hedges the cost of buying gold: To produce Nw widgets, we must purchase λNw ounces of gold. We go long this amount. However, if the price of widgets varies with the price of gold we need to take this into account. The second term on the right-hand side, Nw ρσw /σg , measures the comovement between widget revenue and gold. The term ρσw /σg would typically be referred to as the hedge ratio for hedging widgets with gold.14 The hedge ratio times the number of widgets, Nw , gives us the number of gold contracts to short for the purpose of hedging widget price risk. The net number of gold futures, H ∗, is the difference in the number of contracts needed to hedge gold inputs and widget outputs.

13. This can be derived by differentiating equation (4.2) with respect to H .
14. The term ρσw /σg is often measured as the slope coefficient, β, in a linear regression. A common approach is to regress price changes on price changes:
Pw, t − Pw, t−1 = α + β(Pg, t − Pg, t−1) + ut where the subscript t denotes the value at time t. Other specifications, including the use of percentage changes, or regressing levels on levels, are possible. The correct regression to use depends on context. In general, regressions using changes are more likely to give a correct hedge coefficient since the goal of the hedge is to have changes in the price of the asset matched by changes in futures price. In Chapters 5 and 6, we will present examples of hedging stocks and jet fuel, and the appropriate regressions will be returns on returns (stocks) and changes on changes (jet fuel). Regressions of level on level are problematic in many contexts. For example, in the case of stocks, asset pricing models tell us that stock returns are related, but we would not expect a stable relationship between two prices. The appropriate regression is returns on returns. A comprehensive discussion is Siegel and Siegel (1990, pp. 114–135).

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Example 4.1 Suppose that a firm produces Nw = 350 widgets; that each widget requires as an input λ = 0.5 oz of gold; and that σw = 0.15, σg = 0.25, and ρ = 0.8. Suppose further that a gold futures contract calls for delivery of q = 5 oz of gold. Applying equation (4.3), we have
5H ∗ = 0.5 × 350 − 350

0.8 × 0.15
=7
0.25

We go long 7 oz of gold, or H = 1.4 contracts.
We can parse Example 4.1 to understand the hedge calculation. Producing 350 widgets requires 175 (= 0.5 × 350) oz of gold. If widget prices were uncorrelated with gold (ρ = 0), we would go long 175 oz, which is a long position in 35 contracts (each contract covering
5 oz). However, with ρ = 0.8, if the price of gold rises, we pay more for gold but we also receive a higher price for widgets. The second term on the right-hand side of equation (4.3) tells us that this offsetting effect of higher widget prices reduces the required position by
168 oz. The net result is that we go long 7 oz of gold to hedge profit.
2
When we hedge with H ∗ futures, σhedged is obtained by substituting H ∗ into equation
(4.2):
2
2 2 σhedged = Nw σw 1 − ρ 2

(4.4)

The uncertainty remaining in the hedged position is due to basis risk, which is risk due to the hedging instrument (gold) and the hedged price (widgets) not moving as predicted. The variance of profits ultimately depends upon the ability to hedge the price of widgets, which, since we are using gold to hedge, depends on the correlation, ρ, between widgets and gold.
The larger that ρ is, the less is basis risk.
2
You can see in equation (4.4) that if |ρ| = 1, σhedged = 0. However, if |ρ| = 1, variance will be positive. The optimal hedge strategy will then depend upon how the hedger evaluates the residual risk and return from the imperfectly hedged position. In this case a common
2
benchmark is the hedge that minimizes σhedged . Many other hedging strategies are possible, however. For a discussion of the different strategies and a review of the literature, see (Chen et al., 2003).
In this section we have shown that the ability to cross-hedge depends upon the correlation between the hedging instrument and the asset being hedged, and that we can determine the hedging amount as a regression coefficient. We will see in Section 5.4 that the same analysis obtains when we use stock index futures contracts to cross-hedge a stock portfolio. Quantity Uncertainty
The quantity a firm produces and sells may vary with the prices of inputs or outputs. When this happens, using the “obvious” hedge ratio (for example, by hedging the expected quantity of an input) can increase rather than decrease risk. In this section we examine quantity uncertainty. Agricultural producers commonly face quantity uncertainty because crop size is affected by factors such as weather and disease. Moreover, we expect there to be a correlation between quantity and price, because good weather gives rise to bountiful harvests. What

4.5 Selecting the Hedge Ratio

TABLE 4.9

Three scenarios illustrating different correlations between price and quantity for an agricultural producer. Each row is equally likely. In scenario A, there is no quantity uncertainty. In scenario B, quantity is negatively correlated with price, and in scenario C, quantity is positively correlated with price.

Production Scenario
Corn Price ($) A (Uncorrelated) B (Negative correlation) C (Positive correlation)
3
3
2
2

1.0m
1.0m
1.0m
1.0m

1.0m
0.6m
1.5m
0.8m

1.5m
0.8m
1.0m
0.6m

quantity of forward contracts should a corn producer enter into to minimize the variability of revenue?
We will look at three examples of different relationships between price and quantity: the benchmark case where quantity is certain, an example where quantity and price are negatively correlated, and an example where quantity and price are positively correlated.15
In all the examples, we suppose that the corn forward price is $2.50/bu and that there is a 50% probability that in 1 year the corn price will be $2/bu or $3/bu. In addition, for each possible price of corn there are two equally likely quantities, for a total of four possible price-quantity pairs. Table 4.9 illustrates the three scenarios. Note that in scenario B, average quantity is low when price is high (negative correlation), whereas in scenario C, average quantity is high when price is high (positive correlation).
First, consider scenario A, where quantity is certain: The producer always produces
1m bushels. Let S and Q denote the price and quantity in 1 year. Revenue is SQ. Without hedging, revenue will be either $3m (if the corn price is $3) or $2m (if the corn price is $2).
On the other hand, if the producer sells forward 1m bushels at the forward price
F = 2.50, revenue is
Revenue = (S × 1m) − [1m × (S − 2.50)] = 2.5m
We have guaranteed revenue in this case. The calculation is illustrated explicitly in Table
4.10.
In general, if the producer enters into forward contracts on H units, hedged revenue,
R(H ), will be
Hedged revenue = R(H ) = (S × Q) + [H × (S − F )]

15. There are futures contracts intended to mitigate the problem of quantity uncertainty in an agricultural context. Corn yield futures, for example, traded at the Chicago Board of Trade, permit farmers to hedge variations in regional production quantity, and provide an agricultural example of a “quanto” contract. We discuss quantos further in Chapter 22.

(4.5)

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TABLE 4.10

When the producer is sure to produce 1m bushels (Scenario
A), revenue of $2.5m is assured by selling forward 1m bushels. Corn Price

Quantity
1.0m
1.0m

3
2

TABLE 4.11

Price
$3
$3
$2
$2

Unhedged

Revenue
Sell Forward 1m bu

3.0m
2.0m

2.5m
2.5m

Results in Scenario B (negative correlation between the price of corn and the quantity of production) from shorting 975,000 corn forwards (columns 4 and 5) and from selling forward 100,000 bushels (columns 6 and 7). Each pricequantity combination is equally likely, with a probability of 0.25. Standard deviations are computed using the population estimate of standard deviation.

Quantity
1.0m
0.6m
1.5m
0.8m σtotalrevenue Unhedged
Revenue
$3.0m
$1.8m
$3.0m
$1.6m
$0.654m

Sell Forward 0.975m bu
Futures Gain
Total
−$0.488m
−$0.488m
$0.488m
$0.488m

$2.512m
$1.312m
$3.488m
$2.088m
$0.814m

Sell Forward 0.100m bu
Futures Gain
Total
−$0.050m
−$0.050m
$0.050m
$0.050m

$2.95m
$1.75m
$3.05m
$1.65m
$0.652m

2
Using equation (4.5), when there is uncertainty, the variability of hedged revenue, σR(H ), is
2
2
2
σR(H ) = σSQ + H 2σS + 2HρSQ, S σSQσS

(4.6)

The standard deviation of total revenue, SQ, is σSQ, and the correlation of total revenue with price is ρSQ, S . As in the preceding discussion of cross-hedging, the H that minimizes the variance of hedged revenue will be
H =−

ρSQ, S σSQ σS (4.7)

This is the same as the second term in equation (4.3). The formula for the varianceminimizing hedge ratio in equation (4.7) is the negative of the coefficient from a regression of unhedged revenue on price. We can therefore determine the variance-minimizing hedge ratios for the negative- and positive-correlation scenarios (scenarios B and C) in Table 4.9 either by using equation (4.7) directly or by running a regression of revenue on price.
First, consider what happens in Scenario B if we hedge by shorting the expected quantity of production. As a benchmark, column 3 of Table 4.11 shows that unhedged

Chapter Summary

revenue has variability of $0.654m. From Table 4.9, expected production in the negative correlation scenario, B, is
0.25 × (1 + 0.6 + 1.5 + 0.8) = 0.975
If we short this quantity of corn, column 5 of Table 4.11 shows that there is still variability in hedged revenue. Perhaps more surprising, the variability of total revenue actually increases.
The reason is that price decreases when quantity increases, so nature already provides a degree of hedging: The increase in quantity partially offsets the decrease in price. Hedging by shorting the full expected quantity leaves us overhedged, with a commensurate increase in variability.
The variance-minimizing hedge can be obtained using equation (4.7). By direct calculation, we have ρSQ, S = 0.07647, σS = $0.5, and σSQ = $0.654m.16 Thus, we have
H =−

0.07647 × $0.654m
= −0.100m
$0.5

Column (7) of Table 4.11 shows that variability is reduced to $0.652m when hedging this amount. The optimal hedge quantity is closer to no hedging than to full hedging. In fact, we gain little by hedging optimally, but we increase the standard deviation of revenue by
25% if we adopt the plausible but incorrect hedging strategy of shorting 975,000 bushels.
Problem 4.21 asks you to verify that you obtain the same answer by running a regression of revenue on price.
You might guess by now that when correlation is positive (Scenario C), the optimal hedge quantity exceeds expected quantity. The fact that quantity goes up when price goes up makes revenue that much more variable than when price alone varies, and a correspondingly larger hedge position is required. Problem 4.23 asks you to compute the optimal hedge in scenario C. The answer is to short almost 2 million bushels even though production is never that large.

CHAPTER SUMMARY
A producer selling a risky commodity, such as gold, has an inherent long position in the commodity. Assuming costs are fixed, the firm’s profit increases when the price of the commodity increases. Such a firm can hedge profit with a variety of strategies, including selling forward, buying puts, and buying collars. A firm that faces price risk on inputs has an inherent short position in the commodity, with profit that decreases when the price of the input increases. Hedging strategies for such a firm include buying forward, buying calls, and selling collars. All of the strategies involving options can be customized by changing the option strike prices. Strategies such as a paylater can provide insurance with no initial premium but on which the company has greater losses should the insurance be needed.

16. Because Table 4.11 presents the complete population of outcomes, which are equally likely, it is appropriate to use the population estimate of the standard deviation. In Excel, this is STDEVP as opposed to STDEV. The calculation for σSQ is obtained as STDEVP(3, 1.8, 3, 1.6) = 0.6538.

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Hedging can be optimal for a company when an extra dollar of income received in times of high profits is worth less than an extra dollar of income received in times of low profits. Profits for such a firm are concave, in which case hedging can increase expected cash flow. Concave profits can arise from taxes, bankruptcy costs, costly external finance, preservation of debt capacity, and managerial risk aversion. Such a firm can increase expected cash flow by hedging. Nevertheless, firms may elect not to hedge for reasons including transaction costs of dealing in derivatives, the requirement for expertise, the need to monitor and control the hedging process, and complications from tax and accounting considerations. FURTHER READING
In this and earlier chapters we have examined uses of forwards and options, taking for granted the pricing of those contracts. Two big unanswered questions are: How are those prices determined? How does the market for them work?
In Chapters 5 through 8, we will explore forward and futures contracts discussing pricing as well as how market-makers function. In Chapters 10 through 13, we will answer the same questions for options. Chapter 14 will discuss how exotic options can be used in risk-management strategies in place of the ordinary puts and calls discussed in this chapter.
Wharton and CIBC regularly survey nonfinancial firms to assess their hedging. A recent survey is summarized in Bodnar et al. (1998). Bartram et al. (2004) examine hedging behavior in an international sample of over 7000 firms. Tufano (1996, 1998), Petersen and
Thiagarajan (2000), and Brown et al. (2003) have studied hedging practices in the goldmining industry. Other papers examining hedging include G´ czy et al. (1997), Allayannis e and Weston (2001), Allayannis et al. (2003), and Allayannis et al. (2004). Guay and
Kothari (2003) attempt to quantify derivatives usage using information in firm annual reports from 1997. Brown (2001) provides an interesting and detailed description of the hedging decisions by one (anonymous) firm, and Faulkender (2005) examines interest rate hedging in the chemical industry.
Gastineau et al. (2001) discuss Statement of Financial Accounting Standards 133, which currently governs accounting for derivatives.
Finally, Fleming (1997) relates some of the history of (the fictitious) Auric Enterprises.

PROBLEMS
For the following problems consider the following three firms:
.

.

.

XYZ mines copper, with fixed costs of $0.50/lb and variable cost of $0.40/lb.
Wirco produces wire. It buys copper and manufactures wire. One pound of copper can be used to produce one unit of wire, which sells for the price of copper plus $5.
Fixed cost per unit is $3 and noncopper variable cost is $1.50.
Telco installs telecommunications equipment and uses copper wire from Wirco as an input. For planning purposes, Telco assigns a fixed revenue of $6.20 for each unit of wire it uses.

Problems

The 1-year forward price of copper is $1/lb. The 1-year continuously compounded interest rate is 6%. One-year option prices for copper are shown in the table below.17
Strike

Call

Put

0.9500

$0.0649

$0.0178

0.9750
1.0000

0.0500
0.0376

0.0265
0.0376

1.0250
1.0340
1.0500

0.0274
0.0243
0.0194

0.0509
0.0563
0.0665

In your answers, at a minimum consider copper prices in 1 year of $0.80, $0.90, $1.00,
$1.10, and $1.20.
4.1 If XYZ does nothing to manage copper price risk, what is its profit 1 year from now, per pound of copper? If on the other hand XYZ sells forward its expected copper production, what is its estimated profit 1 year from now? Construct graphs illustrating both unhedged and hedged profit.
4.2 Suppose the 1-year copper forward price were $0.80 instead of $1. If XYZ were to sell forward its expected copper production, what is its estimated profit 1 year from now? Should XYZ produce copper? What if the forward copper price is $0.45?
4.3 Compute estimated profit in 1 year if XYZ buys a put option with a strike of $0.95,
$1.00, or $1.05. Draw a graph of profit in each case.
4.4 Compute estimated profit in 1 year if XYZ sells a call option with a strike of $0.95,
$1.00, or $1.05. Draw a graph of profit in each case.
4.5 Compute estimated profit in 1 year if XYZ buys collars with the following strikes:
a. $0.95 for the put and $1.00 for the call.
b. $0.975 for the put and $1.025 for the call.
c. $1.05 for the put and $1.05 for the call.
Draw a graph of profit in each case.
4.6 Compute estimated profit in 1 year if XYZ buys paylater puts as follows (the net premium may not be exactly zero):
a. Sell one 1.025-strike put and buy two 0.975-strike puts.
b. Sell two 1.034-strike puts and buy three 1.00-strike puts.
Draw a graph of profit in each case.
4.7 If Telco does nothing to manage copper price risk, what is its profit 1 year from now, per pound of copper that it buys? If it hedges the price of wire by buying copper

17. These are option prices from the Black formula assuming that the risk-free rate is 0.06, volatility is 0.1, and time to expiration is 1 year.

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forward, what is its estimated profit 1 year from now? Construct graphs illustrating both unhedged and hedged profit.
4.8 Compute estimated profit in 1 year if Telco buys a call option with a strike of $0.95,
$1.00, or $1.05. Draw a graph of profit in each case.
4.9 Compute estimated profit in 1 year if Telco sells a put option with a strike of $0.95,
$1.00, or $1.05. Draw a graph of profit in each case.
4.10 Compute estimated profit in 1 year if Telco sells collars with the following strikes:
a. $0.95 for the put and $1.00 for the call.
b. $0.975 for the put and $1.025 for the call.
c. $0.95 for the put and $0.95 for the call.
Draw a graph of profit in each case.
4.11 Compute estimated profit in 1 year if Telco buys paylater calls as follows (the net premium may not be exactly zero):
a. Sell one 0.975-strike call and buy two 1.034-strike calls.
b. Sell two 1.00-strike calls and buy three 1.034-strike calls.
Draw a graph of profit in each case.
4.12 Suppose that Wirco does nothing to manage the risk of copper price changes. What is its profit 1 year from now, per pound of copper? Suppose that Wirco buys copper forward at $1. What is its profit 1 year from now?
4.13 What happens to the variability of Wirco’s profit if Wirco undertakes any strategy
(buying calls, selling puts, collars, etc.) to lock in the price of copper next year? You can use your answer to the previous question to illustrate your response.
4.14 Golddiggers has zero net income if it sells gold for a price of $380. However, by shorting a forward contract it is possible to guarantee a profit of $40/oz. Suppose a manager decides not to hedge and the gold price in 1 year is $390/oz. Did the firm earn $10 in profit (relative to accounting break-even) or lose $30 in profit (relative to the profit that could be obtained by hedging)? Would your answer be different if the manager did hedge and the gold price had been $450?
4.15 Consider the example in Table 4.6. Suppose that losses are fully tax-deductible. What is the expected after-tax profit in this case?
4.16 Suppose that firms face a 40% income tax rate on all profits. In particular, losses receive full credit. Firm A has a 50% probability of a $1000 profit and a 50% probability of a $600 loss each year. Firm B has a 50% probability of a $300 profit and a 50% probability of a $100 profit each year.
a. What is the expected pre-tax profit next year for firms A and B?
b. What is the expected after-tax profit next year for firms A and B?
4.17 Suppose that firms face a 40% income tax rate on positive profits and that net losses receive no credit. (Thus, if profits are positive, after-tax income is (1 − 0.4)× profit, while if there is a loss, after-tax income is the amount lost.) Firms A and B have

Problems

the same cash flow distribution as in the previous problem. Suppose the appropriate effective annual discount rate for both firms is 10%.
a. What is the expected pre-tax profit for A and B?
b. What is the expected after-tax profit for A and B?
c. What would Firms A and B pay today to receive next year’s expected cash flow for sure, instead of the variable cash flows described above?
For the following problems use the BSCall option pricing function with a stock price of $420
(the forward price), volatility of 5.5%, continuously compounded interest rate of 4.879%, dividend yield of 4.879%, and time to expiration of 1 year. The problems require you to vary the strike prices.
4.18 Consider the example of Auric.
a. Suppose that Auric insures against a price increase by purchasing a 440-strike call. Verify by drawing a profit diagram that simultaneously selling a 400strike put will generate a collar. What is the cost of this collar to Auric?
b. Find the strike prices for a zero-cost collar (buy high-strike call, sell low-strike put) for which the strikes differ by $30.
4.19 Suppose that LMN Investment Bank wishes to sell Auric a zero-cost collar of width
30 without explicit premium (i.e., there will be no cash payment from Auric to
LMN). Also suppose that on every option the bid price is $0.25 below the BlackScholes price and the offer price is $0.25 above the Black-Scholes price. LMN wishes to earn their spread ($0.25 per option) without any explicit charge to Auric.
What should the strike prices on the collar be? (Note: Since the collar involves two options, LMN is looking to make $0.50 on the deal. You need to find strike prices that differ by 30 such that LMN makes $0.50.)
4.20 Use the same assumptions as in the preceding problem, without the bid-ask spread.
Suppose that we want to construct a paylater strategy using a ratio spread. Instead of buying a 440-strike call, Auric will sell one 440-strike call and use the premium to buy two higher-strike calls, such that the net option premium is zero.
a. What higher strike for the purchased calls will generate a zero net option premium? b. Graph the profit for Auric resulting from this strategy.
4.21 Using the information in Table 4.11, verify that a regression of revenue on price gives a regression slope coefficient of about 100,000.
4.22 Using the information in Table 4.9 about Scenario C:
a. Compute σtotal revenue when correlation between price and quantity is positive.
b. What is the correlation between price and revenue?
4.23 Using the information in Table 4.9 about Scenario C:
a. Using your answer to the previous question, use equation (4.7) to compute the variance-minimizing hedge ratio.

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b. Run a regression of revenue on price to compute the variance-minimizing hedge ratio.
c. What is the variability of optimally hedged revenue?
4.24 Using the information in Table 4.9 about Scenario C:
a. What is the expected quantity of production?
b. Suppose you short the expected quantity of corn. What is the standard deviation of hedged revenue?
4.25 Suppose that price and quantity are positively correlated as in this table:
Price

Quantity

Revenue

$2
$3

0.6m bu
0.934m bu

$1.2m
$2.8m

There is a 50% chance of either price. The futures price is $2.50. Demonstrate the effect of hedging if we do the following:
a. Short the expected quantity.
b. Short the minimum quantity.
c. Short the maximum quantity.
d. What is the hedge position that eliminates variability in revenue? Why?

PART
Forwards, Futures, and Swaps

F

orward contracts permit the purchase of an asset in the future at terms that are set today. In earlier chapters we took forward prices as given. In this part, Chapters 5–8, we explore in detail the pricing of forward and futures contracts on a wide variety of underlying assets: financial assets (such as stocks, currencies, and bonds) and commodities (such as gold, corn, and natural gas). We also examine swaps, which have multiple future settlement dates, as opposed to forward contracts, which settle on a single date. Swaps are in effect a bundle of forward contracts combined with borrowing and lending. As such, swaps are a natural generalization of forward contracts. Forward contracts involve deferring receipt of, and payment for, the underlying asset. Thus, computing the forward price requires you to determine the costs and benefits of this deferral. As in Part 1, present and future value calculations are the primary pricing tool.

5
F

Financial Forwards and Futures

orward contracts—which permit firms and investors to guarantee a price for a future purchase or sale—are a basic financial risk management tool. In this chapter we continue to explore these contracts and study in detail forward and futures contracts on financial instruments, such as stocks, indexes, currencies, and interest rates. Our objectives are to understand more about the use of these contracts, how they are priced, and how marketmakers hedge them.
Questions to keep in mind throughout the chapter include: Who might buy or sell specific contracts? What kinds of firms might use the contract for risk management? Why is the contract designed as it is?

5.1 ALTERNATIVE WAYS TO BUY A STOCK
The purchase of a share of XYZ stock has three components: (1) fixing the price, (2) the buyer making payment to the seller, and (3) the seller transferring share ownership to the buyer. If we allow for the possibility that payment and physical receipt can occur at different times—say, time 0 and time T —then once the price is fixed there are four logically possible purchasing arrangements: Payment can occur at time 0 or T , and physical receipt can occur at time 0 or T . Table 5.1 depicts these four possibilities, along with their customary names.
Let’s discuss these different arrangements.1
Outright purchase. The typical way to think about buying stock. You simultaneously pay the stock price in cash and receive ownership of the stock.
Fully leveraged purchase. A purchase in which you borrow the entire purchase price of the security. Suppose you borrow the share price, S0, and agree to repay the borrowed

1. All of these arrangements can be reversed in the case of the seller. Problem 5.1 asks you to describe them from that perspective.

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TABLE 5.1

Four different ways to buy a share of stock that has price
S0 at time 0. At time 0 you agree to a price, which is paid either today or at time T . The shares are received either at
0 or T . The interest rate is r.

Description

Pay at
Time

Receive Security at Time

Payment

Outright purchase
Fully leveraged purchase
Prepaid forward contract
Forward contract

0
T
0
T

0
0
T
T

S0 at time 0
S0erT at time T
?
? × erT

amount at time T . If the continuously compounded interest rate is r, at time T you would owe erT per dollar borrowed, or S0erT .2
Prepaid forward contract. An arrangement in which you pay for the stock today and receive the stock at an agreed-upon future date.3 The difference between a prepaid forward contract and an outright purchase is that with the former, you receive the stock at time T . We will see that the price you pay is not necessarily the stock price.
Forward contract. An arrangement in which you both pay for the stock and receive it at time T , with the time T price specified at time 0.
From Table 5.1 it is clear that you pay interest when you defer payment. The interesting question is how deferring the physical receipt of the stock affects the price; this deferral occurs with both the forward and prepaid forward contracts. What should you pay for the stock in those cases?4

5.2 PREPAID FORWARD CONTRACTS ON STOCK
A prepaid forward contract entails paying today to receive something—stocks, a foreign currency, bonds—in the future. The sale of a prepaid forward contract permits the owner to sell an asset while retaining physical possession for a period of time.

2. For much of the rest of the book, we will use “r” to denote the interest rate. On any given date, the interest rate is typically different for different maturities, so you should always think of r as representing the rate appropriate for the maturity implied by the context. For example, the expression e−rT should be interpreted as the cost today of a bond that pays $1 at time T , and r is then implicitly the T -period interest rate. In some contexts, we will use a more precise notation to denote interest rates.
3. The term prepaid forward contract, or prepay, is used in practice to describe arrangements in which a party repays a loan with a predetermined number of shares of stock. So-called variable prepaid forwards, discussed in Chapter 15, are commonly used by large shareholders selling their stock.
4. The arrangements also differ with respect to credit risk, which arises from the possibilility that the person on the other side of the transaction will not fulfill his or her end of the deal. (And of course the person on the other side of the deal may be worried about you fulfilling your obligation.)

5.2 Prepaid Forward Contracts on Stock

We will derive the prepaid forward price using three different methods: pricing by analogy, pricing by present value, and pricing by arbitrage.

Pricing the Prepaid Forward by Analogy
Suppose you buy a prepaid forward contract on XYZ. By delaying physical possession of the stock, you do not receive dividends and have no voting or control rights. (We ignore here the value of voting and control.)
In the absence of dividends, whether you receive physical possession today or at time
T is irrelevant: In either case you own the stock, and at time T it will be exactly as if you had owned the stock the whole time.5 Therefore, when there are no dividends, the price of the prepaid forward contract is the stock price today. Denoting the prepaid forward price
P
for an asset bought at time 0 and delivered at time T as F0, T , the prepaid forward price for delivery at time T is
P
F0, T = S0

(5.1)

Pricing the Prepaid Forward by Discounted Present Value
We can also derive the price of the prepaid forward using present value: We calculate the expected value of the stock at time T and then discount that value at an appropriate rate of return. The stock price at time T , ST , is uncertain. Thus in computing the present value of the stock price, we need to use an appropriate risk-adjusted rate.
If the expected stock price at time T based on information we have at time 0 is E0(ST ), then the prepaid forward price is given by
P
F0, T = E0(ST )e−αT

(5.2)

where α, the expected return on the stock, is determined using the CAPM or some other model of expected returns.
How do we compute the expected stock price? By definition of the expected return, we expect that in T years the stock will be worth
E0(ST ) = S0eαT
Thus, equation (5.2) gives
P
F0, T = E0(ST )e−αT = S0eαT e−αT = S0

For a non-dividend-paying stock, the prepaid forward price is the stock price.

Pricing the Prepaid Forward by Arbitrage
Classical arbitrage describes a situation in which we can generate a positive cash flow either today or in the future by simultaneously buying and selling related assets, with no net investment of funds and with no risk. Arbitrage, in other words, is free money. An extremely

5. Suppose that someone secretly removed shares of stock from your safe and returned them 1 year later.
From a purely financial point of view, you would never notice the stock to be missing.

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TABLE 5.2

Cash flows and transactions to undertake arbitrage when
P
the prepaid forward price, F0, T , exceeds the stock price, S0 .

Transaction
Buy stock @ S0
P
Sell prepaid forward @ F0, T
Total

Time 0
−S0
P
+F0, T
P
F0, T − S0

Cash Flows
Time T (expiration)
+ST
−ST
0

important pricing principle, which we will use often, is that the price of a derivative should be such that no arbitrage is possible.
Here is an example of arbitrage. Suppose that the prepaid forward price exceeds the
P
stock price—i.e., F0, T > S0. The arbitrageur will buy low and sell high by buying the
P
stock for S0 and selling the prepaid forward for F0, T . This transaction makes money and it is also risk-free: Selling the prepaid forward requires that we deliver the stock at time
T , and buying the stock today ensures that we have the stock to deliver. Thus, we earn
P
F0, T − S0 today and at expiration we supply the stock to the buyer of the prepaid forward.
We have earned positive profits today and offset all future risk. Table 5.2 summarizes this situation. P
Now suppose on the other hand that F0, T < S0. Then we can engage in arbitrage by
P
buying the prepaid forward and shorting the stock, earning S0 − F0, T . One year from now we acquire the stock via the prepaid forward and we use that stock to close the short position.
The cash flows in the above table are simply reversed.
Throughout the book we will assume that prices are at levels that preclude arbitrage.
This raises a question: If prices are such that arbitrage is not profitable, who can afford to become an arbitrageur, watching out for arbitrage opportunities? We can resolve this paradox with the insight that in order for arbitrageurs to earn a living, arbitrage opportunities must occur from time to time; there must be “an equilibrium degree of disequilibrium.”6
However, you would not expect arbitrage to be obvious or easy to undertake.
The transactions in Table 5.2 are the same as those of a market-maker who is hedging a position. A market-maker would sell a prepaid forward if a customer wished to buy it.
The market-maker then has an obligation to deliver the stock at a fixed price and, in order to offset this risk, can buy the stock. The market-maker thus engages in the same transactions as an arbitrageur, except the purpose is risk management, not arbitrage. Thus, the transaction described in Table 5.2—selling the prepaid forward and buying the stock—also describes the actions of a market-maker.
The no-arbitrage arguments we will make thus serve two functions: They tell us how to take advantage of mispricings, and they describe the behavior of market-makers managing risk. 6. The phrase is from Grossman and Stiglitz (1980), in which this idea was first proposed.

5.2 Prepaid Forward Contracts on Stock

Pricing Prepaid Forwards with Dividends
When a stock pays a dividend, the prepaid forward price is less than the stock price. The owner of stock receives dividends, but the owner of a prepaid forward contract does not. It is necessary to adjust the prepaid forward price to reflect dividends that are received by the shareholder but not by the holder of the prepaid forward contract.
Discrete Dividends. To understand the effect of dividends, consider Stock A, which pays no dividend, and otherwise identical stock B, which pays a $5 dividend 364 days from today.
We know that the 1-year prepaid forward price for stock A is the current stock price. What is the one-year prepaid forward price for stock B?
The $5 dividend is paid just before the prepaid forward expiration date 1 year from today. Thus, on the delivery date stock B will be priced $5 less than stock A. The prepaid forward price for stock B should therefore be lower than that for stock A by the present value of $5.
In general, the price for a prepaid forward contract will be the stock price less the present value of dividends to be paid over the life of the contract. Suppose there are multiple dividend payments made throughout the life of the forward contract: A stock is expected to make dividend payments of Dti at times ti , i = 1, . . . , n. A prepaid forward contract will entitle you to receive the stock at time T but without receiving the interim dividends. Thus, the prepaid forward price is
P
F0, T

n

= S0 − i=1 PV0, ti (Dti )

(5.3)

where PV0, ti denotes the time 0 present value of a time ti payment.
Example 5.1 Suppose XYZ stock costs $100 today and is expected to pay a $1.25 quarterly dividend, with the first coming 3 months from today and the last just prior to the delivery of the stock. Suppose the annual continuously compounded risk-free rate is 10%.
The quarterly continuously compounded rate is therefore 2.5%. A 1-year prepaid forward contract for the stock would cost
P
F0, 1 = $100 −

4

$1.25e−0.025i = $95.30

i=1

The calculation in this example implicitly assumes that the dividends are certain. Over a short horizon this might be reasonable. Over a long horizon we would expect dividend risk to be greater, and we would need to adjust the discount rate in computing the present value of expected dividends.
Continuous Dividends. For stock indexes containing many stocks, it is common to model the dividend as being paid continuously at a rate that is proportional to the level of the index;
i.e., the dividend yield (the annualized dividend payment divided by the stock price) is constant. This is an approximation, but in a large stock index there can be dividend payments

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on a large proportion of days.7 The dividend yield is not likely to be fixed in the short run:
When stock prices rise, the dividend yield falls, at least temporarily. Nevertheless, we will assume a constant proportional dividend yield for purposes of this discussion.
To model a continuous dividend, suppose that the index price is S0 and the annualized daily compounded dividend yield is δ. Then the dollar dividend over 1 day is
Daily dividend =

δ
× S0
365

Now suppose that we reinvest dividends in the index. Because of reinvestment, after T years we will have more shares than we started with. Using continuous compounding to approximate daily compounding, we get
Number of shares = 1 +

δ
365

365×T

≈ eδT

At the end of T years we have approximately eδT times the shares we had initially.
Now suppose we wish to invest today in order to have one share at time T . We can buy e−δT shares today. Because of dividend reinvestment, at time T , we will have eδT more shares than we started with, so we end up with exactly one share. Adjusting the initial quantity in this way in order to offset the effect of income from the asset is called tailing the position. Tailing enables us to offset the effect of continuous dividends. We will encounter the concept of tailing frequently.
Since an investment of e−δT S0 gives us one share at time T , this is the time 0 prepaid forward price for delivery at time T :
P
F0, T = S0e−δT

(5.4)

where δ is the dividend yield and T the time to maturity of the prepaid forward contract.
Example 5.2 Suppose that the index is $125 and the annualized daily compounded dividend yield is 3%. The daily dollar dividend is
Dividend = (0.03 ÷ 365) × $125 = $0.01027 or a little more than one penny per unit of the index. If we start by holding one unit of the index, at the end of 1 year we will have e0.03 = 1.030455 shares. Thus, if we wish to end the year holding one share, we must invest in e−0.03 = 0.970446 shares. The prepaid forward price is
$125e−0.03 = $121.306

7. There is significant seasonality in dividend payments, which can be important in practice. A large number of U.S. firms pay quarterly dividends in February, May, August, and November. German firms, by contrast, pay annual dividends concentrated in May, June, and July.

5.3 Forward Contracts on Stock

BOX

131

5.1: Low Exercise Price Options

I

n some countries, including Australia and
Switzerland, it is possible to buy stock options with very low strike prices—so low that it is virtually certain the option will expire in-the-money.
For example, in Australia, the strike price is a penny. Such an option is called a low exercise price option (LEPO). These often exist in order to avoid taxes or transaction fees associated with directly trading the stock. LEPOs do not pay dividends and do not carry voting rights. As with any call option, a LEPO is purchased outright, and it entitles the option holder to acquire the stock at expiration by paying the (low) strike price. The payoff of a LEPO expiring at time T is

max(0, ST − K)
However, if the strike price, K, is so low that the option is certain to be exercised, this is just
ST − K
This option has a value at time 0 of
P
F0, T − PV(K)

Since the strike price of the option is close to zero, a LEPO is essentially a prepaid forward contract. 5.3 FORWARD CONTRACTS ON STOCK
If we know the prepaid forward price, we can compute the forward price. The difference between a prepaid forward contract and a forward contract is the timing of the payment for the stock, which is immediate with a prepaid forward but deferred with a forward. Because there is deferral of payment, the forward contract is initially costless for both the buyer and seller; the premium is zero. Also, because payment is deferred, the forward price is just the future value of the prepaid forward price:
P
F0, T = FV(F0, T )

(5.5)

This formula holds for any kind of dividend payment.
An important special case that we will use throughout the book is that of a continuous dividend. In that case, equation (5.5) becomes
F0, T = erT S0e−δT or F0, T = S0e(r−δ)T

(5.6)

The r in equation (5.6) is the yield to maturity for a default-free zero coupon bond with the same time to maturity as the forward contract. For each maturity, there is an interest rate, r, such that e−r(T −t) = P (t , T ), where P (t , T ) is the time t price of a zero-coupon bond maturing at time T . We can therefore also write equation (5.6) as the price of the prepaid forward contract divided by the price of a zero-coupon bond:
F0, T = S0e−δT /P (0, T )

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For forward contracts with different maturities, different values of r (and for that matter, δ) will generally be appropriate. For simplicity we will write formulas as if there is a single interest rate for all maturities, but in practice this is unlikely to be true.
As is apparent from equation (5.6), the forward price is generally different from the spot price. The forward premium is the ratio of the forward price to the spot price, defined as
Forward premium =

F0, T

(5.7)

S0

We can annualize the forward premium and express it as a percentage, in which case we have Annualized forward premium =

1 ln T

F0, T
S0

For the case of continuous dividends, equation (5.6), the annualized forward premium is simply the difference between the risk-free rate and the dividend yield, r − δ.
Occasionally it is possible to observe the forward price but not the price of the underlying stock or index. For example, the futures contract for the S&P 500 index trades at times when the NYSE is not open, so it is possible to observe the futures price but not the stock price. The asset price implied by the forward pricing formulas above is said to define fair value for the underlying stock or index. Equation (5.6) is used in this case to infer the value of the index.

Does the Forward Price Predict the Future Spot Price?
It is common to think that the forward price predicts the future spot price. Equation (5.9) tells us that the forward price equals the expected future spot price, but with a discount for the risk premium of the asset. Thus, the forward price systematically errs in predicting the future stock price. If the asset has a positive risk premium, the future spot price will on average be greater than the forward price.
The reason is straightforward. When you buy a stock, you invest money that has an opportunity cost (it could otherwise have been invested in an interest-earning asset), and you are acquiring the risk of the stock. On average you expect to earn interest as compensation for the time value of money. You also expect an additional return as compensation for the risk of the stock—this is the risk premium. Algebraically, the expected return on a stock is α= r
Compensation for time

+

α−r

(5.8)

Compensation for risk

When you enter into a forward contract, there is no investment; hence, you are not compensated for the time value of money. However, the forward contract retains the risk of the stock, so you must be compensated for risk. This means that the forward contract must earn the risk premium. If the risk premium is positive, then on average you must expect a positive return from the forward contract. The only way this can happen is if the forward price predicts too low a stock price. In other words the forward contract is a biased predictor of the future stock price. This explains the discount by the risk premium in equation (5.9).
As an example, suppose that a stock index has an expected return of 15%, while the risk-free rate is 5%. If the current index price is 100, then on average we expect that the index will be 115 in 1 year. The forward price for delivery in 1 year will be only 105, however.

5.3 Forward Contracts on Stock

This means that a holder of the forward contract will on average earn positive profits, albeit at the cost of bearing the risk of the index.8
We can see the bias in another way by using equation (5.2). The prepaid forward price is the present value of the expected future spot price. The forward price is the future value of the prepaid forward price. Thus, we have
P
F0, T = erT F0, T = E0(ST )e−(α−r)T

(5.9)

The forward price is the expected future spot price, discounted at the risk premium.
This bias does not imply that a forward contract is a good investment. Rather, it tells us that the risk premium on an asset can be created at zero cost and hence has a zero value.
Though this seems surprising, it is a result from elementary finance that if we buy any asset and borrow the full amount of its cost—a transaction that requires no investment—then we earn the risk premium on the asset. Since a forward contract has the risk of a fully leveraged investment in the asset, it earns the risk premium. This proposition is true in general, not just for the example of a forward on a non-dividend-paying stock.

Creating a Synthetic Forward Contract
A market-maker or arbitrageur must be able to offset the risk of a forward contract. It is possible to do this by creating a synthetic forward contract to offset a position in the actual forward contract.
In this discussion we will assume that dividends are continuous and paid at the rate δ, and hence that equation (5.6) is the appropriate forward price. We can then create a synthetic long forward contract by buying the stock and borrowing to fund the position. To see how the synthetic position works, recall that the payoff at expiration for a long forward position on the index is
Payoff at expiration = ST − F0, T
In order to obtain this same payoff, we buy a tailed position in the stock, investing S0e−δT .
This gives us one share at time T . We borrow this amount so that we are not required to pay anything additional at time 0. At time T we must repay S0e(r−δ)T and we sell the stock for
ST . Table 5.3 demonstrates that borrowing to buy the stock replicates the expiration payoff to a forward contract.
Just as we can use the stock and borrowing to synthetically create a forward, we can also use the forward to create synthetic stocks and bonds. Table 5.4 demonstrates that we can go long a forward contract and lend the present value of the forward price to synthetically create the stock. The expiration payoff in this table assumes that equation (5.6) holds. Table
5.5 demonstrates that if we buy the stock and short the forward, we create cash flows like those of a risk-free bond. The rate of return on this synthetic bond—the construction of which is summarized in Table 5.5—is called the implied repo rate.
To summarize, we have shown that
Forward = Stock − zero-coupon bond

(5.10)

8. Accounting for dividends in this example would not change the magnitude of the bias, since dividends would lower the expected future price of the index and the forward price by equal amounts.

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Demonstration that borrowing S0 e−δT to buy e−δT shares of the index replicates the payoff to a forward contract,
ST − F0, T .

TABLE 5.3

Transaction
Buy e−δT units of the index
Borrow S0e−δT
Total

TABLE 5.4

Cash Flows
Time T (expiration)

Time 0
−S0e−δT
+S0e−δT
0

+ ST
− S0e(r−δ)T
ST − S0e(r−δ)T

Demonstration that going long a forward contract at the price F0, T = S0 e(r−δ)T and lending the present value of the forward price creates a synthetic share of the index at time T .

Transaction
Long one forward
Lend S0e−δT
Total

Time 0

Cash Flows
Time T (expiration)

0
−S0e−δT
−S0e−δT

ST − F0, T
+S0e(r−δ)T
ST

Demonstration that buying e−δT shares of the index and shorting a forward creates a synthetic bond.

TABLE 5.5

Transaction
Buy e−δT units of the index
Short one forward
Total

Time 0
−S0e−δT
0
−S0e−δT

Cash Flows
Time T (expiration)
+ST
F0, T − ST
F0, T

We can rearrange this equation to derive other synthetic equivalents.
Stock = Forward + zero-coupon bond
Zero-coupon bond = Stock − forward
All of these synthetic positions can be reversed to create synthetic short positions.

5.3 Forward Contracts on Stock

TABLE 5.6

Transactions and cash flows for a cash-and-carry: A marketmaker is short a forward contract and long a synthetic forward contract.

Time 0

Transaction
Buy tailed position in stock, paying S0e−δT
Borrow S0e−δT
Short forward
Total

TABLE 5.7

Cash Flows
Time T (expiration)

−S0e−δT
+S0e−δT
0
0

+ST
−S0e(r−δ)T
F0, T − ST
F0, T − S0e(r−δ)T

Transactions and cash flows for a reverse cash-and-carry:
A market-maker is long a forward contract and short a synthetic forward contract.

Transaction
Short tailed position in stock, receiving S0e−δT
Lend S0e−δT
Long forward
Total

Time 0

Cash Flows
Time T (expiration)

+S0e−δT
−S0e−δT
0
0

−ST
+S0e(r−δ)T
ST − F0, T
S0e(r−δ)T − F0, T

Synthetic Forwards in Market-Making and Arbitrage
Now we will see how market-makers and arbitrageurs use these strategies. Suppose a customer wishes to enter into a long forward position. The market-maker, as the counterparty, is left holding a short forward position. He can offset this risk by creating a synthetic long forward position.
Specifically, consider the transactions and cash flows in Table 5.6. The market-maker is short a forward contract and long a synthetic forward contract, constructed as in Table
5.3. There is no risk because the total cash flow at time T is F0, T − S0e(r−δ)T . All of the components of this cash flow—the forward price, the stock price, the interest rate, and the dividend yield—are known at time 0. The result is a risk-free position.
Similarly, suppose the market-maker wishes to hedge a long forward position. Then it is possible to reverse the positions in Table 5.6. The result is in Table 5.7.
A transaction in which you buy the underlying asset and short the offsetting forward contract is called a cash-and-carry. A cash-and-carry has no risk: You have an obligation to deliver the asset but also own the asset. The market-maker offsets the short forward position with a cash-and-carry. An arbitrage that involves buying the underlying asset and

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selling it forward is called a cash-and-carry arbitrage. As you might guess, a reverse cash-and-carry entails short-selling the index and entering into a long forward position.
If the forward contract is priced according to equation (5.6), then profits on a cashand-carry are zero. We motivated the cash-and-carry in Table 5.6 as risk management by a market-maker. However, an arbitrageur might also engage in a cash-and-carry. If the forward price is too high relative to the stock price—i.e., if F0, T > S0e(r−δ)T —then an arbitrageur or market-maker can use the strategy in Table 5.6 to make a risk-free profit.
An arbitrageur would make the transactions in Table 5.7 if the forward were underpriced relative to the stock—i.e., if S0e(r−δ)T > F0, T .
As a final point, you may be wondering about the role of borrowing and lending in Tables 5.6 and 5.7. When you explicitly account for borrowing, you account for the opportunity cost of investing funds. For example, if we omitted borrowing from Table 5.6, we would invest S0e−δT today and receive F0, T at time T . In order to know if there is an arbitrage opportunity, we would need to perform a present value calculation to compare the time 0 cash flow with the time T cash flow. By explicitly including borrowing in the calculations, this time-value-of-money comparison is automatic.9
Similarly, by comparing the implied repo rate with our borrowing rate, we have a simple measure of whether there is an arbitrage opportunity. For example, if we could borrow at 7%, then there is an arbitrage opportunity if the implied repo rate exceeds 7%.
On the other hand, if our borrowing rate exceeds the implied repo rate, there is no arbitrage opportunity. No-Arbitrage Bounds with Transaction Costs
Tables 5.6 and 5.7 demonstrate that an arbitrageur can make a costless profit if F0, T =
S0e(r−δ)T . This analysis ignores trading fees, bid-ask spreads, different interest rates for borrowing and lending, and the possibility that buying or selling in large quantities will cause prices to change. The effect of such costs will be that, rather than there being a single no-arbitrage price, there will be a no-arbitrage bound: a lower price F − and an upper price
F + such that arbitrage will not be profitable when the forward price is between these bounds.
Suppose that the stock and forward have bid and ask prices of S b < S a and F b < F a , a trader faces a cost k of transacting in the stock or forward, and the interest rates for borrowing and lending are r b > r l . In this example we suppose that there are no transaction costs at time T , with the forward settled by delivery of the stock.
We will first derive F +. An arbitrageur believing the observed bid forward price, F b , is too high will undertake the transactions in Table 5.6: Sell the forward and borrow to buy the stock. For simplicity we will assume the stock pays no dividends. The arbitrageur will a pay the transaction cost k to short the forward and pay (S0 + k) to acquire one share of a stock. The required borrowing to finance the position is therefore S0 + 2k. At time T , the payoff is a −(S0 + 2k)er

bT

Repayment of borrowing

+ F0, T − ST +

ST

Value of forward

Value of stock

9. In general, arbitrageurs can borrow and lend at different rates. A pro forma arbitrage calculation needs to account for the appropriate cost of capital for any particular transaction.

5.3 Forward Contracts on Stock

Arbitrage is profitable if this expression is positive, or a F b > F + = (S0 + 2k)er

bT

(5.11)

Thus, the upper bound reflects the fact that we pay a high price for the stock (the ask price), pay transaction costs on both the stock and forward, and borrow at a high rate.
We can derive F − analogously. Problem 5.14 asks you to verify that the bound below which arbitrage is feasible is l b
F a < F − = (S0 − 2k)er T

(5.12)

This expression assumes that short-selling the stock does not entail costs other than bid-ask transaction costs when the short position is initiated.
Notice that in equations (5.11) and (5.12), the costs all enter in such a way as to make the no-arbitrage region as large as possible (for example, the low lending rate enters F − and the high borrowing rate enters F +). This makes economic sense: Trading costs cannot help an arbitrageur make a profit.
There are additional costs not reflected in equations (5.11) and (5.12). One is that significant amounts of trading can move prices, so that what appears to be an arbitrage may vanish if prices change when the arbitrageur enters a large order. Another challenge can be execution risk. If trades do not occur instantaneously, the arbitrage can vanish before the trades are completed.
It is likely that the no-arbitrage region will be different for different arbitrageurs at a point in time, and different across time for a given arbitrageur. For example, consider the trading transaction cost, k. A large investment bank sees stock order flow from a variety of sources and may have inventory of either long or short positions in stocks. The bank may be able to buy or sell shares at low cost by serving as market-maker for a customer order. It may be inexpensive for a bank to short if it already owns the stocks, or it may be inexpensive to buy if the bank already has a short position.
Borrowing and lending rates can also vary. For a transaction that is explicitly financed by borrowing, the relevant interest rates are the arbitrageur’s marginal borrowing rate (if that is the source of funds to buy stocks) or lending rate (if stocks are to be shorted). However, at other times, it may be possible to borrow at a lower rate or lend at a higher rate. For example, it may be possible to sell T-bills being held for some other purpose as a source of short-term funds. This may effectively permit borrowing at a low rate. Finally, in order to borrow money or securities arbitrageurs must have available capital. Undertaking one arbitrage may prevent undertaking another.
The overall conclusion is not surprising: Arbitrage may be difficult, risky, and costly.
Large deviations from the theoretical price may be arbitraged, but small deviations may or may not represent genuine arbitrage opportunities.

Quasi-Arbitrage
The previous section focused on explicit arbitrage. However, it can also be possible to undertake implicit arbitrage by substituting a low yield position for one with a higher return.
We call this quasi-arbitrage.
Consider, for example, a corporation that can borrow at 8.5% and lend at 7.5%.
Suppose there is a cash-and-carry transaction with an implied repo rate of 8%. There is no pure arbitrage opportunity for the corporation, but it would make sense to divert lending

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from the 7.5% assets to the 8% cash-and-carry. If we attempt explicit arbitrage by borrowing at 8.5% in order to earn 8% on the cash-and-carry, the transaction becomes unprofitable. We can arbitrage only to the extent that we are already lending; this is why it is “quasi”-arbitrage.

An Interpretation of the Forward Pricing Formula
The forward pricing formula for a stock index, equation (5.6), depends on r − δ, the difference between the risk-free rate and the dividend yield. This difference is called the cost of carry.
Suppose you buy a unit of the index that costs S and fund the position by borrowing at the risk-free rate. You will pay rS on the borrowed amount, but the dividend yield will provide offsetting income of δS. You will have to pay the difference, (r − δ)S, on an ongoing basis. This difference is the net cash flow if you carry a long position in the asset; hence, it is called the “cost of carry.”
Now suppose you were to short the index and invest the proceeds at the risk-free rate.
You would receive S for shorting the asset and earn rS on the invested proceeds, but you would have to pay δS to the index lender. We will call δ the lease rate of the index; it is what you would have to pay to a lender of the asset. The lease rate of an asset is the annualized cash payment that the borrower must make to the lender. For a non-dividend-paying stock, the lease rate is zero, while for a dividend-paying stock, the lease rate is the dividend.10
Here is an interpretation of the forward pricing formula:
Forward price = Spot price + Interest to carry the asset − Asset lease rate (5.13)
Cost of carry

The forward contract, unlike the stock, requires no investment and makes no payouts and therefore has a zero cost of carry. One way to interpret the forward pricing formula is that, to the extent the forward contract saves our having to pay the cost of carry, we are willing to pay a higher price. This is what equation (5.13) says.

5.4 FUTURES CONTRACTS
Futures contracts are essentially exchange-traded forward contracts. As with forwards, futures contracts represent a commitment to buy or sell an underlying asset at some future date. Because futures are exchange-traded, they are standardized and have specified delivery dates, locations, and procedures. Futures may be traded either electronically or in trading pits, with buyers and sellers shouting orders to one another (this is called open outcry).
Each exchange has an associated clearinghouse.
Although forwards and futures are similar in many respects, there are differences.
.

Whereas forward contracts are settled at expiration, futures contracts are settled daily. The determination of who owes what to whom is called marking-to-market.
Frequent marking-to-market and settlement of a futures contract can lead to pricing differences between the futures and an otherwise identical forward.

10. This discussion ignores lending fees, which are generally small (several basis points). See the discussion of short-selling in Chapter 1.

5.4 Futures Contracts

.

.

.

.

As a result of daily settlement, futures contracts are liquid—it is possible to offset an obligation on a given date by entering into the opposite position. For example, if you are long the September S&P 500 futures contract, you can cancel your obligation to buy by entering into an offsetting obligation to sell the September S&P 500 contract.
If you use the same broker to buy and to sell, your obligation is officially cancelled.11
Over-the-counter forward contracts can be customized to suit the buyer or seller, whereas futures contracts are standardized. For example, available futures contracts may permit delivery of 250 units of a particular index in only March or June. A forward contract, by contrast, could specify April delivery of 300 units of the index.
Because of daily settlement, the nature of credit risk is different with the futures contract. In fact, futures contracts are structured so as to minimize the effects of credit risk. There are typically daily price limits in futures markets (and on some stock exchanges as well). A price limit is a move in the futures price that triggers a temporary halt in trading. For example, there is an initial 5% limit on down moves in the S&P 500 futures contract. An offer to sell exceeding this limit can trigger a temporary trading halt, after which time a 10% price limit is in effect. If that is exceeded, there are subsequent 15% and 20% limits. The rules can be complicated, but it is important to be aware that such rules exist.

We will illustrate futures contracts with the S&P 500 index futures contract as a specific example.

The S&P 500 Futures Contract
The S&P 500 futures contract has the S&P 500 stock index as the underlying asset. Futures on individual stocks have recently begun trading in the United States. (See the box on page 140.) Figure 2.1 shows a newspaper quotation for the S&P 500 index futures contract along with other stock index futures contracts, and Figure 5.1 shows its specifications. The notional value, or size, of the contract is the dollar value of the assets underlying one contract. In this case it is by definition $250 × 1300 = $325,000.12
The S&P 500 is an example of a cash-settled contract: Instead of settling by actual delivery of the underlying stocks, the contract calls for a cash payment that equals the profit or loss as if the contract were settled by delivery of the underlying asset. On the expiration day, the S&P 500 futures contract is marked-to-market against the actual cash index. This final settlement against the cash index guarantees that the futures price equals the index value at contract expiration.
It is easy to see why the S&P 500 is cash-settled. A physical settlement process would call for delivery of 500 shares (or some large subset thereof ) in the precise percentage they make up the S&P 500 index. This basket of stocks would be expensive to buy and sell. Cash settlement is an inexpensive alternative.

11. Although forward contracts may not be explicitly marketable, it is generally possible to enter into an offsetting position to cancel the obligation to buy or sell.
12. Because the S&P 500 index is a fabricated number—a value-weighted average of individual stock prices—the S&P 500 index is treated as a pure number rather than a price, and the contract is defined at maturity to have a size of $250 × S&P 500 index.

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BOX

Chapter 5. Financial Forwards and Futures

5.2: Single Stock Futures

F

utures contracts on individual stocks trade in various countries. In the United States, single stock futures began trading in November
2002 on OneChicago, an electronic exchange owned jointly by the Chicago Board Options
Exchange, the Chicago Board of Trade, and the
Chicago Mercantile Exchange. Earlier, the trading of single stock futures had been stalled by disagreements among exchanges and by a regulatory turf battle between the Securities and
Exchange Commission, which regulates stocks and stock options, and the Commodity Futures
Trading Commission, which regulates commodity and equity index futures.
Single stock futures were controversial even before trading began, with disagreement about how successful the product would be. What need would single stock futures serve? There was already a well-established market for buying and short-selling stocks, and we saw in

FIGURE 5.1
Specifications for the S&P
500 index futures contract.

Underlying
Where traded
Size
Months
Trading ends
Settlement

Chapter 3 that investors could create synthetic stock forwards using options. Would differences in margin requirements, transaction costs, or contract characteristics make the new product successful? Since 2002, one competitor to OneChicago
(NQLX) has entered and then exited the market for single stock futures in the United States.
Trading volume has proved disappointing for some advocates (see Zwick and Collins, 2004).
In 2010, OneChicago (“OCX”) introduced
OCX.NoDiv, a dividend-protected single-stock futures contract. The new product reduces the futures settlement price by the dividend amount on each ex-dividend day. The new contract should have a price equal to the stock price plus interest, while the standard contract will subtract the present value of dividends. The two contracts trading side-by-side permit investors to speculate on and hedge dividends.

S&P 500 index
Chicago Mercantile Exchange
$250 × S&P 500 index
March, June, September, December
Business day prior to determination of settlement price
Cash-settled, based upon opening price of
S&P 500 on third Friday of expiration month

Margins and Marking to Market
Let’s explore the logistics of holding a futures position. Suppose the futures price is 1100 and you wish to acquire a $2.2 million position in the S&P 500 index. The notional value of one contract is $250 × 1100 = $275,000; this represents the amount you are agreeing to pay at expiration per futures contract. To go long $2.2 million of the index, you would enter into $2.2 million/$0.275 million = 8 long futures contracts. The notional value of eight contracts is 8 × $250 × 1100 = $2,000 × 1100 = $2.2 million.

5.4 Futures Contracts

A broker executes your buy order. For every buyer there is a seller, which means that one or more investors must be found who simultaneously agree to sell forward the same number of units of the index. The total number of open positions (buy/sell pairs) is called the open interest of the contract.
Both buyers and sellers are required to post a performance bond with the broker to ensure that they can cover a specified loss on the position.13 This deposit, which can earn interest, is called margin and is intended to protect the counterparty against your failure to meet your obligations. The margin is a performance bond, not a premium. Hence, futures contracts are costless (not counting, of course, commissions and the bid-ask spread).
To understand the role of margin, suppose that there is 10% margin and weekly settlement (in practice, settlement is daily). The margin on futures contracts with a notional value of $2.2 million is $220,000.
If the S&P 500 futures price drops by 1, to 1099, we lose $2000 on our futures position. The reason is that eight long contracts obligate us to pay $2000 × 1100 to buy
2000 units of the index which we could now sell for only $2000 × 1099. Thus, we lose
(1099 − 1100) × $2000 = −$2000. Suppose that over the first week, the futures price drops
72.01 points to 1027.99, a decline of about 6.5%. On a mark-to-market basis, we have lost
$2000 × −72.01 = −$144,020
We have a choice of either paying this loss directly, or allowing it to be taken out of the margin balance. It doesn’t matter which we do, since we can recover the unused margin balance plus interest at any time by offsetting our position.
If the loss is subtracted from the margin balance, we have earned 1 week’s interest and have lost $144,020. Thus, if the continuously compounded interest rate is 6%, our margin balance after 1 week is
$220,000e0.06×1/52 − $144,020 = $76,233.99
Because we have a 10% margin, a 6.5% decline in the futures price results in a 65% decline in margin. Were we to close out our position by entering into eight short index futures contracts, we would receive the remaining margin balance of $76,233.99.
The decline in the margin balance means the broker has significantly less protection should we default. For this reason, participants are required to maintain the margin at a minimum level, called the maintenance margin. This is often set at 70% to 80% of the initial margin level. In this example, where the margin balance declines 65%, we would have to post additional margin. The broker would make a margin call, requesting additional margin. If we failed to post additional margin, the broker would close the position by selling
2000 units of the index and return to us the remaining margin. In practice, marking-to-market and settling up are performed at least daily.
Since margin you post is the broker’s protection against your default, a major determinant of margin levels is the volatility of the underlying asset. The margin on the S&P
500 contract has generally been less than the 10% we assume in this example. In December

13. The exchange’s clearinghouse determines a minimum margin, but individual brokers can and do demand higher margins from individual customers. The reason is that the broker is liable to the clearing corporation for customer failure to pay.

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Chapter 5. Financial Forwards and Futures

TABLE 5.8

Mark-to-market proceeds and margin balance over 10 weeks from long position in 8 S&P 500 futures contracts.
The last column does not include additional margin payments. The final row represents expiration of the contract. 0
1
2
3
4
5
6
7
8
9
10

Multiplier ($)

Futures
Price

Price
Change

Margin
Balance($)

2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00
2000.00

Week

1100.00
1027.99
1037.88
1073.23
1048.78
1090.32
1106.94
1110.98
1024.74
1007.30
1011.65


−72.01
9.89
35.35
−24.45
41.54
16.62
4.04
−86.24
−17.44
4.35

220,000.00
76,233.99
96,102.01
166,912.96
118,205.66
201,422.13
234,894.67
243,245.86
71,046.69
36,248.72
44,990.57

2010, for example, the initial margin on the S&P 500 futures contract was about 9% of the notional value of the contract. The maintenance margin was 80% of the initial margin.
To illustrate the effect of periodic settlement, Table 5.8 reports hypothetical futures price moves and tracks the margin position over a period of 10 weeks, assuming weekly marking-to-market and a continuously compounded risk-free rate of 6%. As the party agreeing to buy at a fixed price, we make money when the price goes up and lose when the price goes down. The opposite would occur for the seller.
The 10-week profit on the position is obtained by subtracting from the final margin balance the future value of the original margin investment. Week-10 profit on the position in Table 5.8 is therefore
$44,990.57 − $220,000e0.06×10/52 = −$177,562.60
What if the position had been a forward rather than a futures position, but with prices the same? In that case, after 10 weeks our profit would have been
(1011.65 − 1100) × $2000 = −$176,700
Why do the futures and forward profits differ? The reason is that with the futures contract, interest is earned on the mark-to-market proceeds. Given the prices in Table 5.8, the loss is larger for futures than forwards because prices on average are below the initial price and we have to fund losses as they occur. With a forward, by contrast, losses are not funded until expiration. Earning interest on the daily settlement magnifies the gain or loss compared to that on a forward contract. Had there been consistent gains on the position

5.4 Futures Contracts

in this example, the futures profit would have exceeded the forward profit. Appendix 5.B demonstrates that the ultimate payoff to a forward and futures contract can be equated in this example by adjusting the number of futures contracts so as to undo the magnifying effect of interest.

Comparing Futures and Forward Prices
An implication of Appendix 5.B is that if the interest rate were not random, then forward and futures prices would be the same. However, what if the interest rate varies randomly?
Suppose, for example, that on average the interest rate increases when the futures price increases; i.e., the two are positively correlated. Then the margin balance would grow (due to an increased futures price) just as the interest rate was higher. The margin balance would shrink as the interest rate was lower. On average in this case, a long futures position would outperform a long forward contract.
Conversely, suppose that on average the interest rate declined as the futures price rose. Then as the margin balance on a long position grew, the proceeds would be invested at a lower rate. Similarly, as the balance declined and required additional financing, this financing would occur at a higher rate. Here a long futures contract would on average perform worse than a long forward contract.
This comparison of the forward and futures payoffs suggests that when the interest rate is positively correlated with the futures price, the futures price will exceed the price on an otherwise identical forward contract: The investor who is long futures buys at a higher price to offset the advantage of marking-to-market. Similarly, when the interest rate is negatively correlated with the forward price, the futures price will be less than an otherwise identical forward price: The investor who is long futures buys at a lower price to offset the disadvantage of marking-to-market.
As an empirical matter, forward and futures prices are very similar.14 The theoretical difference arises from uncertainty about the interest on mark-to-market proceeds. For short-lived contracts, the effect is generally small. However, for long-lived contracts, the difference can be significant, especially for long-lived interest rate futures, for which there is sure to be a correlation between the interest rate and the price of the underlying asset. For the rest of this chapter we will ignore the difference between forwards and futures.

Arbitrage in Practice: S&P 500 Index Arbitrage
We now illustrate the use of equation (5.6) to determine the theoretical price of an S&P 500 futures contract on a specific day. In order to compute equation (5.6), we need three inputs:
(1) the value of the cash index (S0), (2) the value of dividends to be paid on the index over the life of the contract (δ), and (3) the interest rate (r). The pricing exercise is summarized in Table 5.9.
On December 16, 2010, the closing S&P 500 index price was 1242.87. S&P 500 futures contracts expiring in March 2011 and June 2011 had closing prices of 1238.50 and 1233.60. The dividend yield on the S&P 500 in mid-December was about 1.89%. The remaining input is the risk-free interest rate. What interest rate is appropriate?

14. See French (1983) for a comparison of forward and futures prices on a variety of underlying assets.

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Chapter 5. Financial Forwards and Futures

TABLE 5.9

Actual and theoretical S&P 500 futures prices, December
16, 2010. The S&P 500 index closed at 1242.87 on December 16, and the dividend yield was 1.89%. Theoretical prices are computed using equation (5.6).

Maturity

Closing
Price

T-bill
Yield

Theoretical
Price

LIBOR

Theoretical
Price

Spot
March 2011
June 2011

1242.87
1238.50
1233.60

0.13%
0.19%

1237.41
1232.35

0.30%
0.46%

1237.94
1234.02

Rates and dividend yield from Bloomberg.

Two interest rates that we can easily observe are the yield on U.S. Treasury bills and the London Interbank Offer Rate (LIBOR), which is a borrowing rate for large financial institutions.15 (We will discuss LIBOR further in Section 5.7.) Table 5.9 reports 3-month and 6-month T-bill and LIBOR rates from December 16. LIBOR yields are greater than
Treasury yields for two reasons. First, banks have greater default risk than the government and thus the interest rate on their deposits is greater. Second, Treasury securities are more liquid—they are easier to buy and sell—and consequently their price is greater (their yield is lower).
To think about what interest rate is appropriate, consider the economic characteristics of a futures contract. A long futures contract is like a leveraged position in the underlying asset, where the loan that provides leverage is collateralized with margin that is refreshed daily. The lender will perceive a costly default as unlikely; thus the rate can be close to risk-free. The T-bill rate includes a liquidity premium, however, while the rate implicit in a futures contract would not. The appropriate rate is therefore greater than the T-bill rate and likely lower than LIBOR.
Table 5.9 shows that for the 3-month contract, equation (5.6) yields lower prices than the observed price when using both the T-bill rate and LIBOR. For the 6-month contract, the two rates bracket the observed futures price. The percentage difference between the observed futures price and that computed using LIBOR is small for both contracts, about
0.04%.
There are a number of considerations in interpreting these differences in prices:
.

Future dividends on the S&P 500 stocks are uncertain. For pricing a 3-month futures contract, one could use equation (5.5), with actual recent cash dividends on the underlying stocks for Dti as proxies for forthcoming dividends. There is still a risk that dividends will change over the next 3 months. The risk is greater for longer-dated futures contracts.

15. This example uses reported yields as if they were quoted as continuously compounded yields, which they are not. However, for short periods and low interest rates, there is almost no difference between effective annual and continuously compounded rates. For example, if the effective annual rate is 2%, the continuously compounded equivalent is ln(1.02) = 0.0198, or 1.98%. In practice, interest rates are quoted using a variety of arcane conventions for annualizing the rate, some of which are discussed in Chapter 7.

5.4 Futures Contracts

FIGURE 5.2
Specifications for the Nikkei
225 index futures contract.

Underlying
Where traded
Size
Months
Trading ends
Settlement

.

.

Nikkei 225 Stock Index
Chicago Mercantile Exchange
$5 × Nikkei 225 Index
March, June, September, December
Business day prior to determination of settlement price
Cash-settled, based upon opening Osaka quotation of the Nikkei 225 index on the second Friday of expiration month

There are transaction costs of arbitrage. As illustrated by equations (5.11) and (5.12), transaction costs create no-arbitrage regions, rather than no-arbitrage prices. In practice, a representative bid-ask spread on the index futures contract might be 20 to 30 basis points (a basis point on the S&P futures contract is 0.01—this is roughly 0.02% of the price) and 0.25% to 0.5% on the stocks in the index when traded in significant quantities. These costs are likely larger than the observed pricing difference.
Because of transaction costs, an arbitrageur will usually not buy the entire 500-stock index, but instead either a subset of stocks or an index alternative such as Standard and Poor’s Depository Receipts (SPDRs).16 The futures contract and the offsetting position may not move exactly together. Also, when buying a large number of stocks, there is execution risk—the possibility that prices move during the time between the order being placed and the stock being actually purchased.

Arbitrageurs will need to take into account these considerations and small differences may not be worth arbitraging. Ultimately, the only way to know if arbitrage is profitable is to assess specific prices, trading costs, and borrowing and lending rates.

Quanto Index Contracts
At first glance the Chicago Mercantile Exchange’s Nikkei 225 futures contract—see the details summarized in Figure 5.2—is a stock index contract like the S&P 500 contract.
However, there is one very important difference: Settlement of the contract is in a different currency (dollars) than the currency of denomination for the index (yen).17
To see why this is important, consider a dollar-based investor wishing to invest in the
Nikkei 225 cash index. This investor must undertake two transactions: changing dollars to yen and using yen to buy the index. When the position is sold, the investor reverses these transactions, selling the index and converting yen back to dollars. There are two sources of risk in this transaction: the risk of the index, denominated in yen, and the risk that the

16. SPDRs are unit investment trusts that are backed by a portfolio intended to mimic the S&P 500.
Investors can convert units of 50,000 SPDR shares into the actual stock and can convert stock into SPDRs.
This keeps SPDRs close to the S&P 500 index, but in practice SPDRs may be mispriced relative to the cash S&P 500 just as futures are.
17. There is also a yen-denominated Nikkei 225 futures contract that trades at the Osaka exchange. Since it is purely yen-denominated, this contract is priced according to equation (5.6).

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yen/dollar exchange rate will change. From Figure 5.2, the Nikkei 225 futures contract is denominated in dollars rather than yen. Consequently, the contract insulates investors from currency risk, permitting them to speculate solely on whether the index rises or falls. This kind of contract is called a quanto. Quanto contracts allow investors in one country to invest in a different country without exchange rate risk.
The dollar-denominated Nikkei contract provides an interesting variation on the construction of a futures contract. Because of the quanto feature, the pricing formulas we have developed do not work for the Nikkei 225 contract. We will discuss quantos and the necessary modification to price a quanto futures contract in Chapter 22.

5.5 USES OF INDEX FUTURES
An index futures contract resembles borrowing to buy the index. Why use an index futures contract if you can synthesize one? One answer is that index futures can permit trading the index at a lower transaction cost than actually trading a basket of the stocks that make up the index. If you are taking a temporary position in the index, either for investing or hedging, the transaction cost saving could be significant.
In this section we provide two examples of the use of index futures: asset allocation and cross-hedging a related portfolio.

Asset Allocation
Asset allocation strategies involve switching investments among asset classes, such as stocks, money market instruments, and bonds. Trading the individual securities, such as the stocks in an index, can be expensive. Our earlier discussion of arbitrage demonstrated that we can use forwards to create synthetic stocks and bonds. The practical implication is that a portfolio manager can invest in a stock index without holding stocks, commodities without holding physical commodities, and so on.
Switching from Stocks to T-bills. As an example of asset allocation, suppose that we have an investment in the S&P 500 index and we wish to temporarily invest in T-bills instead of the index. Instead of selling all 500 stocks and investing in T-bills, we can simply keep our stock portfolio and take a short forward position in the S&P 500 index. This converts our investment in the index into a cash-and-carry, creating a synthetic T-bill. When we wish to revert to investing in stocks, we simply offset the forward position, undoing the cash-and-carry. To illustrate this, suppose that the current index price, S0, is $100, and the effective 1year risk-free rate is 10%. The forward price is therefore $110. Suppose that in 1 year, the index price could be either $80 or $130. If we sell the index and invest in T-bills, we will have $110 in 1 year.
Table 5.10 shows that if, instead of selling, we keep the stock and short the forward contract, we earn a 10% return no matter what happens to the value of the stock. In this example 10% is the rate of return implied by the forward premium. If there is no arbitrage, this return will be equal to the risk-free rate.
General Asset Allocation. We can use forwards and futures to perform even more sophisticated asset allocation. Suppose we wish to invest our portfolio in Treasury bonds
(long-term Treasury obligations) instead of stocks. We can accomplish this reallocation

5.5 Uses of Index Futures

TABLE 5.10

Effect of owning the stock and selling forward, assuming that S0 = $100 and F0, 1 = $110.

Transaction

Today

Cash Flows
1 year, S1 = $80
1 year, S1 = $130

Own stock @ $100
Short forward @ $110
Total

−$100
0
−$100

$80
$110 − $80
$110

$130
$110 − $130
$110

with two forward positions: Shorting the forward S&P 500 index and going long the forward T-bond. The first transaction converts our portfolio from an index investment to a T-bill investment. The second transaction converts the portfolio from a T-bill investment to a Tbond investment. This use of futures to convert a position from one asset category (stocks) to another (bonds) is called a futures overlay.
Futures overlays can have benefits beyond reducing transaction costs. Suppose an investment management company has portfolio managers who successfully invest in stocks they believe to be mispriced. The managers are judged on their performance relative to the S&P 500 stock index and consistently outperform the index by 2% per year (in the language of portfolio theory, their “alpha” is 2%). Now suppose that new clients of the company like the performance record, but want to invest in bonds rather than stocks. The investment management company could fire its stock managers and hire bond managers, but its existing investment managers are the reason for the company’s success. The company can use a futures overlay to continue to invest in stocks, but to provide a bond return instead of a stock return to investors. By investing in stocks, shorting index futures, and going long bond futures, the managers continue to invest in stocks, but the client receives a bond return plus 2% rather than a stock return plus 2%. This use of futures to transform an outperforming portfolio on one asset class into an outperforming portfolio on a different asset class is called alpha-porting. Cross-hedging with Index Futures
Index futures are often used to hedge portfolios that are not exactly the index. As discussed in Section 4.5, this is called cross-hedging.
Cross-hedging with Perfect Correlation. Suppose that we have a portfolio that is not the
S&P 500, and we wish to shift the portfolio into T-bills. Can we use the S&P 500 futures contract to do this? The answer depends on the correlation of the portfolio with the S&P
500. To the extent the two are not perfectly correlated, there will be residual risk.
Suppose that we own $100 million of stocks with a beta relative to the S&P 500 of 1.4. Assume for the moment that the two indexes are perfectly correlated. Perfect correlation means that there is a perfectly predictable relationship between the two indexes,

147

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not necessarily that they move one-for-one. Using the Capital Asset Pricing Model (CAPM), the return on our portfolio, rp , is related to its beta, βp , by rp = r + βp (rS&P − r)
Assume also that the S&P 500 is 1100 with a 0 dividend yield and the effective annual risk-free rate is 6%. Hence the futures price is 1100 × 1.06 = 1166.
If we wish to allocate from the index into Treasury bills using futures, we need to short some quantity of the S&P 500. There are two steps to calculating the short futures quantity: 1. Adjust for the difference in the dollar amounts of our portfolio and the S&P 500 contract. In this case, one futures contract has a notional value of $250 × 1166 =
$291,500. This is denominated in future dollars; the value of one contract in current dollars is $291,500/1.06 = $275,000 (this is also the prepaid forward price). Thus, the number of contracts needed to cover $100 million of stock is
$100 million
= 363.636
$0.275 million
Note that both the numerator and denominator are expressed in current dollars.
2. Adjust for the difference in beta. Since the beta of our portfolio exceeds 1, it moves more than the S&P 500 in either direction. Thus we need to further increase our S&P
500 position to account for the greater magnitude moves in our portfolio relative to the S&P 500. This gives us
Final hedge quantity = 363.636 × 1.4 = 509.09
Table 5.11 shows the performance of the hedged position. The result, as you would expect, is that the hedged position earns the risk-free rate, 6%.
Cross-Hedging with Imperfect Correlation. The preceding example assumes that the portfolio and the S&P 500 index are perfectly correlated. In practice, correlations between

TABLE 5.11
S&P 500 Index
900
950
1000
1050
1100
1150
1200

Results from shorting 509.09 S&P 500 index futures against a $100m portfolio with a beta of 1.4.

Gain on 509 Futures

Portfolio Value

Total

33.855
27.491
21.127
14.764
8.400
2.036
−4.327

72.145
78.509
84.873
91.236
97.600
103.964
110.327

106.000
106.000
106.000
106.000
106.000
106.000
106.000

5.5 Uses of Index Futures

two portfolios can be substantially less than one. Using the S&P 500 to hedge such a portfolio would introduce basis risk, creating a hedge with residual risk.18
Denote the return and invested dollars on the portfolio as rp and Ip . Assume that we short H futures contracts, each with a notional value N . The futures position earns the risk premium, rS&P − r. Thus, the return on the hedged position is
Hedged return = rp Ip + H × N × (rS&P − r)
Repeating the analysis in Section 4.5 [in particular, see equation (4.2)], the varianceminimizing hedge position, H ∗, is
H∗ = −
=−

Ip Cov(rp , rS&P)
N
Ip
N

(5.14)

2 σS&P βp

The hedge quantity is denominated in terms of a quantity of futures contracts. The second
2
equality follows because Cov(rp , rS&P )/σS&P is the slope coefficient when we regress the portfolio return on the S&P 500 return; i.e., it is the portfolio beta with respect to the S&P
500 index. Equation (5.14) is also the formula we used in concluding that, with perfect correlation, we should short 509.09 contracts.
Notice that the hedge ratio in equation (5.14) depends on the ratio of the market value of the portfolio, Ip , to the notional value of the S&P 500 contract, N . Thus, as the portfolio changes value relative to the S&P 500 index, it is necessary to change the hedge ratio.
This rebalancing is necessary when we calculate hedge ratios using a relationship based on returns, which are percentage changes.
When we add H ∗ futures to the portfolio, the variance of the hedged portfolio,
2
σhedged , is
2
2 2 σhedged = σp Ip 1 − ρ 2

(5.15)

where ρ is the correlation coefficient between the portfolio and the S&P 500 index. This is the same as equation (4.4). The correlation coefficient, ρ, can be computed directly from rp and rS&P , but it is also the square root of the regression r-squared (R 2) when we regress rP on rS&P in order to estimate β.
Example 5.3 Suppose we are optimistic about the performance of the NASDAQ index relative to the S&P 500 index. We can go long the NASDAQ index and short the S&P
500 futures. We obtain the variance-minimizing position in the S&P 500 by using equation
(5.14). A 5-year regression (from June 1999 to June 2004) of the daily NASDAQ return on the S&P 500 return gives rNASD = − 0.0001 + 1.4784 ×(rS&P − r)
(0.0003)

(0.0262)

R 2 = 0.7188

The regression beta tells us to short a dollar value of the S&P that is 1.4784 times greater than the NASDAQ position we hold. The correlation coefficient between the two returns,

18. There is additional basis risk in such a hedge because, for reasons discussed in Section 5.4, the S&P
500 futures contract and the cash price of the S&P 500 index may not move perfectly together.

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Chapter 5. Financial Forwards and Futures

ρ, is 0.7188 = 0.8478.19 The daily standard deviation of the return on the NASDAQ over this period is 2.24%. Hence, using equation (5.15), for a $1 million investment the variance of the hedged position is
2
2 σNASDIp 1 − ρ 2 = 0.02242 × ($1m)2 × (1 − 0.7188) = ($11, 878)2

Thus, the daily standard deviation of the hedged return is $11,878.
Risk Management for Stock-Pickers. An asset manager who picks stocks is often making a bet about the relative, but not the absolute, performance of a stock. For example, XYZ might be expected to outperform a broad range of stocks on a risk-adjusted basis. If the economy suffers a recession, however, XYZ will decline in value even if it outperforms other stocks. Index futures can be used in this case to help isolate the relative performance of XYZ.
Suppose the return of XYZ is given by the CAPM: rXYZ = αXYZ + r + βXYZ (rm − r)

(5.16)

The term αXYZ in this context represents the expected abnormal return on XYZ. If we use the S&P 500 as a proxy for the market, then we can select H according to equation (5.14).
The result for the hedged position will be that, on average, we earn αXYZ + r. The risk of the position will be given by equation (5.15). Since the correlation of an individual stock and the index will not be close to 1, there will be considerable remaining risk. However, the portfolio will not have market risk.

5.6 CURRENCY CONTRACTS
Currency futures and forwards are widely used to hedge against changes in exchange rates.
The pricing of currency contracts is a straightforward application of the principles we have already discussed. Figure 5.3 reports the specifications for the US dollar/euro futures contract at the CME, and Table 5.12 reports futures price quotes for two dates.
Many corporations use currency futures and forwards for short-term hedging. An importer of consumer electronics, for example, may have an obligation to pay the manufacturer
¥150 million 90 days in the future. The dollar revenues from selling these products are likely known in the short run, so the importer bears pure exchange risk due to the payable being fixed in yen. By buying ¥150 million forward 90 days, the importer locks in a dollar price to pay for the yen, which will then be delivered to the manufacturer.

Currency Prepaid Forward
Suppose that 1 year from today you want to have ¥1. A prepaid forward allows you to pay dollars today to acquire ¥1 in 1 year. What is the prepaid forward price? Suppose the yen-denominated interest rate is ry and the exchange rate today ($/¥) is x0. We can work backward. If we want ¥1 in 1 year, we must have e−ry in yen today. To obtain that many yen today, we must exchange x0e−ry dollars into yen.

19. You can, of course, also compute the correlation coefficient directly from the time series of returns.

5.6 Currency Contracts

FIGURE 5.3

Underlying
Where traded
Size
Months

Specifications for EUR/USD futures contract.

Trading ends
Delivery

TABLE 5.12

Euro currency
Chicago Mercantile Exchange
125,000 euro
March, June, September, December (six consecutive contracts)
9:16a.m. on the second business day prior to the third Wednesday of the month
Cash-settlement

Dollar cost of foreign currencies.

Date

Currency

June

Sept.

Dec.

Mar.

June

Sept.

June 6, 2007

Euro
100 Yen
Sterling
Euro
100 Yen
Sterling

1.351
0.828
1.992
1.224
1.085
1.464

1.355
0.837
1.99
1.225
1.087
1.465

1.358
0.847
1.987
1.226
1.089
1.465

1.361
0.856
1.984
1.228
1.092
1.466

1.363
0.865
1.98
1.229
1.095
1.466

1.365
0.873
1.975
1.231
1.098
1.467

June 2, 2010

Source: CMEGroup via Datastream.

Thus, the prepaid forward price for a yen is
P
F0, T = x0e−ry T

(5.17)

where T is time to maturity of the forward.
The economic principle governing the pricing of a prepaid forward on currency is the same as that for a prepaid forward on stock. By deferring delivery of the underlying asset, you lose income. In the case of currency, if you received the currency immediately, you could buy a bond denominated in that currency and earn interest. The prepaid forward price reflects the loss of interest from deferring delivery, just as the prepaid forward price for stock reflects the loss of dividend income. This is why equation (5.17) is the same as that for a stock paying a continuous dividend, equation (5.4).
Example 5.4 Suppose that the yen-denominated interest rate is 2% and that the current exchange rate is 0.009 dollars per yen. Then in order to have 1 yen in 1 year, we would invest today
0.009$ /¥ × ¥1 × e−0.02 = $.008822

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Chapter 5. Financial Forwards and Futures

Currency Forward
The prepaid forward price is the dollar cost of obtaining 1 yen in the future. Thus, to obtain the forward price, compute the future value using the dollar-denominated interest rate, r:
F0, T = x0e(r−ry )T

(5.18)

The forward currency rate will exceed the current exchange rate when the domestic risk-free rate is higher than the foreign risk-free rate.20
Example 5.5 Suppose that the yen-denominated interest rate is 2% and the dollardenominated rate is 6%. The current exchange rate is 0.009 dollars per yen. The 1-year forward rate is
0.009e0.06−0.02 = 0.009367
Notice that equation (5.18) is just like equation (5.6), for stock index futures, with the foreign interest rate equal to the dividend yield. The interest rate difference r − ry is the cost of carry for a foreign currency (we borrow at the domestic rate r and invest the proceeds in a foreign money-market instrument, earning the foreign rate ry as an offset to our cost). If we wish to borrow foreign currency, ry is the lease rate.

Covered Interest Arbitrage
We can synthetically create a forward contract by borrowing in one currency and lending in the other. If we want to have 1 yen in the future, with the dollar price fixed today, we can pay today for the yen, and borrow in dollars to do so. To have 1 yen in 1 year, we need to invest x0e−ry T in dollars, and we obtain this amount by borrowing. The required dollar repayment is x0e(r−ry )T which is the forward exchange rate.
Example 5.6 Suppose that x0 = 0.009, ry = 2%, and r = 6%. The dollar cost of buying 1 yen today is 0.009 × e−0.02 = 0.008822. We defer the dollar payment by borrowing at 6%, for a cost 1 year from today of 0.008822e0.06 = 0.009367. This transaction is summarized in Table 5.13.
The example shows that borrowing in one currency and lending in another creates the same cash flow as a forward contract. If we offset this borrowing and lending position with an actual forward contract, the resulting transaction is called covered interest arbitrage.

20. Of course if you think about it, every currency transaction can be expressed in terms of either currency— for example, as yen/dollar or dollar/yen. If the forward price exceeds the current exchange rate viewed from the perspective of one currency, it must be less from the perspective of the other.

5.7 Eurodollar Futures

TABLE 5.13

Synthetically creating a yen forward contract by borrowing in dollars and lending in yen. The payoff at time 1 is ¥1 −
$0.009367.

Cash Flows
Year 0
Transaction
−ry

Borrow x0e dollar at 6% ($)
Convert to yen @ 0.009 $/¥
Invest in yen-denominated bill (¥)
Total

$

¥

+0.008822
−0.008822

0


+0.9802
−0.9802
0

Year 1
$
−0.009367


−0.009367

¥


1
1

A carry trade entails borrowing in the currency with low interest rates and lending in the currency with high interest rates, without hedging. The Box on p. 154 explains carry trades.
To summarize, a forward exchange rate reflects the difference in interest rates denominated in different currencies. Imagine that you want to invest $1 for 1 year. You can do so by buying a dollar-denominated bond, or you can exchange the dollar into another currency and buy a bond denominated in that other currency. You can then use currency forwards to guarantee the exchange rate at which you will convert the foreign currency back into dollars. The principle behind the pricing of currency forwards is that a position in foreign risk-free bonds, with the currency risk hedged, pays the same return as domestic risk-free bonds. 5.7 EURODOLLAR FUTURES
Businesses and individuals face uncertainty about future interest rates. A manager planning to borrow money 3 months from today doesn’t know today what the interest rate will be at that time. Forward and futures contracts permit hedging interest rate risk by allowing the manager to lock in now a borrowing rate for 3 months in the future.
The principles underlying interest rate contracts are exactly those we have been discussing, but interest rates seem more complicated because there are so many of them, depending upon whether you invest for 1 day, 1 month, 1 year, or 30 years. There are also implied forward interest rates between any two points in the future.21 Because of this complexity, Chapter 7 is devoted to interest rates. However, the Eurodollar contract is so important that we discuss it briefly here.
The Eurodollar contract, described in Figure 5.4, is based on a $1 million 3-month deposit earning LIBOR (the London Interbank Offer Rate), which is the average borrowing rate faced by large international London banks. The 1-month LIBOR contract is similar.
Suppose that current LIBOR is 1.5% over 3 months. By convention, this is annualized by

21. In addition, there are different rates faced by different classes of borrower: government, private, and municipal. And of course there are different currencies of denomination.

153

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Chapter 5. Financial Forwards and Futures

BOX

5.3: Carry Trades

Suppose that the yen interest rate is 2% and the dollar interest rate is 6%. On the surface, it might seem to you that it would be profitable to borrow at 2% in yen, and lend at 6% in dollars. This strategy of borrowing at a low rate and lending at a high rate is often undertaken in practice and is called a carry trade.
If you borrow yen and invest in dollars, you face the risk that the dollars will become less valuable
(the dollar will depreciate), or to say the same thing differently, that the yen will appreciate (the dollar price of a yen will increase). Thus, while a carry trade may superficially sound like a money machine, the trade has risk.
To illustrate the risk in a currency carry trade, we use the assumptions of Example 5.5. The yen/dollar exchange rate today is 0.009, and we suppose that in 1 year the rate can be
0.0091, 0.009367, or 0.0096. If we plan to invest
¥100,000, the trade entails borrowing $900/0.009
= ¥100,000 at 2% and lending $900 at 6%. Dollar profit at the three exchange rates is
.

x1 = 0.0091. The yen depreciates relative to the forward price, so the trade is profitable:
Profit = $900 × e0.06 − ¥100,000
× e0.02 × 0.0091 = $27.2697

.

x1 = 0.009367. The yen equals the forward price, so the trade breaks even:
Profit = $900 × e0.06 − ¥100,000 × e0.02
× 0.009367 = 0

.

x1 = 0.0096. The yen appreciates relative to the forward price, so the trade loses money:
Profit = $900 × e0.06 − ¥100,000 × e0.02
× 0.0096 = −$23.7404

This example illustrates what we already knew from studying covered interest arbitrage: The carry trade breaks even when the future exchange rate equals the forward rate. (Covered interest arbitrage adds currency futures to the carry trade, ensuring that we can buy yen at the forward price.)
The behavior of the exchange rate determines the profitability of the investment.
Carry trades can be undertaken with different asset classes, e.g., by borrowing short term and lending long term in the same currency. Currency carry trades are popular because historically they have been profitable. Burnside et al. (2008) show that this profitability (relative to risk) persists even when the investor buys options to protect against large losses. The persistent profitability of currency carry trades is surprising.

multiplying by 4, so the quoted LIBOR rate is 6%. Assuming a bank borrows $1 million for
3 months, a change in annualized LIBOR of 0.01% (1 basis point) would raise its borrowing cost by 0.0001/4 × $1 million = $25.
The Eurodollar futures price at expiration of the contract is
100 − Annualized 3-month LIBOR
Thus, if LIBOR is 6% at maturity of the Eurodollar futures contract, the final futures price will be 100 − 6 = 94. It is important to understand that the Eurodollar contract settles based on current LIBOR, which is the interest rate quoted for the next 3 months. Thus, for example, the price of the contract that expires in June reflects the 3-month interest rate between June and September. With the futures contract, as with a $1 million LIBOR deposit, a change of
0.01% in the rate is worth $25.

5.7 Eurodollar Futures

FIGURE 5.4
Specifications for the Eurodollar futures contract.

Where traded
Size
Months
Trading ends

Delivery

Chicago Mercantile Exchange
3-month Eurodollar time deposit, $1 million principal March, June, September, December, out 10 years, plus 2 serial months and spot month
5 a.m. (11 a.m. London) on the second
London bank business day immediately preceding the third Wednesday of the contract month.
Cash settlement based on 100 − British
Banker’s Association Futures Interest
Settlement Rate for 3-Month Eurodollar
Interbank Time Deposits. (This is a 3-month rate annualized by multiplying by 360/90.)

Like most money-market interest rates, LIBOR is quoted assuming a 360-day year.
Thus, the annualized 91-day rate, r91, can be extracted from the futures price, F , by computing the 90-day rate and multiplying by 91/90. The quarterly effective rate is then computed by dividing the result by 4: r91 = (100 − F ) ×

155

1 91
1
× ×
100 4 90

(5.19)

Three-month Eurodollar contracts have maturities out to 10 years, which means that it is possible to use the contract to lock in a 3-month rate as far as 10 years in the future.
If the Eurodollar futures contract maturing 60 months from today has a price of 95.25, this means that the contract can be used to lock in an annualized rate of 4.75% for a period beginning 60 months from today and terminating 63 months from today.
The Eurodollar futures strip (the set of futures prices with different maturities at one point in time) provides basic interest rate information that is commonly used to price other futures contracts and to price swaps. Figure 5.5 depicts the Eurodollar strip for four dates.
Each point on the graph is the annualized 3-month interest rate implied by the Eurodollar futures price maturing that many months in the future.
Let’s see how the Eurodollar contract can be used to hedge interest rate risk. For a borrower, for example, a short position in the contract is a hedge because it pays when the interest rate rises and requires payment when the interest rate falls. To see this, suppose that 7 months from today we plan to borrow $1 million for 90 days, and that our borrowing rate is the same as LIBOR. The Eurodollar futures price for 7 months from today is 94; this implies a 90-day rate of (100 − 94) × 90/360 × 1/100 = 1.5%. Now suppose that 7 months hence, 3-month LIBOR is 8%, which implies a Eurodollar futures price of 92. The implied 90-day rate is 2%. Our extra borrowing expense over 90 days on $1 million will therefore be (0.02 − 0.015) × $1m = $5000.
This extra borrowing expense is offset by gains on the short Eurodollar contract.
The Eurodollar futures price has gone down, giving us a gain of $25 per basis point, or

156

Chapter 5. Financial Forwards and Futures

FIGURE 5.5
Eurodollar futures price strip for four dates.

100 – Futures Price (%)
10

8

6

4
6/2/2004
6/7/2006
6/4/2008
6/2/2010

2

0
0

20

80
40
60
Months to Maturity

100

120

Source: Datastream.

$25 × 100 × (94 − 92) = $5000. The short position in the futures contract compensates us for the increase in our borrowing cost.22 In the same way, a long position can be used to lock in a lending rate.
The Eurodollar futures price is a construct, not the price of an asset. In this sense
Eurodollar futures are different from the futures contracts we have already discussed.
Although Eurodollar LIBOR is closely related to a number of other interest rates, there is no one specific identifiable asset that underlies the Eurodollar futures contract.
LIBOR is quoted in currencies other than dollars, and comparable rates are quoted in different locations. In addition to LIBOR, there are PIBOR (Paris), TIBOR (Tokyo), and
Euribor (the European Banking Federation).
Finally, you might be wondering why we are discussing LIBOR rather than rates on
Treasury bills. Business and bank borrowing rates move more in tandem with LIBOR than with the government’s borrowing rate. Thus, these borrowers use the Eurodollar futures contract to hedge. LIBOR is also a better measure of the cost of funds for a market-maker, so LIBOR is typically used to price forward contracts. We will further discuss Eurodollar futures in Chapter 7.

22. It might occur to you that the Eurodollar contract pays us at the time we borrow, but we do not pay interest until the loan matures, 91 days hence. Since we have time to earn interest on the change in the value of the contract, the hedge ratio should be less than 1 contract per $1 million borrowing. We discuss this complication in Chapter 7.

Chapter Summary

CHAPTER SUMMARY
The purchase of a stock or other asset entails agreeing to a price, making payment, and taking delivery of the asset. A forward contract fixes the price today, but payment and delivery are deferred. The pricing of forward contracts reflects the costs and benefits of this deferred payment and delivery. The seller receives payment later, so the price is higher to reflect interest owed the seller, and the buyer receives possession later, so the price is lower to reflect dividends not received by the buyer. A prepaid forward contract requires payment today; hence, it separates these two effects. The price of a prepaid forward is
Prepaid forward price = S0e−δT
The prepaid forward price is below the asset spot price, S0, due to dividends forgone by deferring delivery. The forward price also reflects deferral of payment, so it is the future value of the prepaid forward price:
Forward price = S0e(r−δ)T
In the case of a currency forward, the dividend yield forgone by holding the forward contract instead of the underlying asset, δ, is the interest rate you could earn by investing in foreigncurrency denominated assets. Thus, for currencies, δ = rf , where rf is the foreign interest rate. A forward contract is equivalent to a leveraged position in an asset—borrowing to buy the asset. By combining the forward contract with other assets it is possible to create synthetic stocks and bonds. These equivalents are summarized in Table 5.14. Since a forward contract is risky but requires no investment, it earns the risk premium. The forward price is therefore a biased predictor of the future spot price of the asset, with the bias equal to the risk premium.
The fact that it is possible to create a synthetic forward has two important implications.
First, if the forward contract is mispriced, arbitrageurs can take offsetting positions in the forward contract and the synthetic forward contract—in effect buying low and selling high— and make a risk-free profit. Second, dealers who make markets in the forward or in the underlying asset can hedge the risk of their position with a synthetic offsetting position.
With transaction costs there is a no-arbitrage region rather than a single no-arbitrage price.
Futures contracts are similar to forward contracts, except that with futures there are margin requirements and daily settlement of the gain or loss on the position. The contractual

TABLE 5.14

Synthetic equivalents assuming the asset pays continuous dividends at the rate δ.

Position

Synthetic Equivalent

Long forward
= Buy e−δT shares of stock + Borrow S0e−δT
Bond paying F0, T = Buy e−δT shares of stock + Short forward
Synthetic stock = Long forward
+ Lend e−rT F0, T

157

158

Chapter 5. Financial Forwards and Futures

differences between forwards and futures can lead to pricing differences, but in most cases forward prices and futures prices are very close.
In addition to hedging, forward and futures contracts can be used to synthetically switch a portfolio invested in stocks into bonds. A portfolio invested in Asset A can remain invested in Asset A but earn the returns associated with Asset B, as long as there are forward or futures contracts on A and B. This is called a futures overlay.
The Eurodollar futures contract, based on LIBOR (London Interbank Offer Rate) is widely used for hedging interest rate risk. Because the Eurodollar futures contract does not represent the price of an asset (at settlement it is 100 − LIBOR), it cannot be priced using the formulas in this chapter.

FURTHER READING
Chapter 6 continues our exploration of forward markets by considering commodity forwards, which are different from financial forwards in important ways. Chapter 7 then examines interest rate forwards. Whereas forward contracts provide a price for delivery at one point in time, swaps, discussed in Chapter 8, provide a price for a series of deliveries over time. Swaps are a natural generalization of forward contracts.
The pricing principles discussed in this chapter will also play important roles when we discuss option pricing in Chapters 10, 11, and 12 and financial engineering in Chapter 15.
To get a sense of the range of traded contracts, explore the websites of futures exchanges: the CME Group (www.cmegroup.com) and the London International Financial
Futures Exchange (www.euronext.com), among others. These sites typically provide current prices, along with information about the contracts: the definition of the underlying asset, how the contracts are settled, and so forth. The site for One Chicago (www.onechicago.com) provides information about single stock futures in the United States.
It is well accepted that forward prices are determined by the models and considerations in this chapter. Early papers that examined futures pricing include Modest and Sundaresan
(1983), Cornell and French (1983), which emphasizes tax effects in futures pricing (see
Appendix 5.A), and French (1983), which compares forwards and futures when both exist on the same underlying asset. Brennan and Schwartz (1990) explore optimal arbitrage when there are transaction costs, and Reinganum (1986) explores the arbitrage possibilities inherent in time travel. There is a more technical academic literature focusing on the difference between forward and futures contracts, including Black (1976a), Cox et al.
(1981), Richard and Sundaresan (1981), and Jarrow and Oldfield (1981).

PROBLEMS
5.1 Construct Table 5.1 from the perspective of a seller, providing a descriptive name for each of the transactions.
5.2 A $50 stock pays a $1 dividend every 3 months, with the first dividend coming 3 months from today. The continuously compounded risk-free rate is 6%.
a. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend?
b. What is the price of a forward contract that expires at the same time?

Problems

5.3 A $50 stock pays an 8% continuous dividend. The continuously compounded riskfree rate is 6%.
a. What is the price of a prepaid forward contract that expires 1 year from today?
b. What is the price of a forward contract that expires at the same time?
5.4 Suppose the stock price is $35 and the continuously compounded interest rate is 5%.
a. What is the 6-month forward price, assuming dividends are zero?
b. If the 6-month forward price is $35.50, what is the annualized forward premium? c. If the forward price is $35.50, what is the annualized continuous dividend yield? 5.5 Suppose you are a market-maker in S&R index forward contracts. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0.
a. What is the no-arbitrage forward price for delivery in 9 months?
b. Suppose a customer wishes to enter a short index futures position. If you take the opposite position, demonstrate how you would hedge your resulting long position using the index and borrowing or lending.
c. Suppose a customer wishes to enter a long index futures position. If you take the opposite position, demonstrate how you would hedge your resulting short position using the index and borrowing or lending.
5.6 Repeat the previous problem, assuming that the dividend yield is 1.5%.
5.7 The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0.
a. Suppose you observe a 6-month forward price of 1135. What arbitrage would you undertake?
b. Suppose you observe a 6-month forward price of 1115. What arbitrage would you undertake?
5.8 The S&R index spot price is 1100, the risk-free rate is 5%, and the continuous dividend yield on the index is 2%.
a. Suppose you observe a 6-month forward price of 1120. What arbitrage would you undertake?
b. Suppose you observe a 6-month forward price of 1110. What arbitrage would you undertake?
5.9 Suppose that 10 years from now it becomes possible for money managers to engage in time travel. In particular, suppose that a money manager could travel to January
1981, when the 1-year Treasury bill rate was 12.5%.
a. If time travel were costless, what riskless arbitrage strategy could a money manager undertake by traveling back and forth between January 1981 and
January 1982?
b. If many money managers undertook this strategy, what would you expect to happen to interest rates in 1981?

159

160

Chapter 5. Financial Forwards and Futures

c. Since interest rates were 12.5% in January 1981, what can you conclude about whether costless time travel will ever be possible?
5.10 The S&R index spot price is 1100 and the continuously compounded risk-free rate is 5%. You observe a 9-month forward price of 1129.257.
a. What dividend yield is implied by this forward price?
b. Suppose you believe the dividend yield over the next 9 months will be only
0.5%. What arbitrage would you undertake?
c. Suppose you believe the dividend yield will be 3% over the next 9 months.
What arbitrage would you undertake?
5.11 Suppose the S&P 500 index futures price is currently 1200. You wish to purchase four futures contracts on margin.
a. What is the notional value of your position?
b. Assuming a 10% initial margin, what is the value of the initial margin?
5.12 Suppose the S&P 500 index is currently 950 and the initial margin is 10%. You wish to enter into 10 S&P 500 futures contracts.
a. What is the notional value of your position? What is the margin?
b. Suppose you earn a continuously compounded rate of 6% on your margin balance, your position is marked to market weekly, and the maintenance margin is 80% of the initial margin. What is the greatest S&P 500 index futures price 1 week from today at which you will receive a margin call?
5.13 Verify that going long a forward contract and lending the present value of the forward price creates a payoff of one share of stock when
a. The stock pays no dividends.
b. The stock pays discrete dividends.
c. The stock pays continuous dividends.
5.14 Verify that when there are transaction costs, the lower no-arbitrage bound is given by equation (5.12).
5.15 Suppose the S&R index is 800, and that the dividend yield is 0. You are an arbitrageur with a continuously compounded borrowing rate of 5.5% and a continuously compounded lending rate of 5%. Assume that there is 1 year to maturity.
a. Supposing that there are no transaction fees, show that a cash-and-carry arbitrage is not profitable if the forward price is less than 845.23, and that a reverse cash-and-carry arbitrage is not profitable if the forward price is greater than 841.02.
b. Now suppose that there is a $1 transaction fee, paid at time 0, for going either long or short the forward contract. Show that the upper and lower no-arbitrage bounds now become 846.29 and 839.97.
c. Now suppose that in addition to the fee for the forward contract, there is also a $2.40 fee for buying or selling the index. Suppose the contract is settled by

5.A Taxes and the Forward Rate

delivery of the index, so that this fee is paid only at time 0. What are the new upper and lower no-arbitrage bounds?
d. Make the same assumptions as in the previous part, except assume that the contract is cash-settled. This means that it is necessary to pay the stock index transaction fee (but not the forward fee) at both times 0 and 1. What are the new no-arbitrage bounds?
e. Now suppose that transactions in the index have a fee of 0.3% of the value of the index (this is for both purchases and sales). Transactions in the forward contract still have a fixed fee of $1 per unit of the index at time 0. Suppose the contract is cash-settled so that when you do a cash-and-carry or reverse cashand-carry you pay the index transaction fee both at time 1 and time 0. What are the new upper and lower no-arbitrage bounds? Compare your answer to that in the previous part. (Hint: To handle the time 1 transaction fee, you may want to consider tailing the stock position.)
5.16 Suppose the S&P 500 currently has a level of 875. The continuously compounded return on a 1-year T-bill is 4.75%. You wish to hedge an $800,000 portfolio that has a beta of 1.1 and a correlation of 1.0 with the S&P 500.
a. What is the 1-year futures price for the S&P 500 assuming no dividends?
b. How many S&P 500 futures contracts should you short to hedge your portfolio? What return do you expect on the hedged portfolio?
5.17 Suppose you are selecting a futures contract with which to hedge a portfolio. You have a choice of six contracts, each of which has the same variability, but with correlations of −0.95, −0.75, −0.50, 0, 0.25, and 0.85. Rank the futures contracts with respect to basis risk, from highest to lowest basis risk.
C
5.18 Suppose the current exchange rate between Germany and Japan is 0.02 = /¥. The euro-denominated annual continuously compounded risk-free rate is 4% and the yen-denominated annual continuously compounded risk-free rate is 1%. What are the 6-month euro/yen and yen/euro forward prices?
5.19 Suppose the spot $/¥ exchange rate is 0.008, the 1-year continuously compounded dollar-denominated rate is 5% and the 1-year continuously compounded yendenominated rate is 1%. Suppose the 1-year forward exchange rate is 0.0084. Explain precisely the transactions you could use (being careful about currency of denomination) to make money with zero initial investment and no risk. How much do you make per yen? Repeat for a forward exchange rate of 0.0083.
5.20 Suppose we wish to borrow $10 million for 91 days beginning next June, and that the quoted Eurodollar futures price is 93.23.
a. What 3-month LIBOR rate is implied by this price?
b. How much will be needed to repay the loan?

Appendix 5.A TAXES AND THE FORWARD RATE
Appendix available at http://www.pearsonhighered.com/mcdonald.

161

162

Chapter 5. Financial Forwards and Futures

TABLE 5.15

Week
0
1
2
3
4
5
6
7
8
9
10

Marking-to-market proceeds and margin balance from long position in the S&P 500 futures contract, where hedge is adjusted on a weekly basis.

Multiplier ($)

Futures Price

Price Change

Margin Balance ($)

1979.34
1981.62
1983.91
1986.20
1988.49
1990.79
1993.09
1995.39
1997.69
2000.00
2000.00

1100.00
1027.99
1037.88
1073.23
1048.78
1090.32
1106.94
1110.98
1024.74
1007.30
1011.65


−72.01
9.89
35.35
−24.45
41.54
16.62
4.04
−86.24
−17.44
4.35

217,727.21
75,446.43
95,131.79
165,372.88
117,001.17
199,738.33
233,055.86
241,377.01
69,573.25
34,813.80
43,553.99

Appendix 5.B EQUATING FORWARDS AND FUTURES
Because the futures price exceeds the prepaid forward price, marking-to-market has the effect of magnifying gains and losses. For example, the futures price on a non-dividendpaying stock is F0, T = S0erT . If the stock price increases by $1 at time 0, the gain on the futures contract at time T is erT . Thus, in order to use futures to precisely hedge a position
(with the hedge being settled at time T ) it is necessary to hold fewer futures than forward contracts, effectively offsetting the extra volatility induced by the future value factor. In the example in Table 5.15, we can go long fewer than eight contracts, to make up for the effect of marking-to-market.
Table 5.15 shows the effect of this adjustment to the futures position and how it is adjusted over time. Initially, we go long
8 × e−0.06×9/52 = 7.91735 contracts. This number of contracts has a multiplier of $250 × 7.91735 = $1979.34, the multiplier in the first row of the table. Reducing the number of contracts offsets the effect of earning interest. Each week there is less time until expiration, so we increase the number of index units we are long.
Profit on this position is
$43,553.99 − $217,727.21e0.06×10/52 = −$176,700 which is exactly the same profit as a forward position. The example in Table 5.15 is unrealistic in the sense that the magnitude is too small for the adjustment to be worth the bother. However, it does demonstrate how to scale the position to offset the magnifying effect of marking-to-market, and the link between the profit on a forward and futures position.

Appendix 5.C FORWARD AND FUTURES PRICES
Appendix available at http://www.pearsonhighered.com/mcdonald.

6

Commodity Forwards and Futures

T

olstoy observed that happy families are all alike; each unhappy family is unhappy in its own way. An analogous idea in financial markets is that financial forwards are all alike; each commodity, however, has unique economic characteristics that determine forward pricing in that market. In this chapter we will see the extent to which commodity forwards on different assets differ from each other, and also how they differ from financial forwards and futures.
We first discuss the pricing of commodity contracts, and then examine specific contracts, including gold, corn, natural gas, and oil. Finally, we discuss hedging.
You might wonder about the definition of a commodity. Gerard Debreu, who won the
Nobel Prize in economics, said this (Debreu, 1959, p. 28):
A commodity is characterized by its physical properties, the date at which it will be available, and the location at which it will be available. The price of a commodity is the amount which has to be paid now for the (future) availability of one unit of that commodity. Notice that with this definition, corn in July and corn in September, for example, are different commodities: They are available on different dates. With a financial asset, such as a stock, we think of the stock as being fundamentally the same asset over time.1 The same is not necessarily true of a commodity, since it can be costly or impossible to transform a commodity on one date into a commodity on another date. This observation will be important. In our discussion of forward pricing for financial assets we relied heavily on the fact that the price of a financial asset today is the present value of the asset at time T , less the value of dividends to be received between now and time T . It follows that the difference between the forward price and spot price of a financial asset reflects the costs and benefits of delaying payment for, and receipt of, the asset. Specifically, the forward price on a financial asset is given by
F0, T = S0e(r−δ)T

(6.1)

1. When there are dividends, however, a share of stock received on different dates can be materially different. 163

164

Chapter 6. Commodity Forwards and Futures

where S0 is the spot price of the asset, r is the continuously compounded interest rate, and δ is the continuous dividend yield on the asset. We will explore the extent to which equation
(6.1) also holds for commodities.

6.1 INTRODUCTION TO COMMODITY FORWARDS
This section provides an overview of some issues that arise in discussing commodity forward and futures contracts. We begin by looking at some commodity futures prices. We then discuss some terms and concepts that will be important for commodities.

Examples of Commodity Futures Prices
For many commodities there are futures contracts available that expire at different dates in the future. Table 6.1 provides illustrative examples; we can examine these prices to see what issues might arise with commodity forward pricing.
First, consider corn. From May to July, the corn futures price rises from 646.50 to 653.75. This is a 2-month increase of 653.75/646.50 − 1 = 1.12%, an annual rate of approximately 7%. As a reference interest rate, 3-month LIBOR on March 17, 2011, was
0.31%, or about 0.077% for 3 months. Assuming that δ ≥ 0, this futures price is greater than that implied by equation (6.1). The discussion in Chapter 5 would suggest an arbitrage strategy: Buy May corn and sell July corn. However, storing corn for 2 months will be costly, a consideration that did not arise with financial futures. Another issue arises with the
December price: The price of corn falls 74.5 cents between July and December. It seems unlikely that this could be explained by a dividend. An alternative, intuitive explanation would be that the fall harvest causes the price of corn to drop, and hence the December

TABLE 6.1

Futures prices for various commodities, March 17, 2011.

Expiration
Month

Corn
(cents/
bushel)

Soybeans
(cents/
bushel)

Gasoline
(cents/
gallon)

Oil (Brent)
(dollars/
barrel)

Gold
(dollars/
ounce)

April
May
June
July
August
September
October
November
December


646.50

653.75

613.00


579.25


1335.25

1343.50

1321.00

1302.25


2.9506
2.9563
2.9491
2.9361
2.8172
2.8958
2.7775
2.7522
2.6444


114.90
114.65
114.38
114.11
113.79
113.49
113.17
112.85

1404.20
1404.90
1405.60

1406.90

1408.20

1409.70
Data from CME Group.

6.1 Introduction to Commodity Forwards

165

futures price is low. But how is this explanation consistent with our results about no-arbitrage pricing of financial forwards?
If you examine the other commodities, you will see similar patterns for soybeans, gasoline, and oil. Only gold, with the forward price rising at approximately $0.70 per month
(about 0.6% annually), has behavior resembling that of a financial contract.
The prices in Table 6.1 suggest that commodities are different than financial contracts.
The challenge is to reconcile the patterns with our understanding of financial forwards, in which explicit expectations of future prices (and harvests!) do not enter the forward price formula. There are many more commodities with traded futures than just those in Table 6.1.
You might think that a futures contract could be written on anything, but it is an interesting bit of trivia, discussed in the box below, that Federal law in the United States prohibits trading on two commodities.

BOX

6.1: Forbidden Futures

I

n the United States, futures contracts on two items are explicitly prohibited by statute: onions and box office receipts for movies. Title 7, Chapter 1, §13-1 of the United States Code is titled
“Violations, prohibition against dealings in onion futures; punishment” and states
(a) No contract for the sale of onions for future delivery shall be made on or subject to the rules of any board of trade in the United States. The terms used in this section shall have the same meaning as when used in this chapter.
(b) Any person who shall violate the provisions of this section shall be deemed guilty of a misdemeanor and upon conviction thereof be fined not more than $5,000.
Along similar lines, Title VII of the Dodd-Frank
Wall Street Reform and Consumer Protection Act of 2010 bans trading in “motion picture box office receipts (or any index, measure, value, or data related to such receipts), and all services, rights, and interests . . . in which contracts for future delivery are presently or in the future dealt in.”
These bans exist because of lobbying by special interests. The onion futures ban was passed in
1959 when Michigan onion growers lobbied their new congressman, Gerald Ford, to ban such

trading, believing that it depressed prices. Today, some regret the law:
Onion prices soared 400% between October
2006 and April 2007, when weather reduced crops, according to the U.S. Department of
Agriculture, only to crash 96% by March 2008 on overproduction and then rebound 300% by this past April.
The volatility has been so extreme that the son of one of the original onion growers who lobbied Congress for the trading ban now thinks the onion market would operate more smoothly if a futures contract were in place.
“There probably has been more volatility since the ban,” says Bob Debruyn of Debruyn
Produce, a Michigan-based grower and wholesaler. “I would think that a futures market for onions would make some sense today, even though my father was very much involved in getting rid of it.”
Source: Fortune magazine on-line, June 27, 2008.

Similarly, futures on movie box office receipts had been approved early in 2010 by the Commodity Futures Trading Commission. After lobbying by Hollywood interests, the ban on such trading was inserted into the Dodd-Frank financial reform bill. 166

Chapter 6. Commodity Forwards and Futures

Differences Between Commodities and Financial Assets
In discussing the commodity prices in Table 6.1, we invoked considerations that did not arise with financial assets, but that will arise repeatedly when we discuss commodities. Among these are:
Storage costs. The cost of storing a physical item such as corn or copper can be large relative to its value. Moreover, some commodities deteriorate over time, which is also a cost of storage. By comparison, financial securities are inexpensive to store.
Consequently, we did not mention storage costs when discussing financial assets.
Carry markets. A commodity for which the forward price compensates a commodity owner for costs of storage is called a carry market. (In such a market, the return on a cash-and-carry, net of all costs, is the risk-free rate.) Storage of a commodity is an economic decision that varies across commodities and that can vary over time for a given commodity. Some commodities are at times stored for later use (we will see that this is the case for natural gas and corn), others are more typically used as they are produced (oil, copper). By contrast, financial markets are always carry markets:
Assets are always “stored” (owned), and forward prices always compensate owners for storage.
Lease rate. The short-seller of an item may have to compensate the owner of the item for lending. In the case of financial assets, short-sellers have to compensate lenders for missed dividends or other payments accruing to the asset. For commodities, a shortseller may have to make a payment, called a lease payment, to the commodity lender.
The lease payment typically would not correspond to dividends in the usual sense of the word.
Convenience yield. The owner of a commodity in a commodity-related business may receive nonmonetary benefits from physical possession of the commodity. Such benefits may be reflected in forward prices and are generically referred to as a convenience yield.
We will discuss all of these concepts in more depth later in the chapter. For now, the important thing to keep in mind is that commodities differ in important respects from financial assets.

Commodity Terminology
There are many terms that are particular to commodities and thus often unfamiliar even to those well acquainted with financial markets. These terms deal with the properties of the forward curve and the physical characteristics of commodities.
Table 6.1 illustrates two terms often used by commodity traders in talking about forward curves: contango and backwardation. If the forward curve is upward sloping—
i.e., forward prices more distant in time are higher—then we say the market is in contango.
We observe this pattern with near-term corn and soybeans, and with gold. If the forward curve is downward sloping, we say the market is in backwardation. We observe this with medium-term corn and soybeans, with gasoline (after 2 months), and with crude oil.
Commodities can be broadly classified as extractive and renewable. Extractive commodities occur naturally in the ground and are obtained by mining and drilling. Examples

6.2 Equilibrium Pricing of Commodity Forwards

include metals (silver, gold, and copper) and hydrocarbons, including oil and natural gas.
Renewable commodities are obtained through agriculture and include grains (corn, soybeans, wheat), livestock (cattle, pork bellies), dairy (cheese, milk), and lumber.
Commodities can be further classified as primary and secondary. Primary commodities are unprocessed; corn, soybeans, oil, and gold are all primary. Secondary commodities have been processed. In Table 6.1, gasoline is a secondary commodity.
Finally, commodities are measured in uncommon units for which you may not know precise definitions. Table 6.1 has several examples. A barrel of oil is 42 gallons. A bushel is a dry measure containing approximately 2150 cubic inches. The ounce used to weigh precious metals, such as gold, is a troy ounce, which is approximately 9.7% greater in weight than the customary avoirdupois ounce.2
Entire books are devoted to commodities (e.g., see Geman, 2005). Our goal here is to understand the logic of forward pricing for commodities and where it differs from the logic of forward pricing for financial assets. We will see that understanding a forward curve generally requires that we understand something about the underlying commodity.

6.2 EQUILIBRIUM PRICING OF COMMODITY FORWARDS
In this section we present definitions relating the prepaid forward price, forward price, and present value of a future commodity price. The same equations appeared in Chapter 5, but the ideas are important for understanding commodities, so we repeat them here.
The prepaid forward price for a commodity is the price today to receive a unit of the commodity on a future date. The prepaid forward price is therefore by definition the present value of the commodity on the future date. Hence, the prepaid forward price is
P
F0, T = e−αT E0[ST ]

(6.2)

where α is the discount rate for the commodity.
The forward price is the future value of the prepaid forward price, with the future value computed using the risk-free rate:
P
F0, T = erT F0, T

(6.3)

Substituting equation (6.2) into equation (6.3), we see that the commodity forward price is the expected spot price, discounted at the risk premium (this is the same as equation (5.9)):
F0, T = E0(ST )e−(α−r)T

(6.4)

We can rewrite equation (6.4) to obtain e−rT F0, T = E0(ST )e−αT

(6.5)

Equation (6.5) deserves emphasis: The time-T forward price discounted at the risk-free rate is the present value of a unit of commodity received at time T . This equation implies that, for example, an industrial producer who buys oil can calculate the present value of future

2. A troy ounce is 480 grains and the more familiar avoirdupois ounce is 437.5 grains. Twelve troy ounces make 1 troy pound, which weighs approximately 0.37 kg.

167

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Chapter 6. Commodity Forwards and Futures

oil costs by discounting oil forward prices at the risk-free rate. This calculation does not depend upon whether the producer hedges. We will see an example of this calculation later in the chapter.

6.3 PRICING COMMODITY FORWARDS BY ARBITRAGE
We now investigate no-arbitrage pricing for commodity forward contracts. We begin by using copper as an example. Copper is durable and can be stored, but it is typically not stored except as needed for production. The primary goal in this section will be to understand the issues that distinguish forward pricing for commodities from forward pricing for financial assets. Figure 6.1 shows specifications for the CME Group copper contract and Figure 6.2 shows forward curves for copper on four dates. The copper forward curve lacks drama: For three of the four curves, the forward price in 1 year is approximately equal to the forward price in the current month. For the fourth curve, the 1-year price is below the current price
(the curve exhibits backwardation).
We saw that for non-dividend-paying financial assets, the forward price rises at the interest rate. How can the forward price of copper on a future date equal the current forward price? At an intuitive level, it is reasonable to expect the price of copper in 1 year to equal the price today. Suppose, for example, that the extraction and other costs of copper production are $3/pound and are expected to remain $3. If demand is not expected to change, or if it is easy for producers to alter production, it would be reasonable to expect that on average the price of copper would remain at $3. The question is how to reconcile this intuition with the behavior of forward prices for financial assets.
While it is reasonable to think that the price of copper will be expected to remain the same over the next year, it is important to recognize that a constant price would not be a reasonable assumption about the price of a non-dividend-paying stock. Investors must expect that a stock will on average pay a positive return, or no one would own it. In equilibrium, stocks and other financial assets must be held by investors, or stored. The stock price appreciates on average so that investors will willingly store the stock. There is no such requirement for copper, which can be extracted and then used in production. The equilibrium condition for copper relates to extraction, not to storage above ground. This distinction between a storage and production equilibrium is a central concept in our discussion of commodities. At the outset, then, there is an obvious difference between copper and a financial asset. It is not necessarily obvious, however, what bearing this difference has on pricing forward contracts.

An Apparent Arbitrage
Suppose that you observe that both the current price and 1-year forward price for copper are $3.00 and that the effective annual interest rate is 10%. For the reasons we have just discussed, market participants could rationally believe that the copper price in 1 year will be $3.00. From our discussion of financial forwards, however, you might think that the forward price should be 1.10 × $3.00 = $3.30, the future value of the current copper price.

6.3 Pricing Commodity Forwards by Arbitrage

FIGURE 6.1
Specifications for the CME
Group/COMEX high-grade copper contract.

Underlying
Where traded
Size
Months
Trading ends
Delivery

169

High-grade (Grade 1) copper
CME Group/COMEX
25,000 pounds
24 consecutive months
Third-to-last business day of the maturing month Exchange-designated warehouse within the
United States
Data from Datastream.

FIGURE 6.2
Forward curves for four dates for the CME Group high-grade copper futures contract. Futures Price (¢/lb)
400

300

200

6/2/2004
6/7/2006
6/4/2008
6/2/2010

100

0
0

5

10
15
Months to Maturity

20

Data from Datastream.

The $3.00 forward price would therefore create an arbitrage opportunity.3 Following the logic in Chapter 5, if the forward price were $3.00 you could buy copper forward and shortsell copper today. Table 6.2 depicts the cash flows in this reverse cash-and-carry arbitrage.

3. We will discuss arbitrage in this section focusing on the forward price relative to the spot price. However, the difference between any forward prices at different dates must also reflect no-arbitrage conditions. So you can apply the discussions in this section to any two points on the forward curve.

170

Chapter 6. Commodity Forwards and Futures

TABLE 6.2

Apparent reverse cash-and-carry arbitrage for copper if the copper forward price is F0, 1 < $3.30. These calculations appear to demonstrate that there is an arbitrage opportunity if the copper forward price is below $3.30. S1 is the spot price of copper in 1 year, and F0, 1 is the copper forward price. There is a logical error in the table.

Transaction

Cash Flows
Time 0
Time 1

Long forward @ F0, 1
Short-sell copper
Lend short-sale proceeds @ 10%
Total

0
+$3.00
−$3.00
0

S1 − F0, 1
−S1
$3.30
$3.30 − F0, 1

The result seems to show that there is an arbitrage opportunity for any copper forward price below $3.30. If the copper forward price is $3.00, it seems that you make a profit of $0.30 per pound of copper.
We seem to be stuck. Common sense suggests that a forward price of $3.00 would be reasonable, but the transactions in Table 6.2 imply that any forward price less than $3.30 leads to an arbitrage opportunity, where we would earn $3.30 − F0, 1 per pound of copper.
If you are puzzled, you should stop and think before proceeding. There is a problem with Table 6.2.
The arbitrage assumes that you can short-sell copper by borrowing it today and returning it in a year. However, in order for you to short-sell for a year, there must be an investor willing to lend copper for that period. The lender must both be holding the asset and willing to give up physical possession for the period of the short-sale. A lender in this case will think: “I have spent $3.00 for copper. Copper that I lend will be returned in 1 year. If copper at that time sells for $3.00, then I have earned zero interest on my
$3.00 investment. If I hedge by selling copper forward for $3.00, I will for certain earn zero interest, having bought copper for $3.00 and then selling it for $3.00 a year later.”
Conversely, from the perspective of the short-seller, borrowing a pound of copper for a year is an arbitrage because it is an interest-free loan of $3.00. The borrower benefits and the lender loses, so no one will lend copper without charging an additional fee. While it is straightforward to borrow a financial asset, borrowing copper appears to be a different matter. To summarize: The apparent arbitrage in Table 6.2 has nothing to do with mispriced forward contracts on copper. The issue is that the copper loan is equivalent to an interest-free loan, and thus generates an arbitrage profit.

Short-selling and the Lease Rate
How do we correct the arbitrage analysis in Table 6.2? We have to recognize that the copper lender has invested $3.00 in copper and must expect to earn a satisfactory return on that investment. The copper lender will require us to make a lease payment so that the commodity

6.3 Pricing Commodity Forwards by Arbitrage

TABLE 6.3

Reverse cash-and-carry arbitrage for copper. This table demonstrates that there is no arbitrage opportunity if the commodity lender requires an appropriate lease payment.

Transaction

Time 0

Long forward @ F0, 1
Short-sell copper
Lease payment
Lend short-sale proceeds @ 10%
Total

Cash Flows
Time 1

0
+$3.00
0
−$3.00
0

S1 − F0, 1
−S1
−($3.30 − F0, 1)
$3.30
0

loan is a fair deal. The actual payment the lender requires will depend on the forward price.
The lender will recognize that it is possible to use the forward market to lock in a selling price for the copper in 1 year, and will reason that copper bought for $3.00 today can be sold for F0, 1 in 1 year. A copper borrower must therefore be prepared to make an extra payment—a lease payment—of
Lease payment = 1.1 × $3.00 − F0, 1
With the lender requiring this extra payment, we can correct the analysis in Table 6.2. Table
6.3 incorporates the lease payment and shows that the apparent arbitrage vanishes.
We can also interpret a lease payment in terms of discounted cash flow. Let α denote the equilibrium discount rate for an asset with the same risk as the commodity. The lender is buying the commodity for S0. One unit returned at time T is worth ST , with a present value of E0(ST )e−αT . If there is a proportional continuous lease payment of δl , the NPV of buying the commodity and lending it is
NPV = E0(ST )e−αT eδl T − S0

(6.6)

The lease rate that makes NPV zero is then δl = α −

1 ln E0(ST )/S0
T

The lease rate is the difference between the discount rate for the commodity and the expected price appreciation. From substituting equation (6.5) into this expression, an equivalent way to write the continuous lease rate is δl = r −

1 ln F0, T /S0
T

(6.7)

It is important to be clear about the reason a lease payment is required for a commodity and not for a financial asset. For a non-dividend-paying financial asset, the price is the present value of the future price, so that S0 = E(ST )e−αT . This implies that the lease payment is zero. For most commodities, the current price is not the present value of the expected future price, so there is no presumption that the lease rate would be zero.

171

172

Chapter 6. Commodity Forwards and Futures

No-Arbitrage Pricing Incorporating Storage Costs
We now consider the effects of storage costs. Storage is not always feasible (for example, fresh strawberries are perishable), and when technically feasible, storage for commodities is almost always costly. If storage is feasible, how do storage costs affect forward pricing?
The intuitive answer is that if it is optimal to store the commodity, then the forward price must be high enough so that the returns on a cash-and-carry compensate for both financing and storage costs. However, if storage is not optimal, storage costs are irrelevant. We will examine both cash-and-carry and reverse cash-and-carry arbitrages to see how they are affected by storage costs.
Cash-and-Carry Arbitrage. Put yourself in the position of a commodity merchant who owns one unit of the commodity, and ask whether you would be willing to store it until time T . You face the choice of selling it today, receiving S0, or selling it at time T . If you guarantee your selling price by selling forward, you will receive F0, T .
It is common sense that you will store only if the present value of selling at time T is at least as great as that of selling today. Denote the future value of storage costs for one unit of the commodity from time 0 to T as λ(0, T ). Table 6.4 summarizes the cash flows for a cash-and-carry with storage costs. The table shows that the cash-and-carry arbitrage is not profitable if
F0, 1 < (1 + R)S0 + λ(0, 1)

(6.8)

If inequality (6.8) is violated, storage will occur because the forward premium is great enough that sale proceeds in the future compensate for the financial costs of storage (RS0) and the physical costs of storage (λ(0, 1)). If there is to be both storage and no arbitrage, then equation (6.8) holds with equality. An implication of equation (6.8) is that when costly storage occurs, the forward curve can rise faster than the interest rate. We can view storage costs as a negative dividend: Instead of receiving cash flow for holding the asset, you have to pay to hold the asset. If there is storage, storage costs increase the upper bound for the forward price. Storage costs can include depreciation of the commodity, which is less a problem for metals such as copper than for commodities such as strawberries and electricity.

TABLE 6.4

Cash-and-carry for copper for 1 year, assuming that there is a 1-year storage cost of λ(0, 1) payable at time 1, and an effective interest rate of R.

Transaction
Buy copper
Pay storage cost
Short forward
Borrow @ R
Total

Time 0
−S0
0
0
+S0
0

Cash Flows
Time 1
S1
−λ(0, 1)
F0, 1 − S1
−(1 + R)S0
F0, 1 − [(1 + R)S0 + λ(0, 1)]

6.3 Pricing Commodity Forwards by Arbitrage

In the special case where continuous storage costs of λ are paid continuously and are proportional to the value of the commodity, storage cost is like a continuous negative dividend. If storage occurs and there is no arbitrage, we have4
F0, T = S0e(r+λ)T

(6.9)

This would be the forward price in a carry market, where the commodity is stored.
Example 6.1 Suppose that the November price of corn is $2.50/bushel, the effective monthly interest rate is 1%, and storage costs per bushel are $0.05/month. Assuming that corn is stored from November to February, the February forward price must compensate owners for interest and storage. The future value of storage costs is
$0.05 + ($0.05 × 1.01) + ($0.05 × 1.012) = ($0.05/.01) × (1 + 0.01)3 − 1
= $0.1515
Thus, the February forward price will be
2.50 × (1.01)3 + 0.1515 = 2.7273
Problem 6.9 asks you to verify that this is a no-arbitrage price.
Keep in mind that just because a commodity can be stored does not mean that it should
(or will) be stored. Copper is typically not stored for long periods, because storage is not economically necessary: A constant new supply of copper is available to meet demand.
Thus, equation (6.8) describes the forward price when storage occurs. We now consider a reverse cash-and-carry arbitrage to see what happens when the forward price is lower than in equation (6.8).
Reverse Cash-and-Carry Arbitrage. Suppose an arbitrageur buys the commodity forward and short-sells it. We have seen that the commodity lender likely requires a lease payment and that the payment should be equal to (1 + R)S0 − F0, 1. The results of this transaction are in Table 6.5. Note first that storage costs do not affect profit because neither the arbitrageur nor the lender is actually storing the commodity. The reverse cash-and-carry is profitable if the lender requires a lease payment below (1 + R)S0 − F0, 1. Otherwise, arbitrage is not profitable. If the commodity lender uses the forward price to determine the lease rate, then the resulting circularity guarantees that profit is zero. This is evident in Table
6.5, where profit is zero if L = (1 + R)S0 − F0, 1.
This analysis has the important implication that the ability to engage in a reverse cashand-carry arbitrage does not put a lower bound on the forward price. We conclude that a forward price that is too high can give rise to arbitrage, but a forward price that is too low need not.
Of course there are economic pressures inducing the forward price to reach the
“correct” level. If the forward price is too low, there will be an incentive for arbitrageurs

4. You might be puzzled by the different ways of representing quantities such as costs and dividends. In some cases we have used discrete values; in others, we have used continuous approximations. All of these represent the same conceptual amount (a present or future value of a cost of cash flow). You should be familiar with different ways of writing the formulas.

173

174

Chapter 6. Commodity Forwards and Futures

TABLE 6.5

Reverse cash-and-carry for copper for 1 year, assuming that the commodity lender requires a lease payment of L.

Transaction
Short-sell copper
Lease payment
Long forward
Invest @ R
Total

Time 0
S0
0
0
−S0
0

Cash Flows
Time 1
−S1
−L
S1 − F0, 1
(1 + R)S0
[(1 + R)S0 − F0, 1] − L

to buy the commodity forward. If it is too high, there is an incentive for traders to sell the commodity, whether or not arbitrage is feasible. Leasing and storage costs complicate arbitrage, however.

Convenience Yields
The discussion of commodities has so far ignored business reasons for holding commodities.
For example, if you are a food producer for whom corn is an essential input, you will hold corn in inventory. If you hold too much corn, you can sell the excess. However, if you hold too little, you may run out of corn, halting production and idling workers and machines.
The physical inventory of corn in this case has value: It provides insurance that you can keep producing in case there is a disruption in the supply of corn.
In this situation, corn holdings provide an extra nonmonetary return called the convenience yield.5 You will be willing to store corn with a lower rate of return than if you did not earn the convenience yield. What are the implications of the convenience yield for the forward price?
The convenience yield is only relevant when the commodity is stored. In order to store the commodity, an owner will require that the forward price compensate for the financial and physical costs of storing, but the owner will accept a lower forward price to the extent there is a convenience yield. Specifically, if the continuously compounded convenience yield is c, proportional to the value of the commodity, the owner will earn an acceptable return from storage if the forward price is
F0, T ≥ S0e(r+λ−c)T
Because we saw that low commodity forward prices cannot easily be arbitraged, this price would not yield an arbitrage opportunity.

5. The term convenience yield is defined differently by different authors. Convenience yield generally means a return to physical ownership of the commodity. In practice it is sometimes used to mean what we call the lease rate. In this book, the two concepts are distinct, and commodities need not have a convenience yield. The lease rate of a commodity can be inferred from the forward price using equation (6.7).

6.4 Gold

What is the commodity lease rate in this case? An owner lending the commodity saves λ and loses c from not storing the commodity. Hence, the commodity borrower would need to pay δl = c − λ in order to compensate the lender for convenience yield less storage cost. The difficulty with the convenience yield in practice is that convenience is hard to observe. The concept of the convenience yield serves two purposes. First, it explains patterns in storage—for example, why a commercial user might store a commodity when the average investor will not. Second, it provides an additional parameter to better explain the forward curve. You might object that we can invoke the convenience yield to explain any forward curve, and therefore the concept of the convenience yield is vacuous.
While convenience yield can be tautological, it is a meaningful economic concept and it would be just as arbitrary to assume that there is never convenience. Moreover, the upper bound in equation (6.8) depends on storage costs but not the convenience yield. Thus, the convenience yield only explains anomalously low forward prices, and only when there is storage. Summary
Much of the discussion in this section was aimed at explaining the differences between commodities and financial assets. The main conclusions are intuitive:
.

.

.

The forward price, F0, T , should not exceed S0e(r+λ)T . If the forward price were greater, you could undertake a simple cash-and-carry and earn a profit after paying both storage costs and interest on the position. Storage costs here includes deterioration of the commodity, so fragile commodities could have large (or infinite) storage costs. In a carry market, the forward price should equal S0e(r−c+λ)T . A user who buys and stores the commodity will then be compensated for interest and physical storage costs less a convenience yield.
In any kind of market, a reverse cash-and-carry arbitrage (attempting to arbitrage too low a forward price) will be difficult, because the terms at which a lender will lend the commodity will likely reflect the forward price, making profitable arbitrage difficult. 6.4 GOLD
Of all commodities, gold is most like a financial asset. Gold is durable, nonreactive, noncorrosive, relatively inexpensive to store (compared to its value), widely held, and actively produced through gold mining. Because of transportation costs and purity concerns, gold often trades in certificate form, as a claim to physical gold at a specific location. There are exchange-traded gold futures, specifications for which are in Figure 6.3.
Figure 6.4 graphs futures prices for all available gold futures contracts—the forward curve—for four different dates. The forward curves all show the forward price steadily increasing with time to maturity.

175

176

Chapter 6. Commodity Forwards and Futures

FIGURE 6.3
Specifications for the CME
Group gold futures contract.

Underlying
Where traded
Size
Months
Trading ends
Delivery

FIGURE 6.4
The forward curve for gold on four dates, from NYMEX gold futures prices.

Refined gold bearing approved refiner stamp
CME Group/NYMEX
100 troy ounces
February, April, August, October, out 2 years. June, December, out 5 years
Third-to-last business day of maturity month
Any business day of the delivery month

Futures Price ($/oz)
1500

1000

500
6/2/2004
6/7/2006
6/4/2008
6/2/2010
0
0

10

20
40
30
Months to Maturity

50

Data from Datastream.

Gold Leasing
From our discussion in Section 6.3, the forward price implies a lease rate for gold. Shortsales and loans of gold are in fact common in the gold market. On the lending side, large gold holders (including some central banks) put gold on deposit with brokers, in order that it may be loaned to short-sellers. The gold lenders earn the lease rate.
The lease rate for gold, silver, and other commodities is typically reported using equation (6.7), with LIBOR as the interest rate. In recent years the lease rate has often been negative, especially for periods of 6 months or less.
As an example of the lease rate computation, consider gold prices on June 2, 2010.
The June, December, and June 2011 futures settlement prices that day were 1220.6, 1226.8, and 1234.3. The return from buying June gold and selling December gold would have been

6.4 Gold

Return6 months =

1226.8
− 1 = 0.00508
1220.6

At the same time, June LIBOR was 99.432 and September LIBOR was 99.2, so the implied
6-month interest rate was (1 + 0.00568/4) × (1 + 0.008/4), a 6-month interest rate of
0.00342. Because the (nonannualized) implied 6-month gold appreciation rate exceeds
(nonannualized) 6-month LIBOR, the lease rate is negative. The annualized lease rate in this calculation is
2 × (0.00342 − 0.00508) = −0.003313
The negative lease rate seems to imply that gold owners would pay to lend gold. With significant demand in recent years for gold storage, the negative lease rate could be measuring increased marginal storage costs. It is also possible that LIBOR is not the correct interest rate to use in computing the lease rate. Whatever the reason for negative lease rates, gold in recent years has been trading at close to full carry.

Evaluation of Gold Production
Suppose we wish to compute the present value of future production for a proposed gold mine. As discussed in Section 6.2, the present value of a unit of commodity received in the future is simply the present value of the forward price, with discounting performed at the risk-free rate. We can thus use the forward curve for gold to compute the value of an operating gold mine.
Suppose that at times ti , i = 1, . . . , n, we expect to extract nti ounces of gold by paying a per-unit extraction cost of x(ti ). We have a set of n forward prices, F0, ti . If the continuously compounded annual risk-free rate from time 0 to ti is r(0, ti ), the value of the gold mine is n PV gold production = i=1 nti F0, ti − x(ti ) e−r(0, ti )ti

(6.10)

This equation assumes that the gold mine is certain to operate the entire time and that the quantity of production is known. Only price is uncertain. (We will see in Chapter 17 how the possibility of mine closings due to low prices affects valuation.) Note that in equation
(6.10), by computing the present value of the forward price, we compute the prepaid forward price. Example 6.2 Suppose we have a mining project that will produce 1 ounce of gold every year for 6 years. The cost of this project is $1100 today, the marginal cost per ounce at the time of extraction is $100, and the continuously compounded interest rate is 6%.
We observe the gold forward prices in the second column of Table 6.6, with implied prepaid forward prices in the third column. Using equation (6.10), we can use these prices to perform the necessary present value calculations.
6

Net present value = i=1 F0, i − 100 e−0.06×i − $1100 = $119.56

(6.11)

177

178

Chapter 6. Commodity Forwards and Futures

TABLE 6.6

Gold forward and prepaid forward prices on 1 day for gold delivered at 1-year intervals, out to 6 years. The continuously compounded interest rate is 6% and the lease rate is assumed to be a constant 1.5%.

Expiration Year

Forward Price ($)

Prepaid Forward Price ($)

1
2
3
4
5
6

313.81
328.25
343.36
359.17
375.70
392.99

295.53
291.13
286.80
282.53
278.32
274.18

FIGURE 6.5
Specifications for the CME
Group/CBOT corn futures contract. Underlying

Where traded
Size
Months
Trading ends
Delivery

#2 Yellow, with #1 Yellow deliverable at a
$0.015 premium and #3 Yellow at a $0.015 discount. CME Group/CBOT
5000 bushels (∼127 metric tons)
March, May, July, September, and December, out 2 years
Business day prior to the 15th day of the month. Second business day following the last trading day of the delivery month

6.5 CORN
Important grain futures in the United States include corn, soybeans, and wheat. In this section we discuss corn as an example of an agricultural product. Corn is harvested primarily in the fall, from September through November. The United States is a leading corn producer, generally exporting rather than importing corn. Figure 6.5 presents specifications for the
CME Group corn futures contract.
Given seasonality in production, what should the forward curve for corn look like?
Corn is produced at one time of the year, but consumed throughout the year. In order to be consumed when it is not being produced, corn must be stored.
As discussed in Section 6.3, storage is an economic decision in which there is a tradeoff between selling today and selling tomorrow. If we can sell corn today for $2/bu and in
2 months for $2.25/bu, the storage decision entails comparing the price we can get today with the present value of the price we can get in 2 months. In addition to interest, we need to include storage costs in our analysis.
An equilibrium with some current selling and some storage requires that corn prices be expected to rise at the interest rate plus storage costs, which implies that there will be an

6.6 Energy Markets

FIGURE 6.6
Forward curves for corn for four years.

179

Futures Price (¢/bushel)
800

600

400

200

6/2/2004
6/7/2006
6/4/2008
6/2/2010

0
0

10

20
Months to Maturity

30

Data from Datastream.

upward trend in the price between harvests. While corn is being stored, the forward price should behave as in equation (6.9), rising at interest plus storage costs.
In a typical year, once the harvest begins, storage is no longer necessary; if supply and demand remain constant from year to year, the harvest price will be the same every year.
Those storing corn will plan to deplete inventory as harvest approaches and to replenish inventory from the new harvest. The corn price will fall at harvest, only to begin rising again after the harvest.
The behavior of the corn forward price, graphed in Figure 6.6, largely conforms with this description. In three of the four forward curves, the forward price of corn rises to reward storage between harvests, and it falls at harvest. An important caveat is that the supply of corn varies from year to year. When there is an unusually large crop, producers will expect corn to be stored not just over the current year but into the next year as well. If there is a large harvest, therefore, we might see the forward curve rise continuously until year 2. This might explain the low price and steady rise in 2006.
Although corn prices vary throughout the year, farmers will plant in anticipation of receiving the harvest price. It is therefore the harvest price that guides production decisions.
The price during the rest of the year should approximately equal the harvest price plus storage, less convenience.

6.6 ENERGY MARKETS
One of the most important and heavily traded commodity sectors is energy. This sector includes oil, oil products (heating oil and gasoline), natural gas, and electricity. These products represent different points on the spectrum of storage costs and carry.

180

Chapter 6. Commodity Forwards and Futures

Electricity
The forward market for electricity illustrates forward pricing when storage is often not possible, or at least quite costly. Electricity is produced in different ways: from fuels such as coal and natural gas, or from nuclear power, hydroelectric power, wind power, or solar power. Once it is produced, electricity is transmitted over the power grid to end-users.
There are several economic characteristics of electricity that are important to understand. First, it is difficult to store; hence it must be consumed when it is produced or else it is wasted.6 Second, at any point in time the maximum supply of electricity is fixed. You can produce less but not more. Third, demand for electricity varies substantially by season, by day of week, and by time of day. Because carry is limited and costly, the electricity price at any time is set by demand and supply at that time.
To illustrate the effects of nonstorability, Table 6.7 displays 1-day-ahead hourly prices for 1 megawatt-hour of electricity in New York City. The 1-day-ahead forward price is
$32.22 at 2 a.m., and $63.51 at 7 p.m. Ideally one would buy 2 a.m. electricity, store it, and sell it at 7 p.m., but there is no way to do so costlessly.
Notice two things. First, the swings in Table 6.7 could not occur with financial assets, which are stored. The 3 a.m. and 3 p.m. forward prices for a stock will be almost identical; if they were not, it would be possible to arbitrage the difference. Second, whereas the forward price for a stock is largely redundant in the sense that it reflects information about the current stock price, interest, and the dividend yield, the forward prices in Table 6.7 provide price discovery, revealing otherwise unobtainable information about the future price of the commodity. The prices in Table 6.7 are best interpreted using equation (6.4).
Just as intraday arbitrage is difficult, there is no costless way to buy winter electricity and sell it in the summer, so there are seasonal variations as well as intraday variations. Peakload power plants operate only when prices are high, temporarily increasing the supply of electricity. However, expectations about supply, storage, and peak-load power generation should already be reflected in the forward price.

Natural Gas
Natural gas is a market in which seasonality and storage costs are important. The natural gas futures contract, introduced in 1990, has become one of the most heavily traded futures contracts in the United States. The asset underlying one contract is 10,000 MMBtu, delivered over one month at a specific location (different gas contracts call for delivery at different locations). Figure 6.7 details the specifications for the Henry Hub contract.
Natural gas has several interesting characteristics. First, gas is costly to transport internationally, so prices and forward curves vary regionally. Second, once a given well has begun production, gas is costly to store. Third, demand for gas in the United States is highly seasonal, with peak demand arising from heating in winter months. Thus, there is a relatively steady stream of production with variable demand, which leads to large and

6. There are costly ways to store electricity. Three examples are pumped storage hydroelectricity (pump water into an uphill reservoir when prices are low, and release the water to flow over turbines when electricity is expensive); night wind storage (refrigerated warehouses are cooled to low temperature when electricity is cheap and the temperature is allowed to rise when electricity is expensive); compressed air energy storage (use wind power to compress air, then use the compressed air to drive turbines when electricity is expensive). All three of these methods entail losses.

6.6 Energy Markets

TABLE 6.7

Day-ahead price, by hour, for 1 megawatt-hour of electricity in New York City, March 21, 2011.

Time

Price

Time

Price

Time

Price

Time

Price

0000
0100
0200
0300
0400
0500

$36.77
$34.43
$32.22
$32.23
$32.82
$35.84

0600
0700
0800
0900
1000
1100

$44.89
$58.05
$52.90
$54.06
$55.06
$55.30

1200
1300
1400
1500
1600
1700

$53.84
$51.36
$50.01
$49.55
$49.71
$51.66

1800
1900
2000
2100
2200
2300

$56.18
$63.51
$54.99
$47.01
$40.26
$37.29
Data from Bloomberg

FIGURE 6.7
Specifications for the
NYMEX Henry Hub natural gas contract.

Underlying
Where traded
Size
Months
Trading ends
Delivery

Natural gas delivered at Sabine Pipe Lines
Co.’s Henry Hub, Louisiana
New York Mercantile Exchange
10,000 million British thermal units
(MMBtu)
72 consecutive months
Third-to-last business day of month prior to maturity month
As uniformly as possible over the delivery month predictable price swings. Whereas corn has seasonal production and relatively constant demand, gas has relatively constant supply and seasonal demand.
Figure 6.8 displays strips of gas futures prices for the first Wednesday in June for 4 years between 2004 and 2010. In all curves, seasonality is evident, with high winter prices and low summer prices. The 2004 and 2006 strips show seasonal cycles combined with a downward trend in prices, suggesting that the market considered prices in that year as anomalously high. For the other years, the long-term trend is upward.
Gas storage is costly and demand for gas is highest in the winter. The steady rise of the forward curve (contango) during the fall months suggests that storage occurs just before the heaviest demand. In the June 2006 forward curve, the October, November, and
December 2006 prices were $7.059, $8.329, and $9.599. The interest rate at that time was about 5.5%, or 0.5%/month. Interest costs would thus contribute at most a few cents to contango. Considering the October and November prices, in a carry market, storage cost would have to satisfy equation (6.8):
8.329 = 7.059e0.005 + λ

181

182

Chapter 6. Commodity Forwards and Futures

FIGURE 6.8
Forward curves for natural gas for four years. Prices are dollars per MMBtu, from
CME Group/NYMEX.

Futures Price ($/MMBtu)
15

10

5
6/2/2004
6/7/2006
6/4/2008
6/2/2010
0
0

150

50
100
Months to Maturity

Data from Datastream.

This calculation implies an estimated expected marginal storage cost of λ = $1.235 in
November 2006. The technologies for storing gas range from pumping it into underground storage facilities to freezing it and storing it offshore in liquified natural gas tankers. By examining Figure 6.8 you will find different imputed storage costs in each year, but this is to be expected if marginal storage costs vary with the quantity stored.
Because of the expense in transporting gas internationally, the seasonal behavior of the forward curve can vary in different parts of the world. In tropical areas where gas is used for cooking and electricity generation, the forward curve is relatively flat because demand is relatively flat. In the Southern hemisphere, where seasons are reversed from the Northern hemisphere, the forward curve will peak in June and July rather than December and January.

Oil
Both oil and natural gas produce energy and are extracted from wells, but the different physical characteristics and uses of oil lead to a very different forward curve than that for gas. Oil is easier to transport than gas, with the result that oil trades in a global market.
Oil is also easier to store than gas. Thus, seasonals in the price of crude oil are relatively unimportant. Specifications for the NYMEX light sweet crude oil contract (also known as
West Texas Intermediate, or WTI) are shown in Figure 6.9.7 The NYMEX forward curve on four dates is plotted in Figure 6.10.

7. Oil is called “sweet” if it has a relatively low sulfur content, and “sour” if the sulfur content is high.

6.6 Energy Markets

FIGURE 6.9
Specifications for the
NYMEX light sweet crude oil contract.

Underlying
Where traded
Size
Months
Trading ends

Delivery

FIGURE 6.10
Multi-year strips of NYMEX crude oil futures prices,
$/barrel, for four different dates. 183

Specific domestic crudes delivered at
Cushing, Oklahoma
New York Mercantile Exchange
1000 U.S. barrels (42,000 gallons)
30 consecutive months plus long-dated futures out 7 years
Third-to-last business day preceding the 25th calendar day of month prior to maturity month As uniformly as possible over the delivery month Futures Price ($/barrel)
150

100

50
●●●●
●●●
●●●●
●●●●●●
●●●●●●●●
●●●●●











6/2/2004
6/7/2006
6/4/2008
6/2/2010

0
0

20

40
60
Months to Maturity

80

100

Data from Datastream.

On the four dates in the figure, near-term oil prices range from $40 to $125. At each price, the forward curves are relatively flat. In 2004, it appears that the market expected oil prices to decline. Obviously, that did not happen. In 2006 and 2008, the early part of the forward curve is steeply sloped, suggesting that there was a return to storage and a temporary surplus supply. During 2009, for example, there was substantial arbitrage activity with traders storing oil on tankers. This is discussed in the box on p. 184
Although oil is a global market, the delivery point for the WTI oil contract is Cushing,
Oklahoma, which is landlocked. Another important oil contract is the Brent crude oil

184

BOX

Chapter 6. Commodity Forwards and Futures

6.2: Tanker-Based Arbitrage

F

rom The Wall Street Journal: The huge floating stockpile of crude oil kept on tankers amid a global supply glut is showing signs of shrinking, as traders struggle to make profits from the once highly lucrative storage play.
The volume being stored at sea has nearly halved from a peak of about 90 million barrels in April last year, according to ship broker ICAP, and [is] expected to fall even further. . . .
The phenomenon of floating storage took off early last year. Oil on the spot market traded at a big discount to forward-dated contracts, in a condition known as contango. Traders took advantage of that by buying crude and putting it into storage on tankers for sale at a higher price at a future date. Profits from the trade more than covered the costs of storage.
At its peak in April last year, there were about
90 million barrels of crude oil in floating storage on huge tankers known as very large crude carriers, or VLCCs, according to ICAP.

But the spread between prompt crude-oil prices and forward prices has narrowed in recent weeks, while freight rates have increased, reducing the incentive to store oil for future delivery.
Contango has narrowed to around 40 cents a barrel, and “to cover your freight and other costs you need at least 90 cents,” said Torbjorn Kjus, an oil analyst at DnB NOR Markets.
J.P. Morgan has said prices could even go into backwardation at the end of the second quarter, where spot prices are higher than those in forward contracts. This would be the first time the spread has been in positive territory since July last year.
ICAP said there were currently 21 trading
VLCCs offshore with some 43 million barrels of crude. Seven of these are expected to discharge in February and one more in March. So far, it appeared those discharged cargoes wouldn’t be replaced by new ones. . . .
Source: Chazan (2010)

contract, based on oil from the North Sea. Historically WTI and Brent traded within a few dollars of each other, and they are of similar quality. In early 2011, however, the price of Brent was at one point almost $20/barrel greater than the price of WTI. Though there is no one accepted explanation for this discrepancy, the difficulty of transporting oil from
Cushing to ports undoubtedly plays a role, and the WTI contract in recent years has lost favor as a global oil benchmark. In particular, in 2009 Saudi Arabia dropped WTI from its export benchmarks. The WTI-Brent price discrepancy illustrates the importance of transportation costs even in an integrated global market.

Oil Distillate Spreads
Some commodities are inputs in the creation of other commodities, which gives rise to commodity spreads. Crude oil is refined to make petroleum products, in particular heating oil and gasoline. The refining process entails distillation, which separates crude oil into different components, including gasoline, kerosene, and heating oil. The split of oil into these different components can be complemented by a process known as “cracking”; hence,

6.7 Hedging Strategies

the difference in price between crude oil and equivalent amounts of heating oil and gasoline is called the crack spread.8
Oil can be processed in different ways, producing different mixes of outputs. The spread terminology identifies the number of gallons of oil as input, and the number of gallons of gasoline and heating oil as outputs. Traders will speak of “5-3-2,” “3-2-1,” and “2-1-1” crack spreads. The 5-3-2 spread, for example, reflects the profit from taking 5 gallons of oil as input, and producing 3 gallons of gasoline and 2 gallons of heating oil. A petroleum refiner producing gasoline and heating oil could use a futures crack spread to lock in both the cost of oil and output prices. This strategy would entail going long oil futures and short the appropriate quantities of gasoline and heating oil futures. Of course there are other inputs to production and it is possible to produce other outputs, such as jet fuel, so the crack spread is not a perfect hedge.
Example 6.3 A refiner in June 2010 planning for July production could have purchased
July oil for $72.86/barrel and sold August gasoline and heating oil for $2.0279/gallon and
$2.0252/gallon. The 3-2-1 crack spread is the gross margin from buying 3 gallons of oil and selling 2 gallons of gasoline and 1 of heating oil. Using these prices, the spread is
2 × $2.0279 + $2.0252 − 3 × $72.86/42 = $0.8767 or $0.8767/3 = $0.29221/gallon.
There are crack spread swaps and options. Most commonly these are based on the difference between the price of heating oil and crude oil, and the price of gasoline and heating oil, both in a 1:1 ratio.

6.7 HEDGING STRATEGIES
In this section we discuss some issues when using commodity futures and forwards to hedge commodity price exposure. First, since commodities are heterogeneous and often costly to transport and store, it is common to hedge a risk with a commodity contract that is imperfectly correlated with the risk being hedged. This gives rise to basis risk: The price of the commodity underlying the futures contract may move differently than the price of the commodity you are hedging. For example, because of transportation cost and time, the price of natural gas in California may differ from that in Louisiana, which is the location underlying the principal natural gas futures contract (see again Figure 6.7). Second, in some cases one commodity may be used to hedge another. As an example of this we discuss the use of crude oil to hedge jet fuel. Finally, weather derivatives provide another example of an instrument that can be used to cross-hedge. We discuss degree-day index contracts as an example of such derivatives.

8. Spreads are also important in agriculture. Soybeans, for example, can be crushed to produce soybean meal and soybean oil (and a small amount of waste). A trader with a position in soybeans and an opposite position in equivalent quantities of soybean meal and soybean oil has a crush spread and is said to be
“trading the crush.”

185

186

Chapter 6. Commodity Forwards and Futures

Basis Risk
Exchange-traded commodity futures contracts call for delivery of the underlying commodity at specific locations and specific dates. The actual commodity to be bought or sold may reside at a different location and the desired delivery date may not match that of the futures contract. Additionally, the grade of the deliverable under the futures contract may not match the grade that is being delivered.
This general problem of the futures or forward contract not representing exactly what is being hedged is called basis risk. Basis risk is a generic problem with commodities because of storage and transportation costs and quality differences. Basis risk can also arise with financial futures, as for example when a company hedges its own borrowing cost with the Eurodollar contract.
Section 5.5 demonstrated how an individual stock could be hedged with an index futures contract. We saw that if we regressed the individual stock return on the index return, the resulting regression coefficient provided a hedge ratio that minimized the variance of the hedged position.
In the same way, suppose we wish to hedge oil delivered on the East Coast with the
NYMEX oil contract, which calls for delivery of oil in Cushing, Oklahoma. The varianceminimizing hedge ratio would be the regression coefficient obtained by regressing the East
Coast price on the Cushing price. Problems with this regression are that the relationship may not be stable over time or may be estimated imprecisely.
Another example of basis risk occurs when hedgers decide to hedge distant obligations with near-term futures. For example, an oil producer might have an obligation to deliver
100,000 barrels per month at a fixed price for a year. The natural way to hedge this obligation would be to buy 100,000 barrels per month, locking in the price and supply on a month-bymonth basis. This is called a strip hedge. We engage in a strip hedge when we hedge a stream of obligations by offsetting each individual obligation with a futures contract matching the maturity and quantity of the obligation. For the oil producer obligated to deliver every month at a fixed price, the hedge would entail buying the appropriate quantity each month, in effect taking a long position in the strip.
An alternative to a strip hedge is a stack hedge. With a stack hedge, we enter into futures contracts with a single maturity, with the number of contracts selected so that changes in the present value of the future obligations are offset by changes in the value of this “stack” of futures contracts. In the context of the oil producer with a monthly delivery obligation, a stack hedge would entail going long 1.2 million barrels using the near-term contract. (Actually, we would want to tail the position and go long fewer than 1.2 million barrels, but we will ignore this.) When the near-term contract matures, we reestablish the stack hedge by going long contracts in the new near month. This process of stacking futures contracts in the near-term contract and rolling over into the new near-term contract is called a stack and roll. If the new near-term futures price is below the expiring near-term price
(i.e., there is backwardation), rolling is profitable.
There are at least two reasons for using a stack hedge. First, there is often more trading volume and liquidity in near-term contracts. With many commodities, bid-ask spreads widen with maturity. Thus, a stack hedge may have lower transaction costs than a strip hedge.
Second, the manager may wish to speculate on the shape of the forward curve. You might decide that the forward curve looks unusually steep in the early months. If you undertake a stack hedge and the forward curve then flattens, you will have locked in all your oil at the

6.7 Hedging Strategies

BOX

187

6.3: Metallgesellschaft A. G.

I

n 1992, a U.S. subsidiary of the German industrial firm Metallgesellschaft A. G. (MG) had offered customers fixed prices on over 150 million barrels of petroleum products, including gasoline, heating oil, and diesel fuel, over periods as long as
10 years. To hedge the resulting short exposure,
MG entered into futures and swaps.
Much of MG’s hedging was done using shortdated NYMEX crude oil and heating oil futures.
Thus, MG was using stack hedging, rolling over the hedge each month.
During much of 1993, the near-term oil market was in contango (the forward curve was upward sloping). As a result of the market remaining in contango, MG systematically lost money when rolling its hedges and had to meet substantial margin calls. In December 1993, the supervisory board of MG decided to liquidate both its supply contracts and the futures positions used to hedge

those contracts. In the end, MG sustained losses estimated at between $200 million and $1.3 billion. The MG case was extremely complicated and has been the subject of pointed exchanges among academics—see in particular Culp and Miller
(1995), Edwards and Canter (1995), and Mello and Parsons (1995). While the case is complicated, several issues stand out. First, was the stack and roll a reasonable strategy for MG to have undertaken? Second, should the position have been liquidated when and in the manner it was?
(As it turned out, oil prices increased—which would have worked in MG’s favor—following the liquidation.) Third, did MG encounter liquidity problems from having to finance losses on its hedging strategy? While the MG case has receded into history, hedgers still confront the issues raised by this case.

relatively cheap near-term price, and implicitly made gains from not having locked in the relatively high strip prices.However, if the curve becomes steeper, it is possible to lose.
The box above recounts the story of Metallgesellschaft A. G. (MG), in which MG’s large losses on a hedged position might have been caused, at least in part, by the use of a stack hedge.

Hedging Jet Fuel with Crude Oil
Jet fuel futures do not exist in the United States, but firms sometimes hedge jet fuel with crude oil futures along with futures for related petroleum products.9 In order to perform this hedge, it is necessary to understand the relationship between crude oil and jet fuel prices. If we own a quantity of jet fuel and hedge by holding H crude oil futures contracts, our markto-market profit depends on the change in the jet fuel price and the change in the futures price: (Pt − Pt−1) + H (Ft − Ft−1)

9. For example, see the box on p. 106 about Southwest Airlines.

(6.12)

188

Chapter 6. Commodity Forwards and Futures

where Pt is the price of jet fuel and Ft the crude oil futures price. We can estimate H by regressing the change in the jet fuel price (denominated in dollars per gallon) on the change in the crude oil futures price (denominated in dollars per gallon, which is the barrel price divided by 42). We use the nearest to maturity oil futures contract. Running this regression using daily data for January 2006–March 2011 gives10 oil Pt − Pt−1 = 0.0004 + 0.8379(Ftoil − Ft−1)
(0.0009)

R 2 = 0.596

(0.0192)

(6.13)

Standard errors are below coefficients. The coefficient on the futures price change tells us that, on average, when the crude futures price increases by $0.01, a gallon of jet fuel increases by $0.008379.11 The R 2 of 0.596 implies a correlation coefficient of about 0.77, so there is considerable variation in the price of jet fuel not accounted for by the price of crude. Because jet fuel is but one product produced from crude oil, it makes sense to see if adding other oil products to the regression improves the accuracy of the hedge. Adding the near term futures prices for heating oil and gasoline, we obtain heating oil

oil
Pt − Pt−1 = 0.0006 + 0.0897 Ftoil − Ft−1 + 0.8476 Ft
(0.0001)

(0.0277)

(0.0278)

gasoline

+ 0.0069 Ft
(0.0222)

gasoline

− Ft−1

heating oil

− Ft−1

(6.14)
R 2 = 0.786

The explanatory power of the regression is improved, with an implied correlation of 0.886 between the actual and predicted jet fuel price. The price of heating oil is more closely related to the price of jet fuel than is the price of crude oil.

Weather Derivatives
Many businesses have revenue that is sensitive to weather: Ski resorts are harmed by warm winters, soft drink manufacturers are harmed by a cold spring, summer, or fall, and makers of lawn sprinklers are harmed by wet summers. In all of these cases, firms could hedge their risk using weather derivatives—contracts that make payments based upon realized characteristics of weather—to cross-hedge their specific risk.
Weather can affect both the price and consumption of energy-related products. If a winter is colder than average, homeowners and businesses will consume extra electricity, heating oil, and natural gas, and the prices of these products will tend to be high as well.
Conversely, during a warm winter, energy prices and quantities will be low. While it is possible to use futures markets to hedge prices of commodities such as natural gas, hedging the quantity is more difficult. Weather derivatives can provide an additional contract with a payoff correlated with the quantity of energy used.

10. This regression omits 4 days: September 11, 12, 15, and 16, 2008. The reported price of jet fuel on those days—a stressful period during the financial crisis—increased by over $1/gallon and then on September
17 returned to its previous price.
11. Recall that in Section 5.5 we estimated a hedge ratio for stocks using a regression based on percentage changes. In that case, we had an economic reason (an asset pricing model) to believe that there was a stable relationship based upon rates of return. In this case, crude is used to produce jet fuel, so it makes sense that dollar changes in the price of crude would be related to dollar changes in the price of jet fuel.

6.8 Synthetic Commodities

An example of a weather contract is the degree-day index futures contract traded at the
CME Group. The contract is based on the premise that heating is used when temperatures are below 65 degrees and cooling is used when temperatures are above 65 degrees. Thus, a heating degree-day is the difference between 65 degrees Fahrenheit and the average daily temperature, if positive, or zero otherwise. A cooling degree-day is the difference between the average daily temperature and 65 degrees Fahrenheit, if positive, and zero otherwise.
The monthly degree-day index is the sum of the daily degree-days over the month. The futures contract then settles based on the cumulative heating or cooling degree-days (the two are separate contracts) over the course of a month. The size of the contract is $100 times the degree-day index. Degree-day index contracts are available for major cities in the
United States, Europe, and Japan. There are also puts and calls on these futures.
With city-specific degree-day index contracts, it is possible to create and hedge payoffs based on average temperatures, or using options, based on ranges of average temperatures. If Minneapolis is unusually cold but the rest of the country is normal, the heating degree-day contract for Minneapolis will make a large payment that will compensate the holder for the increased consumption of energy.

6.8 SYNTHETIC COMMODITIES
Just as it is possible to use stock index futures to create a synthetic stock index, it is also possible to use commodity futures to create synthetic commodities. We can create a synthetic commodity by combining a commodity forward contract and a zero-coupon bond.
Enter into a long commodity forward contract at the price F0, T and buy a zero-coupon bond that pays F0, T at time T . Since the forward contract is costless, the cost of this investment strategy at time 0 is just the cost of the bond, which equals the prepaid forward price: e−rT F0, T . At time T , the strategy pays
ST − F0, T
Forward contract payoff

+

F0, T

= ST

Bond payoff

where ST is the time T price of the commodity. This investment strategy creates a synthetic commodity, which has the same value as a unit of the commodity at time T .
During the early 2000s, indexed commodity investing became popular. Commodity funds use futures contracts and Treasury bills or other bonds to create synthetic commodities and replicate published commodity indexes. Two important indexes are the S&P GSCI index (originally created by Goldman Sachs) and the Dow Jones UBS index (originally created by AIG). Masters (2008) estimates that money invested in commodity funds grew
20-fold between 2003 and 2008, from $13 billion to $260 billion.12 During this same period, commodity prices rose significantly. Figure 6.11 shows the performance of two commodity indexes plotted with the S&P 500. The two indexes diverge sharply in 2009 because they weight commodities differently. The S&P GSCI index, for example, is world-production

12. Index investors have to periodically exchange an expiring futures contract for a new long position.
This transaction is referred to as “rolling” the position. For large index investors, the dollar amount of this futures roll can be substantial. Mou (2010) provides evidence that price effects from the roll are predictable and that front-running it can be profitable.

189

190

Chapter 6. Commodity Forwards and Futures

FIGURE 6.11
Value of S&P GSCI and DJ
UBS indexes from 1991 to
2011, plotted against the
S&P 500 index.

Index
500

400

300

200

100

DJ UBS
S&P GSCI
S&P 500

0
1995

2000
Date

2005

2010

Source: Datastream

weighted and more heavily weights the petroleum sector. The DJ UBS index is designed to be more evenly weighted.13
You might wonder whether a commodity fund should use futures contracts to create synthetic commodities, or whether the fund should hold the physical commodity (where feasible). An important implication of the discussion in Section 6.3 is that it is generally preferable to invest in synthetic commodities rather than physical commodities. To see this, we can compare the returns to owning the physical commodity and owning a synthetic commodity. As before, let λ(0, T ) denote the future value of storage costs.
To invest in the physical commodity for 1 year, we can buy the commodity and prepay storage costs. This costs S0 + λ(0, 1)/(1 + R) initially and one period later pays
S1 + λ(0, 1) − λ(0, 1) = S1.
An investment in the synthetic commodity costs the present value of the forward price,
F0, 1/(1 + R), and pays S1. The synthetic investment will be preferable if

13. Historical commodity and futures data, necessary to estimate expected commodity returns, and thus to evaluate commodity investing as a strategy, are relatively hard to obtain. Bodie and Rosansky (1980) examine quarterly futures returns from 1950 to 1976, while Gorton and Rouwenhorst (2004) examine monthly futures returns from 1959 to 2004. Both studies construct portfolios of synthetic commodities—
T-bills plus commodity futures—and find that these portfolios earn the same average return as stocks, are on average negatively correlated with stocks, and are positively correlated with inflation. These findings imply that a portfolio of stocks and synthetic commodities would have the same expected return and less risk than a diversified stock portfolio alone.

Chapter Summary

F0, 1/(1 + R) < S0 + λ(0, 1)/(1 + R) or F0, 1 < S0(1 + R) + λ(0, 1). Suppose, however, that F0, 1 > S0(1 + R) + λ(0, 1). This is an arbitrage opportunity exploitable by buying the commodity, storing it, paying storage costs, and selling it forward. Thus, if there is no arbitrage, we expect that F0, 1 ≤ S0(1 +
R) + λ(0, 1) and the synthetic commodity will be the less expensive way to obtain the commodity return. Moreover, there will be equality only in a carry market. So investors will be indifferent between physical and synthetic commodities in a carry market, and will prefer synthetic commodities at all other times.

CHAPTER SUMMARY
At a general level, commodity forward prices can be described by the same formula as financial forward prices:
F0, T = S0e(r−δ)T

(6.15)

For financial assets, δ is the dividend yield. For commodities, δ is the commodity lease rate—the return that makes an investor willing to buy and then lend a commodity. Thus, for the commodity owner who lends the commodity, it is like a dividend. From the commodity borrower’s perspective, it is the cost of borrowing the commodity.
Different issues arise with commodity forwards than with financial forwards. For both commodities and financial assets, the forward price is the expected spot price discounted at the risk premium on the asset. (As with financial forwards, commodity forward prices are biased predictors of the future spot price when the commodity return contains a risk premium.) Storage of a commodity is an economic decision in which the investor compares the benefit from selling today with the benefit of selling in the future. When commodities are stored, the forward price must be sufficiently high so that a cash-and-carry compensates the investor for both financing and storage costs (this is called a carry market). When commodities are not stored, the forward price reflects the expected future spot price.
Forward prices that are too high can be arbitraged with a cash-and-carry, while forward prices that are lower may not be arbitrageable, as the terms of a short sale should be based on the forward price. Some holders of a commodity receive a benefit from physical ownership.
This benefit is called the commodity’s convenience yield, and convenience can lower the forward price.
Forward curves provide information about individual commodities, each of which differs in the details. Forward curves for different commodities reflect different properties of storability, storage costs, production, demand, and seasonality. Electricity, gold, corn, natural gas, and oil all have distinct forward curves, reflecting the different characteristics of their physical markets. These idiosyncracies will be reflected in the individual commodity lease rates.
It is possible to create synthetic commodities by combining commodity futures and default-free bonds. In general it is financially preferable to invest in a synthetic rather than a physical commodity. Synthetic commodity indexes have been popular investments in recent years. 191

192

Chapter 6. Commodity Forwards and Futures

FURTHER READING
The concept of a lease rate will appear in later chapters, especially when discussing options (Chapter 10) and commodity-linked notes (Chapter 15). A particularly interesting application of the lease rate arises in the discussion of real options in Chapter 17. We will see there that if an extractable commodity (such as oil or gold) has a zero lease rate, it will never be extracted. Thus, the lease rate is linked in an important way with production decisions. Geman (2005) and Siegel and Siegel (1990) provide a detailed discussion of many commodity futures. There are numerous papers on commodities. Bodie and Rosansky
(1980) and Gorton and Rouwenhorst (2004) examine the risk and return of commodities as an investment. Brennan (1991), Pindyck (1993b), and Pindyck (1994) examine the behavior of commodity prices. Schwartz (1997) compares the performance of different models of commodity price behavior. Jarrow and Oldfield (1981) discuss the effect of storage costs on pricing, and Routledge et al. (2000) present a theoretical model of commodity forward curves. The websites of commodity exchanges are also useful resources, with information about particular contracts and sometimes about trading and hedging strategies.
Finally, Metallgesellschaft engendered a spirited debate. Papers written about that episode include Culp and Miller (1995), Edwards and Canter (1995), and Mello and Parsons
(1995).

PROBLEMS
6.1 The spot price of a widget is $70.00 per unit. Forward prices for 3, 6, 9, and 12 months are $70.70, $71.41, $72.13, and $72.86. Assuming a 5% continuously compounded annual risk-free rate, what are the annualized lease rates for each maturity? Is this an example of contango or backwardation?
6.2 The current price of oil is $32.00 per barrel. Forward prices for 3, 6, 9, and 12 months are $31.37, $30.75, $30.14, and $29.54. Assuming a 2% continuously compounded annual risk-free rate, what is the annualized lease rate for each maturity? Is this an example of contango or backwardation?
6.3 Given a continuously compounded risk-free rate of 3% annually, at what lease rate will forward prices equal the current commodity price? (Recall the copper example in
Section 6.3.) If the lease rate were 3.5%, would there be contango or backwardation?
6.4 Suppose that copper costs $3.00 today and the continuously compounded lease rate for copper is 5%. The continuously compounded interest rate is 10%. The copper price in 1 year is uncertain and copper can be stored costlessly.
a. If you short-sell a pound of copper for 1 year, what payment do you have to make to the copper lender? Would it make sense for a financial investor to store copper in equilibrium?
b. Show that the equilibrium forward price is $3.154.
c. In what sense is $3.316 (= 3 × e0.10) a maximum possible forward price?
d. Explain the circumstances in which any price below $3.316 could be the observed forward price, without giving rise to arbitrage. (Be sure to consider the possibility that the lease rate may not be 5%.)

Problems

6.5 Suppose the gold spot price is $300/oz, the 1-year forward price is 310.686, and the continuously compounded risk-free rate is 5%.
a. What is the lease rate?
b. What is the return on a cash-and-carry in which gold is not loaned?
c. What is the return on a cash-and-carry in which gold is loaned, earning the lease rate?
For the next three problems, assume that the continuously compounded interest rate is 6% and the storage cost of widgets is $0.03 quarterly (payable at the end of the quarter). Here is the forward price curve for widgets:
Year 0

Year 1

Year 2

Dec.

June

Sept.

Dec.

Mar.

June

3.000

6.6

Mar.
3.075

3.152

2.750

2.822

2.894

2.968

a. What are some possible explanations for the shape of this forward curve?
b. What annualized rate of return do you earn on a cash-and-carry entered into in
December of Year 0 and closed in March of Year 1? Is your answer sensible?
c. What annualized rate of return do you earn on a cash-and-carry entered into in December of Year 0 and closed in September of Year 1? Is your answer sensible? 6.7

a. Suppose that you want to borrow a widget beginning in December of Year 0 and ending in March of Year 1. What payment will be required to make the transaction fair to both parties?
b. Suppose that you want to borrow a widget beginning in December of Year 0 and ending in September of Year 1. What payment will be required to make the transaction fair to both parties?

6.8

a. Suppose the March Year 1 forward price were $3.10. Describe two different transactions you could use to undertake arbitrage.
b. Suppose the September Year 1 forward price fell to $2.70 and subsequent forward prices fell in such a way that there is no arbitrage from September
Year 1 and going forward. Is there an arbitrage you could undertake using forward contracts from June Year 1 and earlier? Why or why not?

6.9 Consider Example 6.1. Suppose the February forward price had been $2.80. What would the arbitrage be? Suppose it had been $2.65. What would the arbitrage be? In each case, specify the transactions and resulting cash flows in both November and
February. What are you assuming about the convenience yield?
6.10 Using Table 6.6, what is your best guess about the current price of gold per ounce?
6.11 Consider production ratios of 2:1:1, 3:2:1, and 5:3:2 for oil, gasoline, and heating oil. Assume that other costs are the same per gallon of processed oil.
a. Which ratio maximizes the per-gallon profit if oil costs $80/barrel, gasoline is $2/gallon, and heating oil is $1.80/gallon?

193

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Chapter 6. Commodity Forwards and Futures

b. Suppose gasoline costs $1.80/gallon and heating oil $2.10/gallon. Which ratio maximizes profit?
c. Which spread would you expect to be most profitable during the summer?
Which during the winter?
6.12 Suppose you know nothing about widgets. You are going to approach a widget merchant to borrow one in order to short-sell it. (That is, you will take physical possession of the widget, sell it, and return a widget at time T .) Before you ring the doorbell, you want to make a judgment about what you think is a reasonable lease rate for the widget. Think about the following possible scenarios.
a. Suppose that widgets do not deteriorate over time, are costless to store, and are always produced, although production quantity can be varied. Demand is constant over time. Knowing nothing else, what lease rate might you face?
b. Suppose everything is the same as in (a) except that demand for widgets varies seasonally. c. Suppose everything is the same as in (a) except that demand for widgets varies seasonally and the rate of production cannot be adjusted. Consider how seasonality and the horizon of your short-sale interact with the lease rate.
d. Suppose everything is the same as in (a) except that demand is constant over time and production is seasonal. Consider how production seasonality and the horizon of your short-sale interact with the lease rate.
e. Suppose that widgets cannot be stored. How does this affect your answers to the previous questions?

7
S

Interest Rate Forwards and Futures

uppose you have the opportunity to spend $1 one year from today to receive $2 two years from today. What is the value of this opportunity? To answer this question, you need to know the appropriate interest rates for discounting the two cash flows. This comparison is an example of the most basic concept in finance: using interest rates to compute present values. Once we find a present value for one or more assets, we can compare the values of cash flows from those assets even if the cash inflows and cash outflows occur at different times. In order to perform these calculations, we need information about the set of interest rates prevailing between different points in time.
We begin the chapter by reviewing basic bond concepts—coupon bonds, yields to maturity, and implied forward rates. Any reader of this book should understand these basic concepts. We then look at interest rate forwards and forward rate agreements, which permit hedging interest rate risk. Finally, we look at bond futures and the repo market.

7.1 BOND BASICS
Table 7.1 presents information about current interest rates for bonds maturing in from 1 to
3 years. Identical information is presented in five different ways in the table. Although the information appears differently across columns, it is possible to take the information in any one column of Table 7.1 and reproduce the other four columns.1
In practice, a wide range of maturities exists at any point in time, but the U.S. government issues Treasury securities only at specific maturities—typically 3 months, 6 months, and 1, 2, 5, 10, and 30 years.2 Government securities that are issued with less than
1 year to maturity and that make only a single payment, at maturity, are called Treasury bills.
Notes and bonds pay coupons and are issued at a price close to their maturity value (i.e., they are issued at par). Notes have 10 or fewer years to maturity and bonds have more than 10

1. Depending upon how you do the computation, you may arrive at numbers slightly different from those in Table 7.1. The reason is that all of the entries except those in column 1 are rounded in the last digit, and there are multiple ways to compute the number in any given column. Rounding error will therefore generate small differences among computations performed in different ways.
2. Treasury securities are issued using an auction. In the past the government also issued bonds with maturities of 3 and 7 years. Between 2002 and 2005 the government issued no 30-year bonds.

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TABLE 7.1

Five ways to present equivalent information about default-free interest rates.
All rates but those in the last column are effective annual rates.

(1)
Years to
Maturity
1
2
3

(2)

(3)

(4)

Zero-Coupon
Bond Yield

Zero-Coupon
Bond Price

One-Year Implied
Forward Rate

Par
Coupon

(5)
Continuously
Compounded
Zero Yield

6.00%
6.50
7.00

0.943396
0.881659
0.816298

6.00000%
7.00236
8.00705

6.00000%
6.48423
6.95485

5.82689%
6.29748
6.76586

years to maturity. The distinctions between bills, notes, and bonds are not important for our purposes; we will refer to all three as bonds. Treasury inflation protected securities are bonds for which payments are adjusted for inflation. Finally, the most recently issued government bonds are called on-the-run; other bonds are called off-the-run. These terms are used frequently in talking about government bonds since on-the-run bonds generally have lower yields and greater trading volume than off-the-run bonds. Appendix 7.A discusses some of the conventions used in bond price and yield quotations.
In addition to government bonds there are also STRIPS. A STRIPS—Separate Trading of Registered Interest and Principal of Securities—is a claim to a single interest payment or the principal portion of a government bond. These claims trade separately from the bond. STRIPS are zero-coupon bonds since they make only a single payment at maturity.
“STRIPS” should not be confused with the forward strip, which is the set of forward prices available at a point in time.
We need a way to represent bond prices and interest rates. Interest rate notation is, unfortunately and inevitably, cumbersome, because for any rate we must keep track of three dates: the date on which the rate is quoted, and the period of time (this has beginning and ending dates) over which the rate prevails. We will let rt (t1, t2) represent the interest rate from time t1 to time t2, prevailing on date t. If the interest rate is current—i.e., if t = t1—and if there is no risk of confusion, we will drop the subscript.

Zero-Coupon Bonds
We begin by showing that the zero-coupon bond yield and zero-coupon bond price, columns
(1) and (2) in Table 7.1, provide the same information. A zero-coupon bond is a bond that makes only a single payment at its maturity date. Our notation for zero-coupon bond prices will mimic that for interest rates. The price of a bond quoted at time t0, with the bond to be purchased at t1 and maturing at t2, is Pt0 (t1, t2). As with interest rates, we will drop the subscript when t0 = t1.
The 1-year zero-coupon bond price of P (0, 1) = 0.943396 means that you would pay
$0.943396 today to receive $1 in 1 year. You could also pay P (0, 2) = 0.881659 today to receive $1 in 2 years and P (0, 3) = 0.816298 to receive $1 in 3 years.
The yield to maturity (or internal rate of return) on a zero-coupon bond is simply the percentage increase in dollars earned from the bond. For the 1-year bond, we end up

7.1 Bond Basics

with 1/0.943396 − 1 = 0.06 more dollars per $1 invested. If we are quoting interest rates as effective annual rates, this is a 6% yield.
For the zero-coupon 2-year bond, we end up with 1/0.881659 − 1 = 0.134225 more dollars per $1 invested. We could call this a 2-year effective interest rate of 13.4225%, but it is conventional to quote rates on an annual basis. If we want this yield to be comparable to the
6% yield on the 1-year bond, we could assume annual compounding and get (1 + r(0, 2))2 =
1.134225, which implies that r(0, 2) = 0.065. In general,
P (0, n) =

1
[1 + r(0, n)]n

(7.1)

Note from equation (7.1) that a zero-coupon bond price is a discount factor: A zerocoupon bond price is what you would pay today to receive $1 in the future. If you have a future cash flow at time t, Ct , you can multiply it by the price of a zero-coupon bond,
P (0, t), to obtain the present value of the cash flow. Because of equation (7.1), multiplying by P (0, t) is the same as discounting at the rate r(0, t), i.e.,
Ct × P (0, t) =

Ct
[1 + r(0, t)]t

The inverse of the zero-coupon bond price, 1/P (0, t), provides a future value factor.
In contrast to zero-coupon bond prices, interest rates are subject to quoting conventions that can make their interpretation difficult (if you doubt this, see Appendix 7.A).
Because of their simple interpretation, we can consider zero-coupon bond prices as the building block for all of fixed income.
A graph of annualized zero-coupon yields to maturity against time to maturity is called the zero-coupon yield curve. A yield curve shows us how yields to maturity vary with time to maturity. In practice, it is common to present the yield curve based on coupon bonds, not zero-coupon bonds.

Implied Forward Rates
We now see how column (3) in Table 7.1 can be computed from either column (1) or (2).
The 1-year and 2-year zero-coupon yields are the rates you can earn from year 0 to year
1 and from year 0 to year 2. There is also an implicit rate that can be earned from year 1 to year 2 that must be consistent with the other two rates. This rate is called the implied forward rate.
Suppose we could today guarantee a rate we could earn from year 1 to year 2. We know that $1 invested for 1 year earns [1 + r0(0, 1)] and $1 invested for 2 years earns
[1 + r0(0, 2)]2. Thus, the time 0 forward rate from year 1 to year 2, r0(1, 2), should satisfy
[1 + r0(0, 1)][1 + r0(1, 2)] = [1 + r0(0, 2)]2 or 1 + r0(1, 2) =

[1 + r0(0, 2)]2
1 + r0(0, 1)

(7.2)

Figure 7.1 shows graphically how the implied forward rate is related to 1- and 2-year yields.
If r0(1, 2) did not satisfy equation (7.2), then there would be an arbitrage opportunity.

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FIGURE 7.1
An investor investing for 2 years has a choice of buying a 2-year zero-coupon bond paying
[1 + r0 (0, 2)]2 or buying a 1-year bond paying 1 + r0 (0, 1) for 1 year, and reinvesting the proceeds at the implied forward rate, r0 (1, 2), between years 1 and 2. The implied forward rate makes the investor indifferent between these alternatives.
[1 + r(0, 1)] × [1 + r(1, 2)]
Earn r(0, 1)

Earn implied forward rate, r(1, 2)

Time

0

1

Earn r(0, 2) per year

2

[1 + r(0, 2)]2

Problem 7.15 asks you to work through the arbitrage. In general, we have
1 + r0(t1, t2)

t2 −t1

=

[1 + r0(0, t2)] t2
P (0, t1)
=
t1
[1 + r0(0, t1)]
P (0, t2)

(7.3)

Corresponding to 1-year and 2-year interest rates, r0(0, 1) and r0(0, 2), we have prices of
1-year and 2-year zero-coupon bonds, P0(0, 1) and P0(0, 2). Just as the interest rates imply a forward 1-year interest rate, the bond prices imply a 1-year forward zero-coupon bond price. The implied forward zero-coupon bond price must be consistent with the implied forward interest rate. Rewriting equation (7.3), we have
P0(t1, t2) =

[1 + r0(0, t1)] t1
1
P (0, t2)
=
= t2 −t1 t2 [1 + r0(t1, t2)]
[1 + r0(0, t2)]
P (0, t1)

(7.4)

The implied forward zero-coupon bond price from t1 to t2 is simply the ratio of the zerocoupon bond prices maturing at t2 and t1.
Example 7.1 Using information in Table 7.1, we want to compute the implied forward interest rate from year 2 to year 3 and the implied forward price for a 1-year zero-coupon bond purchased in year 2.
The implied forward interest rate, r0(2, 3), can be computed as
1 + r0(2, 3) =

[1 + r0(0, 3)]3
(1 + 0.07)3
=
= 1.0800705
2
[1 + r0(0, 2)]
(1 + 0.065)2

or equivalently as
1 + r0(2, 3) =

P0(0, 2) 0.881659
=
= 1.0800705
P0(0, 3) 0.816298

7.1 Bond Basics

The implied forward 1-year zero-coupon bond price is
P0(0, 3)
1
=
= 0.925865
P0(0, 2) 1 + r0(2, 3)

Coupon Bonds
Given the prices of zero-coupon bonds—column (1) in Table 7.1—we can price coupon bonds. We can also compute the par coupon—column (4) in Table 7.1—the coupon rate at which a bond will be priced at par. To describe a coupon bond, we need to know the date at which the bond is being priced, the start and end date of the bond payments, the number and amount of the payments, and the amount of principal. Some practical complexities associated with coupon bonds, not essential for our purposes, are discussed in Appendix
7.A.
We will let Bt (t1, t2 , c, n) denote the time t price of a bond that is issued at t1, matures at t2, pays a coupon of c per dollar of maturity payment, and makes n evenly spaced payments over the life of the bond, beginning at time t1 + (t2 − t1)/n. We will assume the maturity payment is $1. If the maturity payment is different than $1, we can just multiply all payments by that amount.
Since the price of a bond is the present value of its payments, at issuance time t the price of a bond maturing at T must satisfy n Bt (t , T , c, n) =

cPt (t , ti ) + Pt (t , T )

(7.5)

i=1

where ti = t + i(T − t)/n, with i being the index in the summation. Using equation (7.5), we can solve for the coupon as c= Bt (t , T , c, n) − Pt (t , T ) n i=1 Pt (t , ti )

A par bond has Bt = 1, so the coupon on a par bond is given by c= 1 − Pt (t , T ) n i=1 Pt (t , ti )

(7.6)

Example 7.2 Using the information in Table 7.1, the coupon on a 3-year coupon bond that sells at par is
1 − 0.816298
0.943396 + 0.881659 + 0.816298
= 6.95485%

c=

Equation (7.5) computes the bond price by discounting each bond payment at the rate appropriate for a cash flow with that particular maturity. For example, in equation (7.5), the coupon occuring at time ti is discounted using the zero-coupon bond price Pt (t , ti ); an alternative way to write the bond price is using the yield to maturity to discount all payments.

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Chapter 7. Interest Rate Forwards and Futures

Suppose the bond makes m payments per year. Denoting the per-period yield to maturity as ym, we have n Bt (t , T , c, n) = i=1 c
1
+
(1 + ym)i
(1 + ym)n

(7.7)

It is common to compute the quoted annualized yield to maturity, y, as y = m × ym.
Government bonds, for example, make two coupon payments per year, so the annualized yield to maturity is twice the semiannual yield to maturity.
The difference between equation (7.5) and equation (7.7) is that in equation (7.5), each coupon payment is discounted at the appropriate rate for a cash flow occurring at that time. In equation (7.7), one rate is used to discount all cash flows. By definition, the two expressions give the same price. However, equation (7.7) can be misleading, since the yield to maturity, ym, is not the return an investor earns by buying and holding a bond. Moreover, equation (7.7) provides no insight into how the cash flows from a bond can be replicated with zero-coupon bonds.

Zeros from Coupons
We have started with zero-coupon bond prices and deduced the prices of coupon bonds. In practice, the situation is often the reverse: We observe prices of coupon bonds and must infer prices of zero-coupon bonds. This procedure in which zero coupon bond prices are deduced from a set of coupon bond prices is called bootstrapping.
Suppose we observe the par coupons in Table 7.1. We can then infer the first zerocoupon bond price from the first coupon bond as follows:
1 = (1 + 0.06)P (0, 1)
This implies that P (0, 1) = 1/1.06 = 0.943396. Using the second par coupon bond with a coupon rate of 6.48423% gives us
1 = 0.0648423P (0, 1) + 1.0648423P (0, 2)
Since we know P (0, 1) = 0.943396, we can solve for P (0, 2):
1 − 0.0648423 × 0.943396
1.0648423
= 0.881659

P (0, 2) =

Finally, knowing P (0, 1) and P (0, 2), we can solve for P (0, 3) using the 3-year par coupon bond with a coupon of 6.95485%:
1 = (0.0695485 × P (0, 1)) + (0.0695485 × P (0, 2)) + (1.0695485 × P (0, 3)) which gives us
1 − (0.0695485 × 0.943396) − (0.0695485 × 0.881659)
1.0695485
= 0.816298

P (0, 3) =

There is nothing about the procedure that requires the bonds to trade at par. In fact, we do not even need the bonds to all have different maturities. For example, if we had a 1-year

7.1 Bond Basics

bond and two different 3-year bonds, we could still solve for the three zero-coupon bond prices by solving simultaneous equations.

Interpreting the Coupon Rate
A coupon rate—for example the 6.95485% coupon on the 3-year bond—determines the cash flows the bondholder receives. However, except in special cases, it does not correspond to the rate of return that an investor actually earns by holding the bond.
Suppose for a moment that interest rates are certain; i.e., the implied forward rates in
Table 7.1 are the rates that will actually occur in years 1 and 2. Imagine that we buy the 3year bond and hold it to maturity, reinvesting all coupons as they are paid. What rate of return do we earn? Before going through the calculations, let’s stop and discuss the intuition. We are going to invest an amount at time 0 and to reinvest all coupons by buying more bonds, and we will not withdraw any cash until time 3. In effect, we are constructing a 3-year zero-coupon bond. Thus, we should earn the same return as on a 3-year zero: 7%. This buyand-hold return is different than the yield to maturity of 6.95485%. The coupon payment is set to make a par bond fairly priced, but it is not actually the return we earn on the bond except in the special case when the interest rate is constant over time.
Consider first what would happen if interest rates were certain, we bought the 3-year bond with a $100 principal and a coupon of 6.95485%, and we held it for 1 year. The price at the end of the year would be
106.95485
6.95485
+
1.0700237 (1.0700237)(1 + 0.0800705)
= 99.04515

B1 =

The 1-period return is thus
6.95485 + 99.04515
−1
100
= 0.06

1-period return =

We earn 6%, since that is the 1-year interest rate. Problem 7.13 asks you to compute your
2-year return on this investment.
By year 3, we have received three coupons, two of which have been reinvested at the implied forward rate. The total value of reinvested bond holdings at year 3 is
6.95485 × [(1.0700237)(1.0800705) + (1.0800705) + 1] + 100 = 122.5043
The 3-year yield on the bond is thus
122.5043
100

1/3

− 1 = 0.07

As we expected, this is equal to the 7% yield on the 3-year zero and different from the coupon rate.
This discussion assumed that interest rates are certain. Suppose that we buy and hold the bond, reinvesting the coupons, and that interest rates are not certain. Can we still expect to earn a return of 7%? The answer is yes if we use interest rate forward contracts to guarantee the rate at which we can reinvest coupon proceeds. Otherwise, the answer in general is no.

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The belief that the implied forward interest rate equals the expected future spot interest rate is a version of the expectations hypothesis. We saw in Chapters 5 and 6 that forward prices are biased predictors of future spot prices when the underlying asset has a risk premium; the same is true for forward interest rates. When you own a coupon bond, the rate at which you will be able to reinvest coupons is uncertain. If interest rates carry a risk premium, then the expected return to holding the bond will not equal the 7% return calculated by assuming interest rates are certain. The expectations hypothesis will generally not hold, and you should not expect implied forward interest rates to be unbiased predictors of future interest rates.
In practice, you can guarantee the 7% return by using forward rate agreements to lock in the interest rate for each of the reinvested coupons. We discuss forward rate agreements in Section 7.2.

Continuously Compounded Yields
Any interest rate can be quoted as either an effective annual rate or a continuously compounded rate. (Or in a variety of other ways, such as a semiannually compounded rate, which is common with bonds. See Appendix 7.A.) Column (5) in Table 7.1 presents the continuously compounded equivalents of the rates in the “zero yield” column.
In general, if we have a zero-coupon bond paying $1 at maturity, we can write its price in terms of an annualized continuously compounded yield, r cc (0, t), as3
P (0, t) = e−r

cc (0, t)t

Thus, if we observe the price, we can solve for the yield as r cc (0, t) =

1 ln[1/P (0, t)] t We can compute the continuously compounded 3-year zero yield, for example, as
1
ln(1/0.816298) = 0.0676586
3
Alternatively, we can obtain the same answer using the 3-year zero yield of 7%: ln(1 + 0.07) = 0.0676586
Any of the zero yields or implied forward yields in Table 7.1 can be computed as effective annual or continuously compounded. The choice hinges on convention and ease of calculation.

7.2 FORWARD RATE AGREEMENTS, EURODOLLAR FUTURES,
AND HEDGING
We now consider the problem of a borrower who wishes to hedge against increases in the cost of borrowing. We consider a firm expecting to borrow $100m for 91 days, beginning 120

3. In future chapters we will denote continuously compounded interest rates simply as r, without the cc superscript. 7.2 Forward Rate Agreements, Eurodollar Futures, and Hedging

days from today, in June. This is the borrowing date. The loan will be repaid in September on the loan repayment date. In the examples we will suppose that the effective quarterly interest rate at that time can be either 1.5% or 2%, and that the implied June 91-day forward rate (the rate from June to September) is 1.8%. Here is the risk faced by the borrower, assuming no hedging:
120 days

211 days rquarterly = 1.5%

Borrow $100m

+100m

rquarterly = 2%

−101.5m

−102.0m

Depending upon the interest rate, there is a variation of $0.5m in the borrowing cost.
How can we hedge this uncertainty?

Forward Rate Agreements
A forward rate agreement (FRA) is an over-the-counter contract that guarantees a borrowing or lending rate on a given notional principal amount. FRAs can be settled either at the initiation or maturity of the borrowing or lending transaction. If settled at maturity, we will say the FRA is settled in arrears. In the example above, the FRA could be settled on day 120, the point at which the borrowing rate becomes known and the borrowing takes place, or settled in arrears on day 211, when the loan is repaid.
FRAs are a forward contract based on the interest rate, and as such do not entail the actual lending of money. Rather, the borrower who enters an FRA is paid if a reference rate is above the FRA rate, and the borrower pays if the reference rate is below the FRA rate. The actual borrowing is conducted by the borrower independently of the FRA. We will suppose that the reference rate used in the FRA is the same as the actual borrowing cost of the borrower.
FRA Settlement in Arrears. First consider what happens if the FRA is settled in September, on day 211, the loan repayment date. In that case, the payment to the borrower should be rquarterly − rFRA × notional principal
Thus, if the borrowing rate is 1.5%, the payment under the FRA should be
(0.015 − 0.018) × $100 m = −$300,000
Since the rate is lower than the FRA rate, the borrower pays the FRA counterparty.
Similarly, if the borrowing rate turns out to be 2.0%, the payment under the FRA should be
(0.02 − 0.018) × $100 m = $200,000
Settling the FRA in arrears is simple and seems like the obvious way for the contract to work. However, settlement can also occur at the time of borrowing.
FRA Settlement at the Time of Borrowing. If the FRA is settled in June, at the time the money is borrowed, payments will be less than when settled in arrears because the borrower has time to earn interest on the FRA settlement. In practice, therefore, the FRA settlement is tailed by the reference rate prevailing on the settlement (borrowing) date. (Tailing in this

203

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Chapter 7. Interest Rate Forwards and Futures

context means that we reduce the payment to reflect the interest earned between June and
September.) Thus, the payment for a borrower is
Notional principal ×

rquarterly − rFRA
1 + rquarterly

(7.8)

If rquarterly = 1.5%, the payment in June is
−$300,000
= −$295,566.50
1 + 0.015
By definition, the future value of this is −$300,000. In order to make this payment, the borrower can borrow an extra $295,566.50, which results in an extra $300,000 loan payment in September. If on the other hand rquarterly = 2.0%, the payment is
$200,000
= $196,078.43
1 + 0.02
The borrower can invest this amount, which gives $200,000 in September, an amount that offsets the extra borrowing cost.
If the forward rate agreement covers a borrowing period other than 91 days, we simply use the appropriate rate instead of the 91-day rate in the above calculations.

Synthetic FRAs
Suppose that today is day 0. By using a forward rate agreement, we will be able to invest
$1 on day 120 and be guaranteed a 91-day return of 1.8%. We can synthetically create the same effect as with an FRA by trading zero-coupon bonds. In order to accomplish this we need to guarantee cash flows of $0 on day 0, −$1 on day 120, and +$1.018 on day 211.4
First, let’s get a general sense of the transaction. To match the FRA cash flows, we want cash going out on day 120 and coming in on day 211. To accomplish this, on day 0 we will need to borrow with a 120-day maturity (to generate a cash outflow on day 120) and lend with a 211-day maturity (to generate a cash inflow on day 211). Moreover, we want the day 0 value of the borrowing and lending to be equal so that there is no initial cash flow.
This description tells us what we need to do.
In general, suppose that today is day 0, and that at time t we want to lend $1 for the period s, earning the implied forward rate r0(t , t + s) over the interval from t to t + s.
To simplify the notation in this section, r0(t , t + s) will denote the nonannualized percent return from time t to time s. Recall first that
1 + r0(t , t + s) =

P (0, t)
P (0, t + s)

The strategy we use is to:
1. Buy 1 + r0(t , t + s) zero-coupon bonds maturing at time t + s.
2. Short-sell 1 zero-coupon bond maturing at time t.

4. The example in the previous section considered locking in a borrowing rate, but in this section we lock in a lending rate; the transactions can be reversed for borrowing.

7.2 Forward Rate Agreements, Eurodollar Futures, and Hedging

TABLE 7.2

Investment strategy undertaken at time 0, resulting in net cash flows of −$1 on day t, and receiving the implied forward rate, 1 + r0 (t , t + s) at t + s. This synthetically creates the cash flows from entering into a forward rate agreement on day 0 to lend at day t.

Cash Flows
Transaction

0

t

t +s

Buy 1 + r0)t , t + s) zeros maturing at t + s
Short 1 zero maturing at t
Total

−P (0, t + s) × (1 + r0(t , t + s))
+P (0, t)
0


01
−1

1 + r0(t , t + s)

1 + r0(t , t + s)

TABLE 7.3

Example of synthetic FRA. The transactions in this table are exactly those in Table 7.2, except that all bonds are sold at time t.

Cash Flows
Transaction

0

t

−P (0, t + s) × [1 + r0(t , t + s)]
+P (0, t)
0

1+r0 (t , t+s)
1+r1(t , t+s)

Buy 1 + r0)t , t + s) zeros maturing at t + s
Short 1 zero maturing at t
Total

−1

r0 (t , t+s)−r1(t , t+s)
1+r1(t , t+s)

The resulting cash flows are illustrated in Table 7.2, which shows that transactions made on day 0 synthetically create a loan commencing on day t and paying the implied forward rate, r0(t , t + s), on day t + s.
This example can be modified slightly to synthetically create the cash flows from a forward rate agreement that settles on the borrowing date, day t. To make this modification, we sell at time t the bond maturing at time t + s. The result is presented in Table 7.3. Note that if we reinvested the FRA proceeds at the market rate prevailing on day t, rt (t , t + s), we would receive r0(t , t + s) − rt (t , t + s) on day t + s.
Example 7.3 Consider the example above and suppose that P (0, 211) = 0.95836 and
P (0, 120) = 0.97561, which implies a 120-day interest rate of 2.5%. In order to receive
$1.018 on day 211, we buy 1.018 211-day zero-coupon bonds. The cost of this is
1.018 × P (0, 211) = $0.97561
In order to have zero cash flow initially and a cash outflow on day 120, we borrow 0.97561, with a 120-day maturity. This entails borrowing one 120-day bond, since

205

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Chapter 7. Interest Rate Forwards and Futures

0.97561
=1
P (0, 120)
The result on day 120 is that we pay $1 to close the short position on the 120-day bond, and on day 211 we receive $1.018 since we bought that many 211-day bonds.
To summarize, we have shown that an FRA is just like the stock and currency forwards we have considered, both with respect to pricing and synthesizing. If at time 0 we want to lock in a lending rate from time t to time t + s, we can create a rate forward synthetically by buying the underlying asset (the bond maturing at t + s) and borrowing (shorting) the bond maturing at day t.
In general, we have the following conclusions concerning a rate forward covering the period t1 to t2:
.

.

The forward rate we can obtain is the implied forward rate—i.e., rt0 (t1, t2) =
Pt0 (t0 , t1)/Pt0 (t0 , t2) − 1. rt (t1 , t2 )−rt (t1 , t2 )

We can synthetically create the payoff to an FRA, 0 1+r (t ,1t ) , by borrowing to t1 1 2 buy a bond maturing at t2, i.e., by:
1. Buying 1 + rt0 (t1, t2) of the zero-coupon bond maturing on day t2, and
2. Shorting 1 zero-coupon bond maturing on day t1.

Eurodollar Futures
Eurodollar futures contracts are similar to FRAs in that they can be used to guarantee a borrowing rate. There are subtle differences between FRAs and Eurodollar contracts, however, that are important to understand.
Let’s consider again the example in which we wish to guarantee a borrowing rate for a $100m loan from June to September. Suppose the June Eurodollar futures price is 92.8.
Implied 3-month LIBOR is 100−92.8 = 1.8% over 3 months. As we saw in Chapter 5, the
4×100
payoff on a single short Eurodollar contract at expiration will be5
[92.8 − (100 − rLIBOR )] × 100 × $25
Thus, the payoff on the Eurodollar contract compensates us for differences between the implied rate (1.8%) and actual LIBOR at expiration.
To illustrate hedging with this contract we again consider two possible 3-month borrowing rates in June: 1.5% or 2%. If the interest rate is 1.5%, borrowing cost on $100m will be $1.5m, payable in September. If the interest rate is 2%, borrowing cost will be $2m.
Suppose that we were to short 100 Eurodollar futures contracts. Ignoring marking-tomarket prior to June, if the 3-month rate in June is 1.5%, the Eurodollar futures price will be 94. The payment is
[(92.8 − 94) × 100 × $25] × 100 = −$300,000

5. This calculation treats the Eurodollar contract as if it were a forward contract, ignoring the issues associated with daily settlement, discussed in Appendix 5.B.

7.2 Forward Rate Agreements, Eurodollar Futures, and Hedging

We multiply by 100 twice: once to account for 100 contracts, and the second time to convert the change in the futures price to basis points. Similarly, if the borrowing rate is 2%, we have [(92.8 − 92) × 100 × $25] × 100 = $200,000
This is like the payment on an FRA paid in arrears, except that the futures contract settles in June, but our interest expense is not paid until September. Thus we have 3 months to earn or pay interest on our Eurodollar gain or loss before we actually have to make the interest payment. Recall that when the FRA settles on the borrowing date, the payment is the present value of the change in borrowing cost. The FRA is thus tailed automatically as part of the agreement. With the Eurodollar contract, by contrast, we need to tail the position explicitly.
We do this by shorting fewer than 100 contracts, using the implied 3-month Eurodollar rate of 1.8% as our discount factor. Thus, we enter into6
Number of Eurodollar contracts = −

100
= −98.2318
1 + 0.018

Now consider the gain on the Eurodollar futures position. If LIBOR = 6% (rquarterly = 1.5%), our total gain on the short contracts when we initiate borrowing on day 120 will be
98.2318 × (92.8 − 94) × $2500 = −$294,695
If LIBOR = 8% (rquarterly = 2.0%), our total gain on the contracts will be
98.2318 × (92.8 − 92) × $2500 = $196,464
Notice that the amounts are different than with the FRA: The reason is that the FRA payment is automatically tailed using the 3-month rate prevailing in June, whereas with the Eurodollar contract we tailed using 1.8%, the LIBOR rate implied by the initial futures price. We can now invest these proceeds at the prevailing interest rate. Here are the results on day 211, when borrowing must be repaid. If LIBOR = 6% (rquarterly = 1.5%), we save
$300,000 in borrowing cost, and the proceeds from the Eurodollar contract are
−$294,695 × (1.015) = −$299,115
If LIBOR = 8% (rquarterly = 2.0%), we owe an extra $200,000 in interest and the invested proceeds from the Eurodollar contract are
$196,464 × (1.02) = $200,393
Table 7.4 summarizes the result from this hedging position. The borrowing cost is close to
1.8%.
Convexity Bias and Tailing. In Table 7.4 the net borrowing cost appears to be a little less than 1.8%. You might guess that this is due to rounding error. It is not. Let’s examine the numbers more closely.

6. We assume here that it is possible to short fractional contracts in order to make the example exact.

207

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Chapter 7. Interest Rate Forwards and Futures

TABLE 7.4

Results from hedging $100m in borrowing with 98.23 short
Eurodollar futures.

Cash Flows
June
Borrowing Rate:
Borrow $100m
Gain on 98.23 short
Eurodollar contracts
Gain plus interest
Net

1.5%

2%

+100m

+100m

−0.294695m

September
1.5%
2%
−101.5m

−102.0m

0.196464m
−0.299115m
−101.799m

0.200393m
−101.799m

If LIBOR = 6% (rquarterly = 1.5%), we pay $1.5m in borrowing cost and we lose
$299,115 on the Eurodollar contract, for a net borrowing expense of $1.799115m. This is a
“profit” from the Eurodollar hedge, relative to the use of an FRA, of $1.8m − $1.799115m
= $884.
If LIBOR = 8% (rquarterly = 2.0%), we pay $2.0m in borrowing cost but make
$200,393 on the Eurodollar contract, for a net borrowing expense of $1.799607m. We make a profit, relative to an FRA, of $1.8m − $1.799607m = $393.
It appears that we systematically come out ahead by hedging with Eurodollar futures instead of an FRA. You are probably thinking that something is wrong.
As it turns out, what we have just shown is that the rate implied by the Eurodollar contract cannot equal the prevailing FRA (implied forward) rate for the same loan. To see this, consider the borrower perspective: When the interest rate turns out to be high, the short
Eurodollar contract has a positive payoff and the proceeds can be reinvested until the loan payment date at the high realized rate. When the interest rate turns out to be low, the short
Eurodollar contract has a negative payoff and we can fund this loss until the loan payment date by borrowing at a low rate. Thus the settlement structure of the Eurodollar contract works systematically in favor of the borrower. By turning the argument around, we can verify that it systematically works against a lender.
The reason this happens with Eurodollars and not FRAs is that we have to make the tailing decision before we know the 3-month rate prevailing on day 120. When we tail by a fixed amount (1.8% in the above example), the actual variations in the realized rate work in favor of the borrower and against the lender. The FRA avoids this by automatically tailing— paying the present value of the change in borrowing cost—using the actual interest rate on the borrowing date.
In order for the futures price to be fair to both the borrower and lender, the rate implicit in the Eurodollar futures price must be higher than a comparable FRA rate. This difference between the FRA rate and the Eurodollar rate is called convexity bias. For the most part in subsequent discussions we will ignore convexity bias and treat the Eurodollar contract and FRAs as if they are interchangeable. The reason is that in many cases the effect is small. In the above example, convexity bias results in a profit of several hundred dollars

7.2 Forward Rate Agreements, Eurodollar Futures, and Hedging

out of a borrowing cost of $1.8m. For short-term contracts, the effect can be small, but for longer-term contracts the effect can be important.7
In practice, convexity bias also matters before the final contract settlement. We saw in Section 5.4 that marking-to-market a futures contract can lead to a futures price that is different from the forward price. When a futures contract is marked to market and interest rates are negatively correlated with the futures price, there is a systematic advantage to being short the futures contract. This leads to a futures price that is greater than the forward price.
This is exactly what happens with the Eurodollar contract in this example. When interest rates rise, the borrower receives a payment that can be invested at the higher interest rate.
When interest rates fall, the borrower makes a payment that can be funded at the lower interest rate. This works to the borrower’s benefit. Marking-to-market prior to settlement is therefore another reason why the rate implied by the Eurodollar contract will exceed that on an otherwise comparable FRA.
LIBOR Versus 3-Month T-Bills. The Eurodollar futures contract is based on LIBOR, but there are other 3-month interest rates. For example, the Treasury-bill futures contract is based on the price of the 3-month Treasury bill. A borrower could use either the Eurodollar contract or the Treasury-bill futures contract to hedge their borrowing rate. Which contract is preferable?
Banks that offer LIBOR time deposits have the potential to default. Thus, LIBOR includes a default premium. (The default premium is an increase in the interest rate that compensates the lender for the possibility the borrower will default.) Private companies that borrow can also default, so their borrowing rates will also include a default premium.
The U.S. government, by contrast, is considered unlikely to default, so it can borrow at a lower rate than firms. In addition, in the United States and other countries, government

7. If future interest rates were known for certain in advance, then it would be possible to perfectly tail the position. However, with uncertainty about rates, the error is due to interest on the difference between the realized rate, r, and the forward rate, rforward . Given that we tail by the forward rate, the error is measured by r (˜ − rforward )
˜ r
1 + rforward
The expected error is r r − rforward
˜ ˜
1 + rforward

=

1
E r 2 − E r rforward
˜
˜
1 + rforward

=

E

σ2
1 + rforward

where σ 2 is the variance of the interest rate. Rates in our example can be 2% or 1.5%, so the standard deviation is approximately 25 basis points, or 0.0025, and the variance is thus 0.00252 = 0.00000625.
Convexity bias is thus
$100m ×

0.00000625
= $613.95
1.018

The actual average convexity error in the example was ($884 + $393)/2, or $638.5.

209

210

Chapter 7. Interest Rate Forwards and Futures

FIGURE 7.2

90-day Interest Rates

The 3-month LIBOR rate and 3-month T-bill rate (top panel) and the difference between the two
(bottom panel), 1982–2011.
3-month T-bill

15

3-month LIBOR
14

5

0
01/04/84

01/03/86

01/05/90

01/06/94

01/07/98

01/08/02

01/09/06

01/08/10

01/08/02

01/09/06

01/08/10

Date

90-day Rate Spread (%)

5
4
3
2
1

0
01/04/84

01/03/86

01/05/90

01/06/94

01/07/98
Date

Source: Datastream.

bonds are more liquid than corporate bonds, and this results in higher prices—a liquidity premium—for government bonds.8
The borrower will want to use the futures contract that has a price that moves in tandem with its own borrowing rate. A private borrower’s interest rate will more closely track LIBOR than the Treasury-bill rate. In fact, the spread between corporate borrowing rates and Treasuries moves around a great deal. A private firm’s borrowing costs can increase even as the T-bill rate goes down; this can occur during times of financial distress, when investors bid up the prices of Treasury securities relative to other assets (a so-called “flight to quality”). Thus, LIBOR is commonly used in markets as a benchmark, high-quality, private interest rate.
Figure 7.2 shows historical 3-month LIBOR along with the difference between LIBOR and the 3-month T-bill yield, illustrating this variability.9 It is obvious that the spread

8. In the United States, another reason for government bonds to have higher prices than corporate bonds is that government bond interest is exempt from state taxation.
9. The TED spread (“T-Bills over Eurodollars”) is obtained by going long T-bill futures and short the
Eurodollar futures contract.

7.3 Duration and Convexity

varies considerably over time: Although the spread has been as low as a few basis points, twice in the 1990s it exceeded 100 basis points. In September of 1982, when Continental
Bank failed, and during the financial crisis of 2008 (see the box on p. 212), the spread exceeded 400 basis points. A private LIBOR-based borrower who had hedged its borrowing rate by shorting T-bill futures in August of 1982 would by September have lost money on the T-bill contract as Treasury rates declined, while the actual cost of borrowing (LIBOR) would have remained close to unchanged. This example illustrates the value of using a hedging contract that reflects the actual cost of borrowing.
The Eurodollar futures contract is far more popular than the T-bill futures contract.
Trading volume and open interest on the two contracts were about equal in the early 1980s.
However, in recent years, open interest on the Eurodollar contract has been millions of contracts, while the T-bill contract has had zero open interest. This is consistent with LIBOR being a better measure of private sector interest rates than the T-bill yield.

7.3 DURATION AND CONVEXITY
An important characteristic of a bond is the sensitivity of its price to interest rate changes, which we measure using duration. Duration tells us approximately how much the bond’s price will change for a given change in the bond’s yield. Duration is thus a summary measure of the risk of a bond, permitting a comparison of bonds with different coupons, times to maturity, and discounts or premiums relative to principal. In this section we also discuss convexity, which is another measure related to bond price risk.

Price Value of a Basis Point and DV01
We first compute the change in the bond price due to a change in the yield. Suppose a bond makes m coupon payments per year for T years in the amount C/m and pays M at maturity. Let y/m be the per-period yield to maturity (by convention, y is the annualized yield to maturity) and n = m × T the number of periods until maturity. The price of the bond, B(y), is given by n B(y) = i=1 C/m
M
+ i (1 + y/m)
(1 + y/m)n

The change in the bond price for a unit change in the yield, y, is10
Change in bond price
=−
Unit change in yield
=−

n i=1 i
C/m
M n − i+1 m (1 + y/m) m (1 + y/m)n+1

1
1 + y/m

n i=1 M i C/m n + i m (1 + y/m) m (1 + y/m)n

(7.9)

Equation (7.9) tells us the dollar change in the bond price for a change of 1.0 in y. It is natural to scale this either to reflect a change per percentage point [in which case we divide equation (7.9) by 100] or per basis point (divide equation (7.9) by 10,000). Equation (7.9) divided by 10,000 is also known as the price value of a basis point (PVBP) or the dollar

10. This is obtained by computing the derivative of the bond price with respect to the yield, dB(y)/dy.

211

212

Chapter 7. Interest Rate Forwards and Futures

BOX

7.1: Was LIBOR Accurate During the Financial Crisis?

Unlike most quoted interest rates, LIBOR is not based on a transaction price. Rather, the British
Bankers’ Association (BBA) conducts a daily survey of 16 large banks and asks about their borrowing rate for a $1 million time deposit. A bank that is regarded as safe will be able to borrow at a lower rate than one thought to be less creditworthy.
During the financial crisis, the BBA became concerned that some of the surveyed banks were not being truthful about their borrowing rates. In
April 2008, the Wall Street Journal reported on
LIBOR:
The world’s most widely used interest rate took its largest jump since the advent of the credit crisis in a sign that banks could be responding to increasing concerns that the rate doesn’t reflect their actual borrowing costs.
Thursday’s sudden jump in the dollardenominated London interbank offered rate, or Libor, comes after a decision Wednesday by the British Bankers’ Association to speed up an inquiry into the daily borrowing rates that banks provide to establish the Libor rate.

The move by the BBA, which oversees Libor, came amid concerns among bankers that their rivals were not reporting the high rates they were paying for short-term loans for fear of appearing desperate for cash. . . .
Some expect Libor to increase further.
William Porter, credit strategist at Credit Suisse, said he believes the three-month dollar rate is 0.4 percentage point below where it should be. That echoes the view of Scott Peng, a Citigroup Inc. analyst who said that Libor understated banks’ true borrowing costs by as much as 0.3 percentage points.
Source: Mollenkamp 2008.

Government investigations can move slowly.
Three years later, the Financial Times (Masters et al., 2011) reported that subpeonas were being issued to a handful of global banks as regulators investigated LIBOR manipulation between 2006 and 2008. The article noted that
LIBOR is used as a benchmark for $350 trillion in financial products, explaining the concern of regulators. value of an 01 (DV01). To interpret PVBP for a bond, we need to know the par value of the bond. Example 7.4 Consider the 3-year zero-coupon bond in Table 7.1 with a yield to maturity of 7%. The bond price per $100 of maturity value is $100/1.073 = $81.62979. At a yield of 7.01%, one basis point higher, the bond price is $100/1.07013 = $81.60691, a change of
−$0.02288 per $100 of maturity value.
As an alternative way to derive the price change, we can compute equation (7.9) with
C = 0, M = $100, n = 3, and m = 1 to obtain


$100
1
×3×
= −$228.87
1.07
1.073

In order for this to reflect a change of 1 basis point, we divide by 10,000 to obtain
−$228.87/10,000 = −$0.02289, almost equal to the actual bond price change. This illustrates the importance of scaling equation (7.9) appropriately.

7.3 Duration and Convexity

Duration
When comparing bonds with different prices and par values, it is helpful to have a measure of price sensitivity expressed per dollar of bond price. We obtain this by dividing equation
(7.9) by the bond price, B(y), and multiplying by −1. This gives us a measure known as modified duration, which is the percentage change in the bond price for a unit change in the yield:
Modified duration = −

=

1
Change in bond price
×
Unit change in yield
B(y)
n

1
1
B(y) 1 + y/m

i=1

i
C/m
M n + m (1 + y/m)i m (1 + y/m)n

(7.10)

We obtain another measure of bond price risk—Macaulay duration—by multiplying equation (7.10) by 1 + y/m.11 This puts both bond price and yield changes in percentage terms and gives us an expression with a clear interpretation:
Macaulay duration = −

=

Change in bond price 1 + y/m
×
Unit change in yield
B(y)
1
B(y)

n i=1 i
C/m
M n + m (1 + y/m)i m (1 + y/m)n

(7.11)

To interpret this expression, note that (C/m)/(1 + y/m)i is the present value of the ith bond payment, which occurs in i/m years. The quantity C/m/[(1 + y/m)i B(y)] is therefore the fraction of the bond value that is due to the ith payment. Macaulay duration is a weighted average of the time (number of periods) until the bond payments occur, with the weights being the percentage of the bond price accounted for by each payment. This interpretation of Macaulay duration as a time-to-payment measure explains why these measures of bond price sensitivity are called “duration.”12 For a zero-coupon bond, equation (7.11) implies that Macaulay duration equals time to maturity.
Macaulay duration illustrates why maturity alone is not a satisfactory risk measure for a coupon bond. A coupon bond makes a series of payments, each with a different maturity.
Macaulay duration summarizes bond price risk as a weighted average of these different maturities. Example 7.5
7% bond is

Returning to Example 7.4, using equation (7.11), Macaulay duration for the



−$228.87
1.07
×
= 3.000
1
$81.62979

11. This measure of duration is named after Frederick Macaulay, who wrote a classic history of interest rates (Macaulay, 1938).
12. The Excel duration functions are Duration for Macaulay duration and MDuration for modified duration.

213

214

Chapter 7. Interest Rate Forwards and Futures

Example 7.6 Consider the 3-year coupon bond in Table 7.1. For a par bond, the yield to maturity is the coupon, 6.95485% in this case. For each payment we have
0.0695485
= 0.065026
1.0695485
0.0695485
= 0.060798
%Payment 2 =
(1.0695485)2
1.0695485
= 0.874176
%Payment 3 =
(1.0695485)3
%Payment 1 =

Thus, with n = 3 and m = 1, Macaulay duration is
(1 × 0.065026) + (2 × 0.060798) + (3 × 0.874176) = 2.80915
The interpretation of the duration of 2.81 is that the bond responds to interest rate changes as if it were a pure discount bond with 2.81 years to maturity. Modified duration is
2.80915/1.0695485 = 2.626482.
Since duration tells us the sensitivity of the bond price to a change in the interest rate, it can be used to compute the approximate bond price change for a given change in interest rates. Suppose the bond price is B(y) and the yield on the bond changes from y to y + , where is a small change in the yield. The formula for Macaulay duration, DMac, can be written DMac = −

B(y + ) − B(y) 1 + y
B(y)

We can therefore rewrite this equation to obtain the new bond price in terms of the old bond price and either duration measure:
B(y + ) = B(y) − [D × B(y) ] = B(y) − [DMac/(1 + y) × B(y) ]

(7.12)

Example 7.7 Consider the 3-year zero-coupon bond with a price of $81.63 per $100 maturity value. The yield is 7% and the bond’s Macaulay duration is 3.0. If the yield were to increase to 7.25%, the predicted price would be
B(7.25%) = $81.63 − (3/1.07) × $81.63 × 0.0025 = $81.058
The actual new bond price is $100/(1.0725)3 = $81.060. The prediction error is about 0.02% of the bond price.
Although duration is an important concept and is frequently used in practice, it has a conceptual problem. We emphasized in the previous section that a coupon bond is a collection of zero-coupon bonds, and therefore each cash flow has its own discount rate.
Yet both duration formulas are computed assuming that all cash flows are discounted by a single artificial number, the yield to maturity. In Chapter 25 we will examine alternative approaches to measuring bond price risk.

Duration Matching
Suppose we own a bond with time to maturity t1, price B1, and Macaulay duration D1. We are considering a short position in a bond with maturity t2, price B2, and Macaulay duration

7.3 Duration and Convexity

D2. How much of the second bond should we short-sell so that the resulting portfolio—long the bond with duration D1 and short the bond with duration D2—is insensitive to interest rate changes?
Equation (7.12) gives us a formula for the change in price of each bond. Let N denote the quantity of the second bond. The value of the portfolio is
B1 + N B2 and, using equation (7.12), the change in price due to an interest rate change of is
B1(y1 + ) − B1(y1) + N B2(y2 + ) − B2(y2)
= −D1B1(y1) /(1 + y1) − N D2B2(y2) /(1 + y2) where D1 and D2 are Macaulay durations. If we want the net change to be zero, we choose
N to set the right-hand side equal to zero. This gives
N =−

D1B1(y1)/(1 + y1)
D2B2(y2)/(1 + y2)

(7.13)

When a portfolio is duration-matched in this fashion, the net investment in the portfolio will typically not be zero. That is, either the value of the short bond is less than the value of the long bond, in which case additional financing is required, or vice versa, in which case there is cash to invest. This residual can be financed or invested in very short-term bonds, with duration approximately zero, in order to leave the portfolio duration matched.
Example 7.8 Suppose we own a 7-year 6% coupon bond with a yield of 7% and want to find the duration-matched short position in a 10-year 8% coupon bond yielding 7.5%.
Assuming annual coupon payments, the Macaulay duration and price of the two bonds is
5.882 years and $94.611, and 7.297 years and $103.432, respectively. Thus, if we own one of the 7-year bonds, we must hold


5.882 × 94.611/(1.07)
= −0.7408
7.297 × 103.432/(1.075)

units of the 10-year bond. The short position in the 10-year bond is not enough to pay for the
7-year bond; hence, investment in the portfolio is 1 × 94.611 − 0.7408 × 103.432 = 17.99.
If the yield on both bonds increases 25 basis points, the price change of the portfolio is
−1.289 + (−0.7408) × −1.735 = −0.004

Convexity
The hedge in Example 7.8 is not perfect, because duration changes as the interest rate changes.13 Convexity measures the extent to which duration changes as the bond’s yield

13. At the original yields, we computed a hedge ratio of 0.7408. Problem 7.19 asks you to compute the hedge ratio that would have exactly hedged the portfolio had both interest rates increased 25 basis points and decreased 25 basis points. The two hedge ratios are different, which means that one hedge ratio would not have worked perfectly.

215

216

Chapter 7. Interest Rate Forwards and Futures

changes. The formula for convexity is14
Convexity =

1
B(y)

n i=1 i(i + 1)
C/m
M n(n + 1)
+
2 i+2 2 m (1 + y/m) m (1 + y/m)n+2

(7.14)

We can use convexity in addition to duration to obtain a more accurate prediction of the new bond price. When we include convexity, the price prediction formula, equation (7.12), becomes15 B(y + ) = B(y) − [D × B(y) × ] + 0.5 × Convexity × B(y) ×

2

(7.15)

where D is modified duration. Here is an example of computing a bond price at a new yield using both duration and convexity.
Example 7.9 Consider again Example 7.7. We want to predict the new price of a 3-year zero-coupon bond when the interest rate changes from 7% to 7.25%. Using equation (7.14) with C = 0, m = 1, and M = $100, convexity of the bond is
Convexity = 3 × 4 ×

1
100
= 10.4812
×
(3+2)
1.07
81.63

Using equation (7.15), the price at a yield of 7.25% is
B(7.25%) = $81.63 − (3/1.07) × $81.63 × 0.0025 + 0.5 × 10.4812 × $81.63 × 0.00252
= $81.060
The predicted price of $81.060 is the same as the actual price at a yield of 7.25%, to an accuracy of three decimal points. In Example 7.7, the predicted price was slightly lower
($81.058) than the actual new price. The difference without a convexity correction occurs because the bond’s sensitivity to the interest rate changes as the interest rate changes.16
Convexity corrects for this effect.
Figure 7.3 illustrates duration and convexity by comparing three bond positions that have identical prices at a yield of 10%. Duration is the slope of the bond price graph at a given yield, and convexity is the curvature of the graph. The 10% 10-year bond has the lowest duration and is the shallowest bond price curve. The other two bonds have almost equal durations at a yield of 10% and their slopes are equal in the figure. However, the 25year bond exhibits greater curvature: Its price is above the 10-year bond at both lower and higher yields. This greater curvature is what it means for the 25-year bond to have greater convexity. 14. This is obtained by taking the second derivative of the bond price with respect to the yield to maturity, d 2 B(y)/dy 2 , and normalizing the result by dividing by the bond price.
15. If you recall calculus, you may recognize equation (7.12) as a Taylor series expansion of the bond price.
See Appendix 13.A.
16. You might wonder about this statement since the bond in Example 7.7 is a zero-coupon bond, for which
Macaulay duration is constant. Notice, however, that the bond price prediction formula, equation (7.12), depends on modified duration, which is DMac/(1 + y). Modified duration does change with the yield on the bond.

7.4 Treasury-Bond and Treasury-Note Futures

FIGURE 7.3

1.4
1.3
Bond Price ($)

Comparison of the value of three bond positions as a function of the yield to maturity: 2.718 10-year zero-coupon bonds, one 10year bond paying a 10% annual coupon, and one 25year bond paying a 10% coupon. The duration (D) and convexity (C) of each bond at a yield of 10% are in the legend.

1.5

1.2

0% coupon 10-year bond
(D = 9.09, C = 90.91)
10% coupon 10-year bond
(D = 6.14, C = 52.79)
10% coupon 25-year bond
(D = 9.08, C = 139.58)

1.1
1.0
0.9
0.8
0.7
0.6
0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15
Yield to Maturity

The idea that using both duration and convexity provides a more accurate model of bond price changes is not particular to bonds, but it pertains to options as well. This is our first glimpse of a crucial idea in derivatives that will appear again in Chapter 13 when we discuss delta-gamma approximations, as well as throughout the book.

7.4 TREASURY-BOND AND TREASURY-NOTE FUTURES
The Treasury-note and Treasury-bond futures contracts are important instruments for hedging interest rate risk.17 The specifications for the T-note contract are listed in Figure 7.4.
The bond contract is similar except that the deliverable bond has a maturity of at least 15 years, or if the bond is callable, has 15 years to first call. The two contracts are similar; we will focus here on the T-note contract. In this discussion we will use the terms “bond” and
“note” interchangeably.
The basic idea of the T-note contract is that a long position is an obligation to buy a
6% bond with between 6.5 and 10 years to maturity. To a first approximation, we can think of the underlying as being like a stock with a dividend yield of 6%. The futures price would then be computed as with a stock index: the future value of the current bond price, less the future value of coupons payable over the life of the futures contract.
This description masks a complication that may already have occurred to you. The delivery procedure permits the short to deliver any note maturing in 6.5 to 10 years. Hence, the delivered note can be one of many outstanding notes, with a range of coupons and maturities. Which bond does the futures price represent?
Of all bonds that could be delivered, there will generally be one that is the most advantageous for the short to deliver. This bond is called the cheapest to deliver. A

17. The interest rate on the 10-year Treasury note is a commonly used benchmark interest rate; hence, the
10-year note futures are important.

217

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Chapter 7. Interest Rate Forwards and Futures

FIGURE 7.4
Specifications for the
Treasury-note futures contract.

Where traded
Underlying
Size
Months
Trading ends

Delivery

CME Group/CBOT
6% 10-year Treasury note
$100,000 Treasury note
March, June, September, December, out 15 months Seventh business day preceding last business day of month. Delivery until last business day of month.
Physical T-note with at least 6.5 years to maturity and not more than 10 years to maturity. Price paid to the short for notes with other than 6% coupon is determined by multiplying futures price by a conversion factor. The conversion factor is the price of the delivered note ($1 par value) to yield
6%. Settlement until last business day of the month. description of the delivery procedure will demonstrate the importance of the cheapest-todeliver bond.
In fulfilling the note futures contract, the short delivers the bond in exchange for payment. The payment to the short—the invoice price for the delivered bond—is the futures price times the conversion factor. The conversion factor is the price of the bond if it were priced to yield 6%. Thus, the short delivering a bond is paid18
Invoice price = (Futures price × conversion factor) + accrued interest
Example 7.10 Consider two bonds making semiannual coupon payments. Bond A is a
7% coupon bond with exactly 8 years to maturity, a price of 103.71, and a yield of 6.4%.
This bond would have a price of 106.28 if its yield were 6%. Thus its conversion factor is
1.0628.
Bond B has 7 years to maturity and a 5% coupon. Its current price and yield are 92.73 and 6.3%. It would have a conversion factor of 0.9435, since that is its price at a 6% yield.

Now suppose that the futures contract is close to expiration, the observed futures price is 97.583, and the only two deliverable bonds are Bonds A and B. The short can decide which bond to deliver by comparing the market value of the bond to its invoice price if delivered.
For Bond A we have
Invoice price − market price = (97.583 × 1.0628) − 103.71 = 0.00

18. Appendix 7.A contains a definition of accrued interest.

7.4 Treasury-Bond and Treasury-Note Futures

TABLE 7.5

Prices, yields, and the conversion factor for two bonds.
The futures price is 97.583. The short would break even delivering the 8-year 7% bond, and lose money delivering the 7-year 5% bond. Both bonds make semiannual coupon payments. 8-Year 7% Coupon,
6.4% Yield

Description
Market price
Price at 6% (conversion factor)
Invoice price (futures × conversion factor)
Invoice − market

7-Year 5% Coupon,
6.3% Yield

103.71
106.28

92.73
94.35

103.71
0

92.09
−0.66

For Bond B we have
Invoice price − market price = (97.583 × 0.9435) − 92.73 = −0.66
These calculations are summarized in Table 7.5.
Based on the yields for the two bonds, the short breaks even delivering the 8-year 7% bond and would lose money delivering the 7-year 5% coupon bond (the invoice price is less than the market price). In this example, the 8-year 7% bond is thus the cheapest to deliver.
In general there will be a single cheapest-to-deliver bond. You might be wondering why both bonds are not equally cheap to deliver. The reason is that the conversion factor is set by a mechanical procedure (the price at which the bond yields 6%), taking no account of the current relative market prices of bonds. Except by coincidence, two bonds will not be equally cheap to deliver.
Also, all but one of the bonds must have a negative delivery value. If two bonds had a positive delivery value, then arbitrage would be possible. The only no-arbitrage configuration in general has one bond worth zero to deliver (Bond A in Example 7.10) and the rest lose money if delivered. To avoid arbitrage, the futures price is
Futures price =

Price of cheapest to deliver
Conversion factor for cheapest to deliver

(7.16)

This discussion glosses over subtleties involving transaction costs (whether you already own a bond may affect your delivery profit calculation) and uncertainty before the delivery period about which bond will be cheapest to deliver. Also the T-note is deliverable at any time during the expiration month, but trading ceases with 7 business days remaining.
Consequently, if there are any remaining open contracts during the last week of the month, the short has the option to deliver any bond at a price that might be a week out of date. This provides a delivery option for the short that is also priced into the contract.
The T-bond and T-note futures contracts have been extremely successful. The contracts illustrate some important design considerations for a futures contract. Consider first how the contract is settled. If the contract designated a particular T-bond as the underlying

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asset, that T-bond could be in short supply, and in fact it might be possible for someone to corner the available supply. (A market corner occurs when someone buys most or all of the deliverable asset or commodity.) A short would then be unable to obtain the bond to deliver. In addition, the deliverable T-bond would change from year to year and the contract would become more complicated, since traders would have to price the futures differently to reflect different underlying bonds for different maturity dates.
An alternative scheme could have had the contract cash-settle against a T-bond index, much like the S&P 500. This arrangement, however, introduces basis risk, as the T-bond futures contract might then track the index but fail to track any particular bond.
In the end, settlement procedures for the T-bond and T-note contracts permitted a range of bonds and notes to be delivered. Since a high-coupon bond is worth more than an otherwise identical low-coupon bond, there had to be a conversion factor, in order that the short is paid more for delivering the high-coupon bond.
The idea that there is a cheapest to deliver is not exclusive to Treasury bonds. The same issue arises with commodities, where a futures contract may permit delivery of commodities at different locations or of different qualities.

7.5 REPURCHASE AGREEMENTS
A repurchase agreement, or repo, is the sale of a security, with the seller agreeing to buy the security back at a prespecified price at a later date. The counterparty to a repo enters into a reverse repurchase agreement, or reverse repo, which is the purchase of a security, with the buyer agreeing to sell it at the prespecified price at the later date. The repo is a reverse cash-and-carry—a sale coupled with a long forward position. The reverse repo is a cash-and-carry—a purchase coupled with a short forward position.
A repo is at bottom a simple idea. The borrower (the repo) owns a security and would like to temporarily obtain cash. Similarly, the lender (the reverse repo) has cash and would like to lend the cash to earn a return. The security owner is able to borrow cash using the security as collateral for the loan. While the repo is in place, however, the lender owns the bond. The party who repos, however, owns the bond both before and after the repo and therefore bears the long-run risk of owning the security. Although the mechanics of a repo can seem arcane, it is a collateralized loan. We will use the language of loans to talk about repos. The implicit interest rate in the transaction is the repo rate: the annualized percentage difference between the sale and purchase price. Repos are common in bond markets, but a repurchase agreement can be used for any asset. Most repos are overnight. A longer-term repurchase agreement is called a term repo.
Example 7.11 Suppose you enter into a 1-week repurchase agreement for a 9-month $1m
Treasury bill. The current price of the T-bill is $956,938, and you agree to repurchase it in 1 week for $958,042. You have borrowed money at a 1-week rate of 958,042/956,938 − 1 =
0.115%, receiving cash today and promising to repay cash plus interest in a week. The security provides collateral for the loan.
An important feature of a repo is that the actual cash amount exchanged for the security can be less than the market value of the security serving as collateral. The amount by which the value of the collateral exceeds the amount of the loan is called a haircut. The haircut provides a cushion should the security fall in value and the borrower fail to repay the loan.

7.5 Repurchase Agreements

The size of the haircut reflects the credit risk of the borrower as well as the risk of the collateral. A 2% haircut would mean that a borrower repoing a security worth $102 would receive a loan of only $100. Collateral with a more variable price and a less liquid market is lower quality from the perspective of the lender and typically would require a greater haircut. Repurchase agreements played an important role in the financial crisis in 2008. The box on p. 223 discusses this.
Repurchase agreements often use government securities as collateral, and can be negotiated to require a specific security as collateral—called a special collateral repurchase agreement—or with any of a variety of government securities as collateral—called a general collateral repurchase agreement. General collateral repos have greater flexibility and hence lower transaction costs.
The repo rate on special collateral repos will generally be below that on general collateral repos. (The borrower obtains a more favorable rate because the collateral is more desirable.) With a low enough repo rate, the original bondholder can earn interest on the cash received for the bond that exceeds the repo rate. The borrower thereby profits from the specialness of the bond.
When two parties transact a repo directly with each other, the transaction is known as a bilateral repo. It is also common for the two parties to agree to a repo but then to use a tri-party agent to provide operational assistance, such as the transferring of collateral and cash between the two parties, and valuing collateral. This is called a tri-party repo.19
Repurchase agreements can be used by dealers as a form of financing. The purchase of a security requires funds. A dealer can buy a bond and then repo it overnight. The money raised with the repo, together with the haircut, provides the cash needed to pay the bond seller. The dealer then has a cost of carrying the bond equal to the repo rate plus the capital cost of the haircut. The counterparty on this transaction is a lender with cash to invest shortterm, such as a money-market fund.
The same techniques can be used to finance speculative positions. A hedge fund speculating on the price difference between two Treasury bonds—a transaction known as a
“convergence trade”—can finance the transaction with repos. The hedge fund will undertake the following two transactions simultaneously:
The long position. Buy bond A and repo it, using the cash received in the repo to pay for the bond. When it is time to unwind the repo, close the repo position and sell the bond, using the cash raised from the sale to repay the loan (think of the bond sale and close of the repo as happening simultaneously). Note that a low repo rate for this bond works to the arbitrageur’s advantage, since it means that the interest rate on the loan is low. The arbitrageur also benefits from a price increase on the bond.
The short position. Borrow bond B by entering into a reverse repurchase agreement.
Obtain the bond (collateral for the loan) via the reverse repo, sell it, and use the proceeds to lend to the counterparty. At the termination of the agreement, buy the bond back in the open market and return it, being paid the repo rate. Since we receive interest in this transaction, a high repo rate works to the arbitrageur’s advantage as does a price decrease on the bond.

19. For addditional details about the workings of the repo market, with a focus on tri-party repo, see Federal
Reserve Bank of New York (2010).

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One well-known convergence trade involves newly issued on-the-run 30-year Trea1 sury bonds, which typically sell at a lower yield than the almost identical off-the-run 29 2 20 One would expect that the yields of the 30-year and 29 1 -year bonds year Treasury bond.
2
would converge as the 30-year bond aged and became off-the-run. As described above, traders making this bet would reverse repo the on-the-run bond (betting that its price will fall) and repo the off-the-run bond (betting that its price will rise). Traders profit from a convergence in price.
The arbitrageur in this situation would like a low repo rate on the purchased bond and a high repo rate on the sold bond, as well as a price increase of the purchased bond relative to the short-sold bond.21 Even if the price gap between the two bonds closes, the arbitrage can be prohibitively costly if the difference in repo rates on the two bonds is sufficiently great. Finally, the size of the haircut will affect the desirability of the trade. The box on
p. 224 describes the use of repos by Long-Term Capital Management, a hedge fund.

CHAPTER SUMMARY
The price of a zero-coupon bond with T years to maturity tells us the value today of $1 to be received at time T . The set of these bond prices for different maturities is the zerocoupon yield curve and is the basic input for present value calculations. There are equivalent ways to express the same information about interest rates, including the par coupon rate and implied forward rates.
Forward rate agreements (FRAs) permit borrowers and lenders to hedge the interest rate by locking in the implied forward rate. If the interest rate changes, FRAs require a payment reflecting the change in the value of the interest rate as of the loan’s maturity day. Eurodollar contracts are an alternative to FRAs as a hedging mechanism. However,
Eurodollar contracts make payment on the initiation date for the loan rather than the maturity date, so there is a timing mismatch between the Eurodollar payment and the interest payment date. This gives rise to convexity bias, which causes the rate implied by the Eurodollar contract to be greater than that for an otherwise equivalent FRA. Treasury bill contracts are yet another possible hedging vehicle, but suffer from basis risk since the change in the government’s borrowing rate may be different from the change in the borrowing rate for a firm or individual.
PVBP and DV01 measure the dollar change in a bond’s price when the yield increases by one basis point.
Modified duration is the percentage change in the bond price for a unit change in the interest rate. Macaulay duration is the percentage change in the bond price for a percentage change in the discount factor. Duration is not a perfect measure of bond price risk. A portfolio

20. An “on-the-run” Treasury bond is the most recently issued and generally the most heavily traded.
Other bonds are “off-the-run.” Krishnamurthy (2002) studies the behavior of the spread between the onand off-the-run bonds.
21. Cornell and Shapiro (1989) document one well-known episode of on-the-run/off-the-run arbitrage in which the repo rate on an on-the-run (short-sold) bond went to zero, making the arbitrage position costly.
Moreover, the price gap remained when the on-the-run bond became off-the-run.

Chapter Summary

BOX

223

7.2: Repo in the 2008 Financial Crisis

Repurchase agreements are not as familiar as stocks and bonds. During and after the 2008 financial crisis, however, repo transactions made headlines. Three aspects of repo were noteworthy.
First, the volume of repo was enormous. Second, most repo agreements are for short time periods, often overnight. This contributed to the fragility of financial institutions that were heavily reliant on repo financing. Third, repo transactions were used to manipulate accounting reports, most notably at Lehman Brothers and Bank of America.
Surprisingly, no one knows exactly how big the repo market is. There is agreement that the outstanding amount of repo in the U.S. and Europe is in the trillions of dollars, with estimates as great as $10 trillion in each locale (H¨ rdahl and King, o 2008; Gorton and Metrick, 2010). One difficulty of measuring repo is that a given security may be repoed multiple times. For example, a moneymarket fund may have cash to lend and a dealer may have a bond to use as collateral. The dealer may repo the bond to another dealer who in turn repos it to the money market fund. The act of repoing a security that is itself serving as collateral is called rehypothecation. From an economic perspective, the same bond has been loaned twice for the same ultimate economic purpose; the multiple transactions due to rehypothecation should count as one transaction (the loan of one security for cash) rather than two.
Repurchase agreements are widely used as a way to finance the holding of a long-term asset.
A bank could buy a bond (e.g., a mortgagebacked obligation) and finance the bond with a repurchase agreement. If the haircut were 5%,

the bank could buy a $105 bond by investing $5 of capital and repoing the bond for $100. When the repo expires, the bank must either renew the repo or find a new repo counterparty. A risk of repo finance is that if the bank gets into trouble, or if the bond’s risk increases, the counterparty might refuse to continue lending or might increase the haircut. Gorton and Metrick (2009) argue that increases in haircuts on mortgage-backed instruments caused a “run on repo.” If haircuts increase significantly, a bank that is heavily repo-financed will have to sell assets, possibly realizing losses, contributing to (or revealing) price declines for the assets, and causing lenders to flee. Lenders to Bear Stears and Lehman, for example, withdrew repo financing in the days before they failed.
Finally, the bankruptcy of Lehman provided a glimpse into the use of repo to manipulate accounting statements. The Lehman bankruptcy examiner found that Lehman systematically understated its leverage on its accounting reports
(Valukas, 2010). At each quarter end, Lehman engaged in a strategy known as “Repo 105,” in which it would use repos to sell assets, receive cash, and use the cash to buy down debt. It would reverse the transaction at the beginning of the next quarter, borrowing to buy the assets back. The repos were reported as asset sales rather than repos.
It was later disclosed that Citigroup and Bank of America did the same thing, although both banks claimed that they did not intend to create misleading financial statements (see Rapoport,
2010).

is said to be duration-matched if it consists of short and long bond positions with equal valueweighted durations. Convexity is a measure of the change in duration as the bond’s yield to maturity changes.
Treasury-note and Treasury-bond futures contracts have Treasury notes and bonds as underlying assets. A complication with these contracts is that a range of bonds are

224

Chapter 7. Interest Rate Forwards and Futures

BOX

7.3: Long-Term Capital Management

Repurchase agreements achieved particular notoriety during the Long-Term Capital Management (LTCM) crisis in 1998. LTCM was a hedge fund with a luminous roster of partners, including star bond trader John Meriwether, former Federal
Reserve Vice Chairman David Mullins, and academics Robert Merton and Myron Scholes, who won the Nobel Prize in Economics while associated with LTCM.
Many of LTCM’s strategies involved convergence trades. In his book about LTCM, Lowenstein (2000, p. 45) described one trade like this:
“No sooner did Long-Term buy the off-the-run bonds than it loaned them to some other Wall
Street firm, which then wired cash to Long-Term

as collateral. Then Long-Term turned around and used this cash as collateral on the bonds that it borrowed. The collateral it paid equaled the collateral it collected. In other words, Long-Term pulled off the entire $2 billion trade without using a dime of its own cash.” (Emphasis in original.) When LTCM failed in the fall of 1998, it had many such transactions and thus potentially many creditors. The difficulty of unwinding all of these intertwined positions was one of the reasons the Fed brokered a buyout of LTCM by other banks, rather than have LTCM explicitly declare bankruptcy. deliverable, and there is a cheapest to deliver. The futures price will reflect expectations about which bond is cheapest to deliver.
Repurchase agreements and reverse repurchase agreements are synthetic short-term borrowing and lending, the equivalent of reverse cash-and-carry and cash-and-carry transactions.

FURTHER READING
Basic interest rate concepts are fundamental in finance and are used throughout this book.
Some of the formulas in this chapter will appear again as swap rate calculations in Chapter 8. Chapter 15 shows how to price bonds that make payments denominated in foreign currencies or commodities, and how to price bonds containing options. While the bond price calculations in this chapter are useful in practice, concepts such as duration have conceptual problems. In Chapter 25, we will see how to build a coherent, internally consistent model of interest rates and bond prices.
Useful references for bond and money market calculations are Stigum (1990) and
Stigum and Robinson (1996). Veronesi (2010), Sundaresan (2009) and Tuckman (1995) are fixed-income texts that go into topics in this chapter in more depth. Convexity bias is studied by Burghardt and Hoskins (1995) and Gupta and Subrahmanyam (2000). Grinblatt and Longstaff (2000) discuss the market for STRIPS and study the pricing relationships between Treasury bonds and STRIPS. The repo market is discussed in Fleming and Garbade
(2002, 2003, 2004).

Problems

PROBLEMS
7.1 Suppose you observe the following zero-coupon bond prices per $1 of maturity payment: 0.96154 (1-year), 0.91573 (2-year), 0.87630 (3-year), 0.82270 (4-year),
0.77611 (5-year). For each maturity year compute the zero-coupon bond yields
(effective annual and continuously compounded), the par coupon rate, and the 1year implied forward rate.
7.2 Using the information in the previous problem, find the price of a 5-year coupon bond that has a par payment of $1,000.00 and annual coupon payments of $60.00.
7.3 Suppose you observe the following effective annual zero-coupon bond yields: 0.030
(1-year), 0.035 (2-year), 0.040 (3-year), 0.045 (4-year), 0.050 (5-year). For each maturity year compute the zero-coupon bond prices, continuously compounded zero-coupon bond yields, the par coupon rate, and the 1-year implied forward rate.
7.4 Suppose you observe the following 1-year implied forward rates: 0.050000 (1year), 0.034061 (2-year), 0.036012 (3-year), 0.024092 (4-year), 0.001470 (5-year).
For each maturity year compute the zero-coupon bond prices, effective annual and continuously compounded zero-coupon bond yields, and the par coupon rate.
7.5 Suppose you observe the following continuously compounded zero-coupon bond yields: 0.06766 (1-year), 0.05827 (2-year), 0.04879 (3-year), 0.04402 (4-year),
0.03922 (5-year). For each maturity year compute the zero-coupon bond prices, effective annual zero-coupon bond yields, the par coupon rate, and the 1-year implied forward rate.
7.6 Suppose you observe the following par coupon bond yields: 0.03000 (1-year),
0.03491 (2-year), 0.03974 (3-year), 0.04629 (4-year), 0.05174 (5-year). For each maturity year compute the zero-coupon bond prices, effective annual and continuously compounded zero-coupon bond yields, and the 1-year implied forward rate.
7.7 Using the information in Table 7.1,
a. Compute the implied forward rate from time 1 to time 3.
b. Compute the implied forward price of a par 2-year coupon bond that will be issued at time 1.
7.8 Suppose that in order to hedge interest rate risk on your borrowing, you enter into an
FRA that will guarantee a 6% effective annual interest rate for 1 year on $500,000.00.
On the date you borrow the $500,000.00, the actual interest rate is 5%. Determine the dollar settlement of the FRA assuming
a. Settlement occurs on the date the loan is initiated.
b. Settlement occurs on the date the loan is repaid.
7.9 Using the same information as the previous problem, suppose the interest rate on the borrowing date is 7.5%. Determine the dollar settlement of the FRA assuming
a. Settlement occurs on the date the loan is initiated.
b. Settlement occurs on the date the loan is repaid.

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Chapter 7. Interest Rate Forwards and Futures

Use the following zero-coupon bond prices to answer the next three questions:
Days to
Maturity

Zero-Coupon
Bond Price

90

0.99009

180
270

0.97943
0.96525

360

0.95238

7.10 What is the rate on a synthetic FRA for a 90-day loan commencing on day 90? A
180-day loan commencing on day 90? A 270-day loan commencing on day 90?
7.11 What is the rate on a synthetic FRA for a 180-day loan commencing on day 180?
Suppose you are the counterparty for a borrower who uses the FRA to hedge the interest rate on a $10m loan. What positions in zero-coupon bonds would you use to hedge the risk on the FRA?
7.12 Suppose you are the counterparty for a lender who enters into an FRA to hedge the lending rate on $10m for a 90-day loan commencing on day 270. What positions in zero-coupon bonds would you use to hedge the risk on the FRA?
7.13 Using the information in Table 7.1, suppose you buy a 3-year par coupon bond and hold it for 2 years, after which time you sell it. Assume that interest rates are certain not to change and that you reinvest the coupon received in year 1 at the 1-year rate prevailing at the time you receive the coupon. Verify that the 2-year return on this investment is 6.5%.
7.14 As in the previous problem, consider holding a 3-year bond for 2 years. Now suppose that interest rates can change, but that at time 0 the rates in Table 7.1 prevail. What transactions could you undertake using forward rate agreements to guarantee that your 2-year return is 6.5%?
7.15 Consider the implied forward rate between year 1 and year 2, based on Table 7.1.
a. Suppose that r0(1, 2) = 6.8%. Show how buying the 2-year zero-coupon bond and borrowing at the 1-year rate and implied forward rate of 6.8% would earn you an arbitrage profit.
b. Suppose that r0(1, 2) = 7.2%. Show how borrowing the 2-year zero-coupon bond and lending at the 1-year rate and implied forward rate of 7.2% would earn you an arbitrage profit.
7.16 Suppose the September Eurodollar futures contract has a price of 96.4. You plan to borrow $50m for 3 months in September at LIBOR, and you intend to use the
Eurodollar contract to hedge your borrowing rate.
a. What rate can you secure?
b. Will you be long or short the Eurodollar contract?
c. How many contracts will you enter into?
d. Assuming the true 3-month LIBOR is 1% in September, what is the settlement in dollars at expiration of the futures contract? (For purposes of this question, ignore daily marking-to-market on the futures contract.)

Problems

7.17 A lender plans to invest $100m for 150 days, 60 days from today. (That is, if today is day 0, the loan will be initiated on day 60 and will mature on day 210.) The implied forward rate over 150 days, and hence the rate on a 150-day FRA, is 2.5%. The actual interest rate over that period could be either 2.2% or 2.8%.
a. If the interest rate on day 60 is 2.8%, how much will the lender have to pay if the FRA is settled on day 60? How much if it is settled on day 210?
b. If the interest rate on day 60 is 2.2%, how much will the lender have to pay if the FRA is settled on day 60? How much if it is settled on day 210?
7.18 Consider the same facts as the previous problem, only now consider hedging with the
3-month Eurodollar futures. Suppose the Eurodollar futures contract that matures 60 days from today has a price on day 0 of 94.
a. What issues arise in using the 3-month Eurodollar contract to hedge a 150-day loan? b. If you wish to hedge a lending position, should you go long or short the contract? c. What 3-month LIBOR is implied by the Eurodollar futures price? Approximately what lending rate should you be able to lock in?
d. What position in Eurodollar futures would you use to lock in a lending rate? In doing this, what assumptions are you making about the relationship between
90-day LIBOR and the 150-day lending rate?
7.19 Consider the bonds in Example 7.8. What hedge ratio would have exactly hedged the portfolio if interest rates had decreased by 25 basis points? Increased by 25 basis points? Repeat assuming a 50-basis-point change.
7.20 Compute Macaulay and modified durations for the following bonds:
a. A 5-year bond paying annual coupons of 4.432% and selling at par.
b. An 8-year bond paying semiannual coupons with a coupon rate of 8% and a yield of 7%.
c. A 10-year bond paying annual coupons of 6% with a price of $92 and maturity value of $100.
7.21 Consider the following two bonds which make semiannual coupon payments: a 20year bond with a 6% coupon and 20% yield, and a 30-year bond with a 6% coupon and a 20% yield.
a. For each bond, compute the price value of a basis point.
b. For each bond, compute Macaulay duration.
c. “For otherwise identical bonds, Macaulay duration is increasing in time to maturity.” Is this statement always true? Discuss.
7.22 An 8-year bond with 6% annual coupons and a 5.004% yield sells for $106.44 with a Macaulay duration of 6.631864. A 9-year bond has 7% annual coupons with a
5.252% yield and sells for $112.29 with a Macaulay duration of 7.098302. You wish

227

228

Chapter 7. Interest Rate Forwards and Futures

to duration-hedge the 8-year bond using a 9-year bond. How many 9-year bonds must we short for every 8-year bond?
7.23 A 6-year bond with a 4% coupon sells for $102.46 with a 3.5384% yield. The conversion factor for the bond is 0.90046. An 8-year bond with 5.5% coupons sells for $113.564 with a conversion factor of 0.9686. (All coupon payments are semiannual.) Which bond is cheaper to deliver given a T-note futures price of 113.81?
7.24

a. Compute the convexity of a 3-year bond paying annual coupons of 4.5% and selling at par.
b. Compute the convexity of a 3-year 4.5% coupon bond that makes semiannual coupon payments and that currently sells at par.
c. Is the convexity different in the two cases? Why?

7.25 Suppose a 10-year zero-coupon bond with a face value of $100 trades at $69.20205.
a. What is the yield to maturity and modified duration of the zero-coupon bond?
b. Calculate the approximate bond price change for a 50-basis-point increase in the yield, based on the modified duration you calculated in part (a). Also calculate the exact new bond price based on the new yield to maturity.
c. Calculate the convexity of the 10-year zero-coupon bond.
d. Now use the formula (equation (7.15)) that takes into account both duration and convexity to approximate the new bond price. Compare your result to that in part (b).

Appendix 7.A INTEREST RATE AND BOND
PRICE CONVENTIONS
This appendix will focus on conventions for computing yields to maturity for different kinds of bonds, and the conventions for quoting bond prices. When discussing yields to maturity, it is necessary to distinguish on the one hand between notes and bonds, which make coupon payments and are issued with more than 1 year to maturity, and on the other hand bills, which have no coupons and are issued with 1 year or less to maturity. The quotation conventions are different for notes and bonds than for bills. For a full treatment of bond pricing and quoting conventions, see Stigum and Robinson (1996).

Bonds
We first consider notes and bonds, which we will refer to as just “bonds.” Bond coupons and yields are annualized. If a bond is described as paying a 6% semiannual coupon, this means that the bond pays 6%/2 = 3% every 6 months. Further, if the bond yield is 7%, this means that the bond’s 6-month yield to maturity is 7%/2 = 3.5%. Bond coupons and yields are annualized by multiplying by 2 rather than by compounding.

7.A Interest Rate and Bond Price Conventions

Suppose a bond makes semiannual coupon payments of C/2, and has a semiannual yield of y/2. The quoted coupon and yield are C and y.22 Let d be the actual number of days until the next coupon, and d the number of days between the previous and next coupon.
We take into account a fractional period until the next coupon by discounting the cash flows for that fractional period. The price of the bond is n B(y) = i=1 C/2
M
+ i−1+d/d (1 + y/2)
(1 + y/2)n−1+d/d

This can be rewritten as
B(y) =

1
1 + y/2

d/d

C/2 +

1
C/2
M
1−
+ y/2 (1 + y/2)n−1
(1 + y/2)n−1

(7.17)

In the special case when the bond has just paid a coupon, then d = d , and equation (7.17) becomes n

B(y) = i=1 C/2
M
+
(1 + y/2)i
(1 + y/2)n

(7.18)

This formula assumes there is one full period until the next coupon.
Example 7.12 Consider a 7% $100 maturity coupon bond that makes semiannual coupon payments on February 15 and August 15 and matures on August 15, 2012. Suppose it is
August 15, 2004, and the August coupon has been paid. There are 16 remaining payments.
If the semiannual yield, y, is 3.2%, then using equation (7.18), the price of the bond is
$3.5
1
$100
1−
= $103.71
+
16
0.032
(1 + 0.032)
(1 + 0.032)16
Example 7.13 Consider the same bond as in Example 7.12. Suppose that on November
11, 2004, the semiannual yield is still 3.2%. There are 96 days until the February coupon payment and 184 days between the August and February payments. Using equation (7.17), the price for the bond at a 6.4% yield (3.2% semiannual) is
1
1.032

96/184

$3.5 +

$100
1
$3.5
+
1−
0.032
(1.032)15
(1.032)15

= $105.286

The bond-pricing formulas in Examples 7.12 and 7.13 illustrate that even with a constant yield to maturity, the bond price will vary with the time until the next coupon payment. This occurs because equation (7.17) computes a bond price that fully reflects the coming coupon payment. Using this formula, the bond price rises over time as a coupon payment approaches, then falls on the coupon payment date, and so forth. The bond price quoted in this fashion is called the dirty price.
Intuitively, if you buy a bond three-fourths of the way from one coupon payment to the next, the price you pay should reflect three-fourths of the coming coupon payment.

22. If a bond makes coupon payments m times a year, the convention is to quote the coupon rate as m times the per-period payment. The yield to maturity is computed per payment period and multiplied by m to obtain the annual quoted yield.

229

230

Chapter 7. Interest Rate Forwards and Futures

TABLE 7.6

Treasury bill quotations.

Maturity
February 13
December 18

Days to Maturity

Ask Discount

Ask Yield

43
351

3.65
3.87

3.72
4.04

This prorated amount is accrued interest, which is included in the price in equation (7.17).
Accrued interest is calculated as the prorated portion of the coupon since the last coupon date. With d − d days since the last coupon, accrued interest is C × (d − d)/d .
In practice, bond prices are quoted net of accrued interest. The dirty price less accrued interest is the clean price, which does not exhibit the predictable rise and fall in price due to the coming coupon payment.23
Example 7.14 Consider the bond in Example 7.13. Accrued interest as of November 11 would be 3.5 × (184 − 96)/184 = 1.674. Thus, the clean price for the bond would be
Clean price = Dirty price − accrued interest
= $105.286 − $1.674 = $103.612

Bills
Table 7.6 presents typical Treasury-bill quotations. Suppose today is January 1. A bond maturing February 13 has 43 days to maturity and one maturing December 18 has 351 days to maturity (assuming it is not a leap year).
The “ask yields” in this table are “bond-equivalent yields” (Stigum and Robinson,
1996), intended to make Treasury-bill yields comparable to Treasury-bond yields. To obtain the yields, we first find the market prices of the T-bills. A T-bill price is quoted on an annualized discount basis. The discount is the number subtracted from 100 to obtain the invoice price for the T-bill, P . The formula, normalizing the face value of the T-bill to be
100, is
P = 100 −

discount × days
360

The T-bills in Table 7.6 have invoice prices of
100 − 3.65 × 43/360 = 99.5640
100 − 3.87 × 351/360 = 96.2268
Thus, an investor pays 0.995640 per dollar of maturity value for the 43-day bill and 0.962268 for the 351-day bill. Note that these prices give us “true” 43-day and 351-day discount factors. Given the prices, what are the yields?
100
A 43-day bill yields 99.5640 = 1.004379 or 0.4379% over 43 days, while the 351-day bill yields 3.9212% over 351 days. The bond-equivalent yield calculations annualize these

23. Because accrued interest is amortized linearly rather than geometrically, this statement is not precisely true; see Smith (2002).

7.A Interest Rate and Bond Price Conventions

yields in a way that makes them more comparable to bond yields. This necessarily involves making arbitrary assumptions.
For bills less than 182 days from maturity, a bill is directly comparable to a maturing bond since neither makes a coupon payment over that period. In this case we need only to adjust for the fact that bonds are quoted using the actual number of days (i.e., a 365-day basis) and bills are quoted on a 360-day basis: rbe =

365 × discount/100
360 − discount/100 × days

where rbe stands for “bond-equivalent yield.” Applying this formula to the 43-day T-bill, we see that
365 × 0.0365
= 0.0372
360 − 0.0365 × 43
If you use this formula for the 351-day bill, however, you obtain a yield of 4.078 rather than the 4.04 listed in Table 7.6. The bond-equivalent yield calculation for this bill takes into account that a bond with more than 182 days to maturity would make a coupon payment. Hence, to make the bill yield comparable to that for a bond, we need to account for the imaginary coupon. The formula from Stigum and Robinson (1996) is


2×days
365

+2

days
365

rbe =

2



2×days
365

2×days
365

−1

1−

100
P

(7.19)

−1

Applying this to the 351-day bond gives
− 2×351 + 2
365

351
365

2



2×351
365

2×351
365

−1

−1

1−

1
0.962268

= 0.040384

This matches the quoted yield in Table 7.6. In Excel, the function TBILLEQ provides the bond-equivalent yield for a T-bill.

231

8

Swaps

T

hus far we have talked about derivatives contracts that settle on a single date. A forward contract, for example, fixes a price for a transaction that will occur on a specific date in the future. However, many transactions occur repeatedly. Firms that issue bonds make periodic coupon payments. Multinational firms frequently exchange currencies. Firms that buy commodities as production inputs or that sell them make payments or receive income linked to commodity prices on an ongoing basis.
These situations raise the question: If a manager seeking to reduce risk confronts a risky payment stream—as opposed to a single risky payment—what is the easiest way to hedge this risk? One obvious answer is that we can enter into a separate forward contract for each payment we wish to hedge. However, it might be more convenient, and entail lower transaction costs, if we could hedge a stream of payments with a single transaction.
A swap is a contract calling for an exchange of payments over time. One party makes a payment to the other depending upon whether a reference price turns out to be greater or less than a fixed price that is specified in the swap contract. A swap thus provides a means to hedge a stream of risky payments. By entering into an oil swap, for example, an oil buyer confronting a stream of uncertain oil payments can lock in a fixed price for oil over a period of time. The swap payments would be based on the difference between a fixed price for oil and a market price that varies over time.
From this description, you can see that there is a relationship between swaps and forward contracts. In fact, a forward contract is a single-payment swap. It is possible to price a multi-date swap—determine the fixed price for oil in the above example—by using information from the set of forward prices with different maturities (i.e., the strip). We will see that swaps are nothing more than forward contracts coupled with borrowing and lending money. 8.1 AN EXAMPLE OF A COMMODITY SWAP
We begin our study of swaps by presenting an example of a simple commodity swap. Our purpose here is to understand how a swap is related to forwards, why someone might use a swap, and how market-makers hedge the risk of swaps. In later sections we present swapprice formulas and examine interest rate swaps, total return swaps, and more complicated commodity swap examples.

233

234

Chapter 8. Swaps

An industrial producer, IP Inc., is going to buy 100,000 barrels of oil 1 year from today and 2 years from today. Suppose that the forward price for delivery in 1 year is $110/barrel and in 2 years is $111/barrel. Suppose that, as in Table 7.1 (see page 196), the 1- and 2-year annual zero-coupon bond yields are 6% and 6.5%.
IP can use forward contracts to guarantee the cost of buying oil for the next 2 years.
Specifically, IP could enter into long forward contracts for 100,000 barrels in each of the next 2 years, committing to pay $110/barrel in 1 year and $111/barrel in 2 years. The present value of this cost is
$111
$110
+
= $201.638
1.06
1.0652
IP could invest this amount today and ensure that it had the funds to buy oil in 1 and 2 years. Alternatively, IP could pay an oil supplier $201.638, and the supplier would commit to delivering one barrel in each of the next 2 years. A single payment today for a single delivery of oil in the future is a prepaid forward. A single payment today to obtain multiple deliveries in the future is a prepaid swap.
Although it is possible to enter into a prepaid swap, buyers might worry about the resulting credit risk: They have fully paid for oil that will not be delivered for up to 2 years.
(The prepaid forward has the same problem.) For the same reason, the swap counterparty would worry about a postpaid swap, where the oil is delivered and full payment is made after 2 years. A more attractive solution for both parties is to defer payment until the oil is delivered, while still fixing the total price.
Note that there are many feasible ways to have the buyer pay. Typically, however, a swap will call for equal payments in each year. The payment per year per barrel, x, will then have to be such that x x
+
= $201.638
1.06 1.0652
To satisfy this equation, the payments must be $110.483 in each year. We then say that the 2year swap price is $110.483. However, any payments that have a present value of $201.638 are acceptable.

Physical Versus Financial Settlement
Thus far we have described the swap as if the swap counterparty supplied physical oil to the buyer. Figure 8.1 shows a swap that calls for physical settlement. In this case $110.483 is the per-barrel cost of oil.

FIGURE 8.1
Illustration of a swap where the oil buyer pays
$110.483/year and receives one barrel of oil each year.

$110.483
Oil Buyer

Swap Counterparty
Oil

8.1 An Example of a Commodity Swap

However, we could also arrange for financial settlement of the swap. With financial settlement, the oil buyer, IP, pays the swap counterparty the difference between $110.483 and the spot price (if the difference is negative, the counterparty pays the buyer), and the oil buyer then buys oil at the spot price. For example, if the market price is $115, the swap counterparty pays IP
Spot price − swap price = $115 − $110.483 = $4.517
If the market price is $108, the spot price less the swap price is
Spot price − swap price = $108 − $110.483 = −$2.483
In this case, the oil buyer, IP, makes a payment to the swap counterparty. Whatever the market price of oil, the net cost to the buyer is the swap price, $110.483:
Spot price − swap price −
Swap payment

= −Swap price

spot price
Spot purchase of oil

Figure 8.2 depicts cash flows and transactions when the swap is settled financially. The results for the buyer are the same whether the swap is settled physically or financially. In both cases, the net cost to the oil buyer is $110.483.
We have discussed the swap on a per-barrel basis. For a swap on 100,000 barrels, we simply multiply all cash flows by 100,000. In this example, 100,000 is the notional amount of the swap, meaning that 100,000 barrels is used to determine the magnitude of the payments when the swap is settled financially.
To illustrate how a commodity swap would be specified in practice, Figure 8.3 is an abbreviated example of a term sheet for an oil swap. Term sheets are commonly used by broker-dealers to succinctly convey the important terms of a financial transaction. The specific example is hypothetical, but the language is from a real term sheet. This particular example is a 3-month oil swap with settlement each month based on the difference between a fixed price and the average over the month of the NYMEX near-month futures price.
As you would expect, the complete documentation for such a deal is lengthy. Transaction confirmations typically make reference to standard documentation supplied by the International Swaps and Derivatives Association (ISDA). The use of standard documentation makes swaps less costly to trade and prices more comparable across dealers.

FIGURE 8.2
Cash flows from a transaction where the oil buyer enters into a financially settled 2-year swap. Each year the buyer pays the spot price for oil and receives spot price − $110.483. The buyer’s net cost of oil is
$110.483/barrel.

Spot Price – $110.483

Swap
Counterparty

Oil Buyer
Spot Price
Oil Seller
Oil

235

236

Chapter 8. Swaps

FIGURE 8.3
Illustrative example of the terms for an oil swap based on West Texas Intermediate (WTI) crude oil.

Fixed-Price Payer:
Floating-Price Payer:
Notional Amount:
Trade Date:
Effective Date:
Termination Date:
Period End Date:
Fixed Price:
Commodity Reference Price:
Floating Price:

Calculation Period:
Method of Averaging:
Settlement and Payment:

Payment Date:

Broker-dealer
Counterparty
100,000 barrels
April 18, 2011
July 1, 2011
September 31, 2011
Final Pricing Date of each Calculation Period as defined in the description of the Floating Price.
110.89 USD per barrel
OIL-WTI-NYMEX
The average of the first nearby NYMEX WTI Crude Oil Futures settlement prices for each successive day of the Calculation Period during which such prices are quoted
Each calendar month during the transaction
Unweighted
If the Fixed Amount exceeds the Floating Amount for such Calculation
Period, the Fixed Price Payer shall pay the Floating Price Payer an amount equal to such excess. If the Floating Amount exceeds the
Fixed Amount for such Calculation Period, the Floating Price Payer shall pay the Fixed Price Payer an amount equal to such excess.
5 business days following each Period End Date

Why Is the Swap Price Not $110.50?
The swap price, $110.483, is close to the average of the two oil forward prices, $110.50.
However, it is not exactly the same. Why?
Suppose that the swap price were $110.50. The oil buyer would then be committing to pay $0.50 more than the forward price the first year and would pay $0.50 less than the forward price the second year. Thus, relative to the forward curve, the buyer would have made an interest-free loan to the counterparty. There is implicit lending in the swap.
Now consider the actual swap price of $110.483/barrel. Relative to the forward curve prices of $110 in 1 year and $111 in 2 years, we are overpaying by $0.483 in the first year and we are underpaying by $0.517 in the second year. Therefore, the swap is equivalent to being long the two forward contracts, coupled with an agreement to lend $0.483 to the swap counterparty in 1 year, and receive $0.517 in 2 years. This loan has the effect of equalizing the net cash flow on the two dates.
The interest rate on this loan is 0.517/0.483 − 1 = 7%. Where does 7% come from?
We assumed that 6% is the 1-year zero yield and 6.5% is the 2-year yield. Given these interest rates, 7% is the 1-year implied forward yield from year 1 to year 2. (See Table 7.1.)
By entering into the swap, we are lending the counterparty money for 1 year beginning in 1

8.1 An Example of a Commodity Swap

FIGURE 8.4
Cash flows from a transaction where an oil buyer and seller each enters into a financially settled 2year swap. The buyer pays the spot price for oil and receives the spot price − $110.483 each year as a swap payment. The oil seller receives the spot price for oil and receives $110.483 − spot price as a swap payment.

Spot Price – $110.483

Swap
Counterparty

Spot Price – $110.483

Oil Buyer

Oil Seller
Spot Price

Oil

Spot Oil
Market

Spot Price

Oil

year. If the deal is priced fairly, the interest rate on this loan should be the implied forward interest rate.

The Swap Counterparty
The swap counterparty is a dealer who hedges the oil price risk resulting from the swap.
The dealer can hedge in several ways. First, imagine that an oil seller would like to lock in a fixed selling price of oil. In this case, the dealer locates the oil buyer and seller and serves as a go-between for the swap, receiving payments from one party and passing them on to the other. In practice the fixed price paid by the buyer exceeds the fixed price received by the seller. This price difference is a bid-ask spread and is the dealer’s fee.
Figure 8.4 illustrates how this transaction would work with financial settlement. The oil seller receives the spot price for oil and receives the swap price less the spot price, on net receiving the swap price. The oil buyer pays the spot price and receives the spot price less the swap price. The situation where the dealer matches the buyer and seller is called a back-to-back transaction or “matched book” transaction. The dealer bears the credit risk of both parties but is not exposed to price risk.
A more interesting situation occurs when the dealer serves as counterparty and hedges the transaction using forward markets. Let’s see how this would work.
After entering the swap with the oil buyer, the dealer has the obligation to pay the spot price and receive the swap price. If the spot price rises, the dealer can lose money. The dealer has a short position in 1- and 2-year oil.
The natural hedge for the dealer is to enter into long forward or futures contracts to offset this short exposure. Table 8.1 illustrates how this strategy works. As we discussed earlier, there is an implicit loan in the swap and this is apparent in Table 8.1. The net cash flow for the hedged dealer is a loan, where the dealer receives cash in year 1 and repays it in year 2.

237

238

Chapter 8. Swaps

TABLE 8.1

Year
1
2

Positions and cash flows for a dealer who has an obligation to receive the fixed price in an oil swap and who hedges the exposure by going long year 1 and year 2 oil forwards.

Payment from Oil Buyer

Long Forward

Net

$110.483 − year 1 spot price
$110.483 − year 2 spot price

Year 1 spot price − $110
Year 2 spot price − $111

$0.483
−$0.517

This example shows that hedging the oil price risk in the swap, with forwards only, does not fully hedge the position. The dealer also has interest rate exposure. If interest rates fall, the dealer will not be able to earn a sufficient return from investing $0.483 in year 1 to repay $0.517 in year 2. Thus, in addition to entering oil forwards, it would make sense for the dealer to use Eurodollar contracts or forward rate agreements to hedge the resulting interest rate exposure.

The Market Value of a Swap
When the buyer first enters the swap, its market value is zero, meaning that either party could enter or exit the swap without having to pay anything to the other party (apart from commissions and bid-ask spreads). From the oil buyer’s perspective, the swap consists of two forward contracts plus an agreement to lend money at the implied forward rate of 7%.
The forward contracts and forward rate agreement have zero value, so the swap does as well. Once the swap is struck, however, its market value will generally no longer be zero, for two reasons. First, the forward prices for oil and interest rates will change over time.
New swaps would no longer have a fixed price of $110.483; hence, one party will owe money to the other should one party wish to exit or unwind the swap.
Second, even if oil and interest rate forward prices do not change, the value of the swap will remain zero only until the first swap payment is made. Once the first swap payment is made, the buyer has overpaid by $0.483 relative to the forward curve, and hence, in order to exit the swap, the counterparty would have to pay the oil buyer $0.483. Thus, even if prices do not change, the market value of swaps can change over time due to the implicit borrowing and lending.
A buyer wishing to exit the swap could negotiate terms with the original counterparty to eliminate the swap obligation. An alternative is to leave the original swap in place and enter into an offsetting swap with the counterparty offering the best price. The original swap called for the oil buyer to pay the fixed price and receive the floating price; the offsetting swap has the buyer receive the fixed price and pay floating. The original obligation would be cancelled except to the extent that the fixed prices are different. However, the difference is known, so oil price risk is eliminated. (There is still credit risk when the original swap counterparty and the counterparty to the offsetting swap are different. This could be a reason for the buyer to prefer offsetting the swap with the original counterparty.)
To see how a swap can change in value, suppose that immediately after the buyer enters the swap, the forward curve for oil rises by $2 in years 1 and 2. Thus, the year1 forward price becomes $112 and the year-2 forward price becomes $113. The original swap will no longer have a zero market value.

8.1 An Example of a Commodity Swap

BOX

239

8.1: Enron’s Hidden Debt

W

hen energy giant Enron collapsed in the fall of 2001, there were charges that other companies had helped Enron mislead investors. In July
2003, the Securities and Exchange Commission announced that J. P. Morgan Chase and Citigroup had each agreed to pay more than $100 million to settle allegations that they had helped Enron commit fraud. Specifically, the SEC alleged that both banks had helped Enron characterize loan proceeds as operating income.
The basic outline of the transaction with J. P.
Morgan Chase is as follows. Enron entered into
“prepaid forward sales contracts” (essentially a prepaid swap) with an entity called Mahonia; Enron received a lump-sum payment and agreed to deliver natural gas in the future. Mahonia in turn received a lump-sum payment from Chase and agreed to deliver natural gas in the future.
Chase, which controlled Mahonia, then hedged

its Mahonia transaction with Enron. With all transactions netted out, Enron had no commodity exposure, and received its lump-sum initial payment from Mahonia in exchange for making future fixed installment payments to Chase.
In other words, Enron in effect had a loan with
Chase. Not only did Enron not record debt from these transactions, but the company reported operating income. The transaction is illustrated in
Figure 8.5.
The SEC complaint included a revealing excerpt from internal Chase e-mail:
WE ARE MAKING DISGUISED LOANS,
USUALLY BURIED IN COMMODITIES OR
EQUITIES DERIVATIVES (AND I’M SURE
IN OTHER AREAS). WITH AFEW [sic] EXCEPTIONS, THEY ARE UNDERSTOOD TO
BE DISGUISED LOANS AND APPROVED
AS SUCH. (Capitalization in the original.)

Assuming interest rates are unchanged, the new swap price is $112.483. (You should verify this.) The buyer could unwind the swap at this point by agreeing to sell oil at $112.483, while the original swap still calls for buying oil at $110.483. Thus, the net swap payments in each year are
(Spot price − $110.483) + ($112.483 − spot price) = $2
Original swap

New swap

The present value of this difference is
$2
$2
+
= $3.650
1.06 (1.065)2
The buyer can receive a stream of payments worth $3.65 by offsetting the original swap with a new swap. Thus, $3.65 is the market value of the swap.
If interest rates had changed, we would have used the new interest rates in computing the new swap price.
As a practical matter, swaps and other derivatives can cause problems for regulators, accountants, and investors, all of whom would like an accurate depiction of activities within a firm. Box 8.1 shows an extreme example of a hedged transaction—allegedly used to hide debt and manipulate earnings—involving Enron and J. P. Morgan Chase. Figure 8.5, which was taken from an SEC account of the transaction, illustrates the transactions and flows.
The examples we have analyzed in this section illustrate the fundamental characteristics of swaps and their cash flows. In the rest of the chapter, we will examine swaps based

240

Chapter 8. Swaps

FIGURE 8.5
Enron’s swaps with Mahonia and Chase.

Up-front fixed payment
Enron

Future commodity

Mahonia

deliveries
Floating
payments
Future
Floating, installment commodity deliveries payments

Future commodity deliveries

Up-front, fixed payment

Fixed installment payments

Chase

Source: Securities and Exchange Commission.

on interest rates, currencies, and commodities and see some of the ways in which we can modify the terms of a swap.

8.2 COMPUTING THE SWAP RATE IN GENERAL
We now present a general formula for computing the swap rate. As we saw in the previous section, the swap rate calculation equates the value of a prepaid swap with the present value of the fixed swap payments. This principle for computing the swap rate is the same for any underlying asset.

Fixed Quantity Swaps
We first consider swaps with a notional amount that is fixed over time.1 Suppose there are n swap settlements, occurring on dates ti , i = 1, . . . , n. The forward prices on these dates are given by F0, ti . We will account for interest rates by using the bond price notation introduced in Chapter 7. The price of a zero-coupon bond maturing on date ti is P (0, ti ). This price is the factor for discounting a fixed payment from date ti to date 0.
If the buyer at time zero were to enter into forward contracts to purchase one unit on each of the n dates, the present value of payments would be the present value of the forward prices, which equals the price of the prepaid swap:

1. A swap with payments based on 100,000 barrels of oil, as discussed before and shown in Figure 8.3, would be such a swap.

8.2 Computing the Swap Rate in General

n

Prepaid swap = i=1 F0, ti P (0, ti )

(8.1)

We determine the fixed swap price, R, by requiring that the present value of the swap payments equal the value of the prepaid swap. Thus, we have n n

RP (0, ti ) = i=1 i=1

F0, ti P (0, ti )

(8.2)

Solving for R gives
R=

n i=1 P (0, ti )F0, ti n i=1 P (0, ti )

(8.3)

The expression n P (0, ti )F0, ti is the present value of payments implied by the strip of i=1 forward prices. The expression n P (0, ti ) is the present value of a $1 annuity. Thus, the i=1 swap rate annuitizes the interest payments on the floating-rate bond.
A different way to motivate the swap price calculation is to note that the present value of the differences between the swap price and the forward prices must equal zero: n i=1

P (0, ti )[R − F0, ti ] = 0

(8.4)

This also gives rise to equation (8.3).
We can rewrite equation (8.3) to make it easier to interpret: n R= i=1 P (0, ti ) n j =1 P (0, tj )

F0, ti

The terms in square brackets sum to 1. This form of equation (8.3) emphasizes that the fixed swap rate is a weighted average of the forward prices, where zero-coupon bond prices are used to determine the weights.
Figure 8.6 displays a swap curve for natural gas, constructed using equation 8.3.
We saw in Chapter 6 that the natural gas price exhibits seasonality. The swap price is a weighted average of natural gas forward prices over the life of the swap, and thus exhibits less variation. In Figure 8.6, the average natural gas futures price climbs so the swap curve climbs as well.
You will observe from Figure 8.6 that it is easier to describe the general level of the natural gas price at different horizons by referring to the swap price rather than the individual prices. The swap price is about $5.50 at a horizon of 3 years and $6 at a horizon of 7 years.

Swaps with Variable Quantity and Price
A commodity buyer might prefer a swap in which quantities vary over time. For example, a natural gas buyer could enter into a swap supplying a greater quantity of gas during the heating season. A buyer might also wish to fix different prices in different seasons. For example, there could be seasonal variation in the price of the output produced using natural gas as an input. How do we determine the swap price with seasonally varying quantities and prices?

241

242

Chapter 8. Swaps

FIGURE 8.6

Price per MMBtu ($)
7.5

Natural gas swap curve, June
2, 2010. The swap curve displays the fixed price for a natural gas swap beginning
June 2010 and continuing, with monthly settlement, for the number of months specified on the x-axis.

7.0

6.5

6.0

5.5

5.0
Natural gas strip
Swap price

4.5
0

20

40
Months to Maturity

60

80

Let Qti denote the quantity of the commodity to be purchased at time ti . If a buyer pays in advance, the prepaid swap price is n Prepaid swap = i=1 Qti F0, ti P (0, ti )

(8.5)

Consider a swap in which the buyer pays RQti for Qti units of the commodity. The present value of these fixed payments (fixed per unit of the commodity) must equal the prepaid swap price, so that n i=1

n

Qti F0, ti P (0, ti ) =

i=1

Qti RP (0, ti )

Solving this equation for R gives
R=

n i=1 Qti P (0, ti )F0, ti n i=1 Qti P (0, ti )

(8.6)

The equation shows that if we are going to buy more gas when the forward price is high, we have to weight more heavily the forward price in those months. When Qt = 1, the formula is the same as equation (8.3).
Once again, another way to derive this equation is to require that the present value of the quantity-weighted difference between the swap price and the forward prices is zero:

8.3 Interest Rate Swaps

n i=1 P (0, ti )Qti R − F0, ti = 0

We can also permit prices to be time-varying. If, for example, we let the summer swap price be denoted by Rs and the winter price by Rw , then the summer and winter swap prices can be any prices for which the value of the prepaid swap equals the present value of the fixed swap payments: n n

Rs i ∈ summer

P (0, ti )Qti + Rw

i ∈ winter

n

P (0, ti )Qti =

i=1

P (0, ti )Qti F0, ti

The notations i ∈ summer and i ∈ winter mean to sum over only the months in those seasons.
This gives us one equation and two unknowns, Rs and Rw . Once we fix one of the two prices, the equation will give us the other.

8.3 INTEREST RATE SWAPS
Companies use interest rate swaps to modify their interest rate exposure. In this section we will begin with a simple example of an interest rate swap, similar to the preceding oil swap example. We will then present general pricing formulas and discuss ways in which the basic swap structure can be altered.

A Simple Interest Rate Swap
Suppose that XYZ Corp. has $200m of floating-rate debt at LIBOR—meaning that every year XYZ pays that year’s current LIBOR—but would prefer to have fixed-rate debt with
3 years to maturity. There are several ways XYZ could effect this change.
First, XYZ could change their interest rate exposure by retiring the floating-rate debt and issuing fixed-rate debt in its place. However, an actual purchase and sale of debt has transaction costs.
Second, they could enter into a strip of forward rate agreements (FRAs) in order to guarantee the borrowing rate for the remaining life of the debt. Since the FRA for each year will typically carry a different interest rate, the company will lock in a different rate each year and, hence, the company’s borrowing cost will vary over time, even though it will be fixed in advance.
A third alternative is to obtain interest rate exposure equivalent to that of fixed rate debt by entering into a swap. XYZ is already paying a floating interest rate. They therefore want to enter a swap in which they receive a floating rate and pay the fixed rate, which we will suppose is 6.9548%. This swap is illustrated in Figure 8.7. Notice the similarity to the oil swap.
In a year when the fixed 6.9548% swap rate exceeds 1-year LIBOR, XYZ pays
6.9548% − LIBOR to the swap counterparty. Conversely, when the 6.9548% swap rate is less than LIBOR, the swap counterparty pays LIBOR − 6.9548% to XYZ. On net, XYZ pays 6.9548%. Algebraically, the net interest payment made by XYZ is
XYZ net payment = −LIBOR + LIBOR − 6.9548% = −6.9548%
Floating payment

Swap payment

243

244

Chapter 8. Swaps

FIGURE 8.7
Illustration of cash flows for a company that borrows at
LIBOR and swaps to fixedrate exposure at 6.9548%.

Pay LIBOR
(floating)

Borrower
(XYZ)

Lender

Pay 6.9548%
(fixed)
Swap
Counterparty
Receive LIBOR
(floating)

The notional principal of the swap is $200m: It is the amount on which the interest payments—and, hence, the net swap payment—is based. The life of the swap is the swap term or swap tenor.
There are timing conventions with a swap similar to those for a forward rate agreement. At the beginning of a year, the borrowing rate for that year is known. However, the interest payment on the loan is due at the end of the year. The interest rate determination date for the floating interest payment would therefore occur at the beginning of the period.
As with an FRA we can think of the swap payment being made at the end of the period
(when interest is due).
With the financially settled interest rate swap, only net swap payments—in this case the difference between LIBOR and 6.9548%—are actually made between XYZ and the counterparty. If one party defaults, they owe to the other party at most the present value of net swap payments they are obligated to make at current market prices. This means that a swap generally has less credit risk than a bond: Whereas principal is at risk with a bond, only net swap payments are at risk in a swap.
The swap in this example is a construct, making payments as if there were an exchange of payments between a fixed-rate and floating-rate bond. In practice, a fund manager might own fixed-rate bonds and wish to have floating-rate exposure while continuing to own the bonds. A swap in which a fund manager receives a floating rate in exchange for the payments on bonds the fund continues to hold is called an asset swap.

Pricing and the Swap Counterparty
To understand the pricing of the swap, we will examine it from the perspective of both the counterparty and the firm. We first consider the perspective of the counterparty, who we assume is a market-maker.
The market-maker is a counterparty to the swap in order to earn fees, not to take on interest rate risk. Therefore, the market-maker will hedge the transaction. The marketmaker receives the fixed rate from the company and pays the floating rate; the danger for the market-maker is that the floating rate will rise. The risk in this transaction can be hedged by entering into forward rate agreements. We express the time 0 implied forward rate between time ti and tj as r0(ti , tj ) and the realized 1-year rate as rti . The current 1-year rate, 6%, is known. With the swap rate denoted R, Table 8.2 depicts the risk-free (but time-varying) cash flows faced by the hedged market-maker.

8.3 Interest Rate Swaps

TABLE 8.2

Cash flows faced by a market-maker who receives fixed and pays floating and hedges the resulting exposure using forward rate agreements.

Year

Payment on Forward

Net Swap Payment

1
2
3

— r2 − 7.0024% r3 − 8.0071%

R − 6%
R − r2
R − r3

TABLE 8.3

Net
R − 6%
R − 7.0024%
R − 8.0071%

Cash flows faced by a floating-rate borrower who enters into a 3-year swap with a fixed rate of 6.9548%.

Year

Floating-Rate Debt Payment

Net Swap Payment

1
2
3

−6%
−r2
−r3

6% − 6.9548% r2 − 6.9548% r3 − 6.9548%

Net
−6.9548%
−6.9548%
−6.9548%

How is R determined? Obviously a market-maker receiving the fixed rate would like to set a high swap rate, but the swap market is competitive. We expect R to be bid down by competing market-makers until the present value of the hedged cash flows is zero.
In computing this present value, we need to use the appropriate rate for each cash flow: the 1-year rate for R − 6%, the 2-year rate for R − 7.0024%, and so forth. Using the rate information from Table 7.1, we compute
R − 6% R − 7.0024% R − 8.0071%
+
+
=0
1.06
1.0652
1.073
This formula gives us an R of 6.9548%, which from Table 7.1 is the same as the par coupon rate on a 3-year bond! In fact, our swap-rate calculation is a roundabout way to compute a par bond yield. On reflection, this result should be no surprise. Once the borrower has entered into the swap, the net effect is exactly like borrowing at a fixed rate. Thus the fixed swap rate should be the rate on a coupon bond.
Notice that the unhedged net cash flows in Table 8.2 (the “net swap payment” column) can be replicated by borrowing at a floating rate and lending at a fixed rate. In other words, an interest rate swap is equivalent to borrowing at a floating rate to buy a fixed-rate bond.
The borrower’s calculations are just the opposite of the market-maker’s. The borrower continues to pay the floating rate on its floating-rate debt, and receives floating and pays fixed in the swap. Table 8.3 details the cash flows.
Since the swap rate is the same as the par 3-year coupon rate, the borrower is indifferent between the swap and a coupon bond, ignoring transaction costs. Keep in mind that the borrower could also have used forward rate agreements, locking in an escalating interest rate over time: 6% the first year, 7.0024% the second, and 8.0071% the third. By

245

246

Chapter 8. Swaps

using interest rate forwards the borrower would have eliminated uncertainty about future borrowing rates and created an uneven but certain stream of interest payments over time.
The swap provides a way to both guarantee the borrowing rate and lock in a constant rate in a single transaction.

Swap Rate and Bond Calculations
The interest rate swap calculation we just illustrated is an application of equation (8.3), with the implied forward interest rate used as the forward price. There is, however, an equivalent way to express the swap rate that is helpful in the case of an interest rate swap.
Recall from Chapter 7, equation (7.4), that the implied forward rate between times t1 and t2, r0(t1, t2), is given by the ratio of zero-coupon bond prices, i.e., r0(t1, t2) = P (0, t1)/P (0, t2) − 1
Therefore equation (8.4) can be rewritten n n

P (0, ti )[R − r(ti−1, ti )] = i=1 P (0, ti ) R − i=1 P (0, ti−1)
+1
P (0, ti )

Setting this equation equal to zero and solving for R gives us
R=

1 − P0(0, tn) n i=1 P0 (0, ti )

(8.7)

You may recognize this as the formula for the coupon on a par coupon bond, equation (7.6), from Chapter 7. This in turn can be rewritten as n R

P (0, ti ) + P (0, tn) = 1 i=1 This is the valuation equation for a bond priced at par with a coupon rate of R.
The conclusion is that the swap rate is the coupon rate on a par coupon bond. This result is intuitive since a firm that swaps from floating-rate to fixed-rate exposure ends up with the economic equivalent of a fixed-rate bond.
The correspondence between swap and bond calculations also extends to the change in value of the swap when interest rates change. An interest rate increase reduces the present value of the fixed payments. (The fixed-rate receiver is in the position of a bond owner and suffers a loss in value.) It is common to use the DV01 (PVBP) calculation from Chapter 7 to describe the price sensitivity of a swap.
Example 8.1 Consider again the swap described in Table 8.3. Suppose that the notional value is $100 million. We saw in Chapter 7 that the Macaulay duration of a 6.95485% par bond is 2.80915 years. By rearranging equation (7.11), DV01 (the change in bond price for a 1-basis-point change in yield) would therefore be
DV01 = Duration × 0.0001 × 100,000, 000
= 2.80915 × 0.0001 × $100,000,000 = $28,091.50

8.3 Interest Rate Swaps

The Swap Curve
As discussed in Chapter 5, the Eurodollar futures contract provides a set of 3-month forward
LIBOR rates extending out 10 years. It is possible to use this set of forward interest rates to compute the interest rate swap curve. As discussed in Chapter 7, zero-coupon bond prices can be constructed from implied forward rates.
There is an over-the-counter market in interest rate swaps, which is widely quoted.
The swap curve should be consistent with the interest rate curve implied by the Eurodollar futures contract, which is used to hedge swaps.2
We now use equation (8.3) to construct the interest rate swap curve using Eurodollar prices.3 To compute swap rates we need forward interest rates, which we obtain from the
Eurodollar futures prices, and we need zero coupon bond prices, which we build from the forward interest rates.
Column 2 of Table 8.4 lists near-term Eurodollar futures prices from June 2010. The third column shows the implied 91-day interest rate for a loan terminating in the month in that row. For example, the June Eurodollar futures price was 99.432. The implied quarterly
(nonannualized) interest rate for June to September was
(100 − 99.432)

91 1
= 0.14358%
90 400

The implied June price for a $1 cash flow in September was therefore
1
= 0.998566
1 + 0.0014358
The implied June price for a $1 cash flow in December is obtained by first finding the forward interest rate from September to December:
(100 − 99.200)

91 1
= 0.20222%
90 400

The price of a zero-coupon bond paying $1 in December is then
0.998566 ×

1
= 0.99655
1 + 0.0020222

We can then use equation (8.3) to determine the swap rate. For the June to December swap rate, for example, we have
0.998566 × 0.0014358 + 0.99655 × 0.0020222
= 0.17287%
0.998566 + 0.99655

2. The Eurodollar contract is a futures contract, while a swap is a set of forward rate agreements. Because of convexity bias, discussed in Chapter 7, the swap curve constructed from Eurodollar futures contracts following the procedure described in this section will be somewhat greater than the observed swap curve.
This is discussed by Burghardt and Hoskins (1995) and Gupta and Subrahmanyam (2000).
3. Collin-Dufresne and Solnik (2001) point out that the credit risk implicit in the LIBOR rate underlying the Eurodollar futures contract is different than the credit risk of an interest rate swap. LIBOR is computed as an average 3-month borrowing rate for international banks with good credit. Banks that experience credit problems are dropped from the sample. Thus, by construction, the pool of banks represented in the Eurodollar contract never experience a credit downgrade. A firm with a swap, by contrast, could be downgraded. 247

248

Chapter 8. Swaps

TABLE 8.4

Eurodollar futures prices and implied interest rate swap rates, June 2, 2010. The second column is the Eurodollar futures price for the eurodollar contract maturing in the month in the first column. The third column is the implied quarterly interest rate for a 3-month loan maturing in that month. The fourth column is the implied June price for a zero coupon bond paying $1 in that month, and the final column is the annualized swap rate for a swap maturing in that month.

Maturity

Eurodollar
Futures

Implied
Rate

Zero
Price

Swap
Rate

Jun
Sep
Dec
Mar
Jun

99.432
99.200
99.035
98.875
98.665

0.00144
0.00202
0.00244
0.00284

0.99857
0.99655
0.99413
0.99131

0.57431
0.69148
0.78601
0.87354

We multiply this rate by 4 to annualize it, giving the entry 0.69148% in Table 8.4.
In Figure 8.8 we graph the entire swap curve against quarterly forward rates implied by the Eurodollar curve. Figure 8.8 also displays yields on government bonds. The swap spread is the difference between swap rates and Treasury-bond yields for comparable maturities. As you would expect, it is positive in Figure 8.8.4

The Swap’s Implicit Loan Balance
An interest rate swap behaves much like the oil swap in Section 8.1. At inception, the swap has zero value to both parties. If interest rates change, the present value of the fixed payments and, hence, the swap rate will change. The market value of the swap is the difference in the present value of payments between the old swap rate and the new swap rate. For example, consider the 3-year swap in Table 8.3 (see page 245). If interest rates rise after the swap is entered into, the value of the existing 6.9548% swap will fall for the party receiving the fixed payment.
Even in the absence of interest rate changes, however, the swap in Table 8.3 changes value over time. Once the first swap payment is made, the swap acquires negative value for the market-maker (relative to the use of forwards) because in the second year the marketmaker will make a net cash payment. Similarly, the swap will have positive value for the borrower (again relative to the use of forwards) after the first payment is made. In order to smooth payments, the borrower pays “too much” (relative to the forward curve) in the

4. Beginning in late 2008, and up through the date of this writing in early 2012, 30-year swap spreads (and, less frequently, 7- and 10-year swap spreads) have been negative. That is, the yield on 30-year Treasury bonds has been greater than the 30-year swap rate. This is surprising because Treasury yields should be less than private yields.

8.3 Interest Rate Swaps

FIGURE 8.8
Forward 3-month interest rate curve implied by the
Eurodollar strip, swap rates, and constant maturity
Treasury yields for June 2,
2010.

Rate (Percent)
5

4

3

2

Eurodollar strip
Swap rate
Treasury curve

1

0
0

20

40
60
80
Months to Maturity

100

120

first year and receives a refund in the second year. The swap is equivalent to entering into forward contracts and undertaking some additional borrowing and lending.
The 9.5-year swap rate in Figure 8.8 is 3.547%. We can use this value to illustrate the implicit borrowing and lending in the swap. Consider an investor who pays fixed and receives floating. This investor is paying a high rate in the early years of the swap, and, hence, is lending money. About halfway through the life of the swap, the Eurodollar forward rate exceeds the swap rate and the loan balance declines, falling to zero by the end of the swap.
The fixed rate recipient has a positive loan balance over the life of the swap because the
Eurodollar futures rate is below the swap initially—so the fixed-rate recipient is receiving payments—and crosses the swap price once. The credit risk in this swap is therefore borne, at least initially, by the fixed-rate payer, who is lending to the fixed-rate recipient. The implicit loan balance in the swap is illustrated in Figure 8.9.

Deferred Swaps
We can construct a swap that begins at some date in the future, but for which the swap rate is agreed upon today. This type of swap is called a deferred swap. To demonstrate this type of swap, we can use the information in Table 7.1 to compute the value of a 2-period swap that begins 1 year from today. The reasoning is exactly as before: The swap rate will be set by the market-maker so that the present value of the fixed and floating payments is the same. This gives us
R − 0.070024 R − 0.080071
+
=0
1.073
1.0652

249

250

Chapter 8. Swaps

FIGURE 8.9
Eurodollar strip and the 9.5year swap rate (3.547%) are plotted in the top panel, and implicit lending from being a fixed-rate recipient in the bottom panel. Swap payment dates are on the horizontal axis.

Rate (Percent)
5
4
3
2
1
0

Eurodollar strip
9.5-year swap rate
0

20

Loan Balance
30
25
20
15
10
5
0
0
20

40
60
80
Months to Maturity

100

120

40
60
80
Months to Maturity

100

120

Solving for R, the deferred swap rate is 7.4854%. Using equation (8.3), with F0, ti = r0(ti−1, ti ), the fixed rate on a deferred swap beginning in k periods (instead of 1 period) is computed as
R=

n i=k P0(0, ti )r0(ti−1, ti ) n i=k

P (0, ti )

(8.8)

This can also be written as
R=

P (0, tk−1) − P (0, tn) n i=k P (0, ti )

(8.9)

Equation (8.8) is equal to equation (8.3) when k = 1.

Related Swaps
We have so far described plain vanilla interest rate swaps in this section. Two related swaps are overnight indexed swaps (OIS) and inflation swaps.
Overnight lending is common in financial markets. One of the most important overnight markets in the U.S. is the fed funds market, in which banks lend to one another. The interest rate in this market is called the fed funds rate. An overnight index swap makes payments based on this rate. Rather than settle the swap each day, the swap is settled quarterly based on the geometric average of the overnight rates. The fixed rate on the swap therefore is effectively the forward price for 3 months’ worth of overnight borrowing in the fed funds market.

8.3 Interest Rate Swaps

During the financial crisis, the spread between LIBOR and OIS was an important indicator of stress in the interbank lending market. Taylor and Williams (2009) document that the LIBOR-OIS spread was approximately 10 basis points during early 2007, but spiked to almost 100 basis points in August 2007, at the onset of the crisis. The spread exceeded
400 basis points during the fall of 2008.
Inflation swaps are also of interest in fixed income markets. The vast majority of bonds are nominal bonds: they make dollar payments that do not depend on the inflation rate.
(Treasury inflation-protected securities (TIPS) are a notable exception.) Lenders holding a nominal bond are harmed when the inflation rate increases and borrowers issuing a nominal bond are harmed when the inflation rate decreases. The payment on an inflation swap depends upon whether the realized inflation rate is greater or less than the fixed rate specified in the swap. Therefore, an inflation swap permits borrowers and lenders to hedge their exposure to inflation. Of course, it also permits speculation on the inflation rate. Why Swap Interest Rates?
Managers sometimes say that they would like to borrow short-term because short-term interest rates are on average lower than long-term interest rates. Leaving aside the question of whether this view makes sense theoretically, let’s take for granted the desire to borrow at short-term interest rates. The problem facing the manager is that the firm may be unable to borrow significant amounts by issuing short-term debt.
When a firm borrows by issuing long-term debt, bondholders bear both interest rate risk and the credit risk of the firm. If the firm borrows short-term (for example, by issuing commercial paper), lenders primarily bear credit risk.
In practice, short-term lenders appear unwilling to absorb large issues from a single borrower because of credit risk. For example, money-market mutual funds that hold commercial paper will not hold large amounts of any one firm’s commercial paper, preferring instead to diversify across firms. This diversification minimizes the chance that a single bankruptcy will significantly reduce the fund’s rate of return.
Because short-term lenders are sensitive in this way to credit risk, a firm cannot borrow a large amount of money short-term without lenders demanding a higher interest rate. By contrast, long-term lenders to corporations—for example, pension funds and insurance companies—willingly assume both interest rate and credit risk. Thus there are borrowers who wish to issue short-term debt and lenders who are unwilling to buy it. Swaps provide a way around this problem, permitting the firm to separate credit risk and interest rate risk. Suppose, for example, that a firm borrows long-term and then swaps into short-rate exposure. The firm receives the fixed rate, pays the fixed rate to bondholders, and pays the floating rate on the swap. The net payment is the short-term rate, which is the rate the firm wanted to pay.
Credit risk does not vanish; it is still mostly held by the long-term bondholders. The swap counterparty faces credit risk since the firm could go bankrupt when the value of the swap is positive to the counterparty (this would occur if interest rates had risen). The notional principal of the loan is not at risk for the swap counterparty, however, so the credit risk in the swap is less than for a short-term loan. Thus, by swapping its interest rate exposure, the firm pays the short-term interest rate, but the long-term bondholders continue to bear most of the credit risk.

251

252

Chapter 8. Swaps

If it seems odd to you that the firm can use a swap to convert a high fixed rate into a low floating rate, recognize that any time there is an upward-sloping yield curve, the shortterm interest rate is below the long-term interest rate. If you reduce the period for which your borrowing rate is fixed (which happens when you swap fixed for floating), you borrow at the lower short-term interest rate instead of the higher long-term interest rate.
Swaps thus permit separation of two aspects of borrowing: credit risk and interest rate risk. To the extent these risks are acquired by those most willing to hold them, swaps increase efficiency.

Amortizing and Accreting Swaps
We have assumed that the notional value of the swap remains fixed over the life of the swap.
However, it is also possible to engage in a swap where the notional value is changing over time. For example, consider a floating-rate mortgage, for which every payment contains an interest and principal component. Since the outstanding principal is declining over time, a swap involving a mortgage would need to account for this. Such a swap is called an amortizing swap because the notional value is declining over time. It is also possible for the principal in a swap to grow over time. This is called an accreting swap.
Let Qt be the relative notional amount at time t. Then the basic idea in pricing a swap with a time-varying notional amount is the same as with a fixed notional amount—the present value of the fixed payments should equal the present value of the floating payments: n i=1

Qti P (0, ti )[R − r(ti−1, ti )] = 0

(8.10)

where, as before, there are n payments on dates t1, t2 , . . . , tn. Equation (8.10) can be rewritten as
R=

n i=1 Qti P (0, ti )r(ti−1, ti ) n i=1 Qti P (0, ti )

(8.11)

The fixed swap rate is still a weighted average of implied forward rates, only now the weights also involve changing notional principal.
Many other structures are possible for swaps based on interest rates or other prices.
One infamous swap structure is described in the box on page 253, which recounts the 1993 swap between Procter & Gamble and Bankers Trust.

8.4 CURRENCY SWAPS
Firms sometimes issue debt denominated in a foreign currency. A firm may do this as a hedge against revenues received in that currency, or because perceived borrowing costs in that currency are lower. Whatever the reason, if the firm later wants to change the currency to which they have exposure, there are a variety of ways to do so. The firm can use forward contracts to hedge exchange rate risk, or it can use a currency swap, in which payments are based on the difference in debt payments denominated in different currencies.
To understand these alternatives, let’s consider the example of a dollar-based firm that has euro-denominated 3-year fixed-rate debt. The annual coupon rate is ρ. The firm is

8.4 Currency Swaps

BOX

253

8.2: The Procter & Gamble Swap

I

n November 1993, consumer products company
Procter & Gamble (P&G) entered into a 5-year
$200m notional value swap with Bankers Trust.
The contract called for P&G to receive a 5.3% fixed rate from Bankers Trust and pay the 30day commercial paper rate less 75 basis points, plus a spread. Settlements were to be semiannual.
The spread would be zero for the first settlement, and thereafter be fixed at the spread as of May 4,
1994.
The spread was determined by the difference between the 5-year constant maturity treasury
(CMT) rate (the yield on a 5-year Treasury bond, but a constructed rate since there is not always a
Treasury bond with exactly 5 years to expiration) and the price per $100 of maturity value of the
6.25% 30-year Treasury bond. The formula for the spread was
Spread = max
5-year CMT %
× 98.5 − price of 30-year bond
0.0578
100

,0

At inception in November 1993, the 5-year CMT rate was 5.02% and the 30-year Treasury price was 102.57811. The expression in the max function evaluated to −0.17 (−17%), so the spread was zero.

If the spread were zero on May 4, 1994, P&G would save 75 basis points per year on $200m for 4.5 years, an interest rate reduction worth approximately $7m. However, notice something important: If interest rates rise before the spread determination date, then the 5-year CMT goes up and the price of the 30-year bond goes down.
Thus, the swap is really a bet on the direction of interest rates, not the difference in rates!
The swap is recounted in Smith (1997) and
Srivastava (1998). Interest rates rose after P&G entered the swap. P&G and Bankers Trust renegotiated the swap in January 1994, and P&G liquidated the swap in March, with a loss of about
$100m. P&G sued Bankers Trust, complaining in part that the risks of the swap had not been adequately disclosed by Bankers Trust.
In the end, P&G and Bankers Trust settled, with P&G paying Bankers Trust about $35m.
(Forster (1996) and Horwitz (1996) debate the implications of the trial and settlement.) The notion that Procter & Gamble might have been uninformed about the risk of the swap, and if so, whether this should have mattered, was controversial. U.S. securities laws are often said to protect “widows and orphans.” Nobel Prize– winning economist Merton Miller wryly said of the case, “Procter is the widow and Gamble is the orphan.” obligated to make a series of payments that are fixed in euro terms but variable in dollar terms. Since the payments are known, eliminating euro exposure is a straightforward hedging problem using currency forwards. We have cash flows of −ρ each year, and −(1 + ρ) in the maturity year. If currency forward prices are F0, t , we can enter into long euro forward contracts to acquire at a known exchange rate the euros we need to pay to the lenders.
Hedged cash flows in year t are −ρF0, t .
As we have seen in other examples, the forward transactions eliminate risk but leave the firm with a variable (but riskless) stream of cash flows. The variability of hedged cash flows is illustrated in the following example.

254

Chapter 8. Swaps

TABLE 8.5

Unhedged and hedged cash flows for a dollar-based firm with euro-denominated debt.

Year

Unhedged
Euro Cash Flow

Forward
Exchange Rate

Hedged
Dollar Cash Flow

1
2
3

C
−= 3.5
= 3.5
−C
−= 103.5
C

0.922
0.944
0.967

−$3.226
−$3.304
−$100.064

Example 8.2 Suppose the effective annual euro-denominated interest rate is 3.5% and the
C
dollar-denominated rate is 6%. The spot exchange rate is $0.90/= . A dollar-based firm has
C
C a 3-year 3.5% euro-denominated bond with a = 100 par value and price of = 100. The firm wishes to guarantee the dollar value of the payments. Since the firm will make debt payments in euros, it buys the euro forward to eliminate currency exposure. Table 8.5 summarizes the transaction and reports the currency forward curve and the unhedged and hedged cash flows.
The value of the hedged cash flows is
$3.226 $3.304 $100.064
+
+
= $90
1.06
1.062
1.063
Example 8.2 verifies what we knew had to be true: Hedging does not change the value of the debt. The initial value of the debt in euros is = 100. Since the exchange rate is $0.90/= ,
C
C the debt should have a dollar value of $90, which it has.
As an alternative to hedging each euro-denominated payment with a forward contract, a firm wishing to change its currency exposure can enter into a currency swap, which entails making debt payments in one currency and receiving debt payments in a different currency.
There is typically an exchange of principal at both the start and end of the swap. Compared with hedging the cash flows individually, the currency swap generates a different cash flow stream, but with equivalent value. We can examine a currency swap by supposing that the firm in Example 8.2 uses a swap rather than forward contracts to hedge its euro exposure.
Example 8.3 Make the same assumptions as in Example 8.2. The dollar-based firm enters
C
into a swap where it pays dollars (6% on a $90 bond) and receives euros (3.5% on a = 100 bond). The firm’s euro exposure is eliminated. The market-maker receives dollars and pays euros. The position of the market-maker is summarized in Table 8.6. The present value of the market-maker’s net cash flow is
$2.174 $2.096 $4.664
+

=0
1.06
1.062
1.063
The market-maker’s net exposure in this transaction is long a dollar-denominated bond and short a euro-denominated bond. Table 8.6 shows that after hedging there is a series of net cash flows with zero present value. As in all the previous examples, the effect of the swap is equivalent to entering into forward contracts, coupled with borrowing or lending.
In this case, the firm is lending to the market-maker in the first 2 years, with the implicit loan repaid at maturity.

8.4 Currency Swaps

TABLE 8.6

Unhedged and hedged cash flows for a dollar-based firm with euro-denominated debt. The effective annual dollardenominated interest rate is 6% and the effective annual euro-denominated interest rate is 3.5%.

Year

Forward Exchange
Rate ($/= )
C

Receive Dollar
Interest

1
2
3

0.9217
0.9440
0.9668

$5.40
$5.40
$95.40

Pay Hedged
Euro Interest

Net Cash
Flow

C
−= 3.5 × 0.9217
−= 3.5 × 0.9440
C
= 103.5 × 0.9668
−C

$2.174
$2.096
−$4.664

The fact that a currency swap is equivalent to borrowing in one currency and lending in the other is familiar from our discussion of currency forwards in Chapter 5. There we saw the same is true of currency forwards.

Currency Swap Formulas
Currency swap calculations are the same as those for the other swaps we have discussed.
To see this, consider a swap in which a dollar annuity, R, is exchanged for an annuity in another currency, R ∗. Given the foreign annuity, R ∗, what is R?
We start with the observation that the present value of the two annuities must be the same. There are n payments and the time-0 forward price for a unit of foreign currency delivered at time ti is F0, ti . This gives n i=1

RP0, ti − R ∗F0, ti P0, ti = 0

In calculating the present value of the payment R ∗, we first convert to dollars by multiplying by F0, ti . We can then compute the present value using the dollar-denominated zero-coupon bond price, P0, ti . Solving for R gives
R=

n

i=1 P0, ti R F0, ti n i=1 P0, ti

(8.12)

This expression is exactly like equation (8.3), with the implied forward rate, r0(ti−1, ti ), replaced by the foreign-currency-denominated annuity payment translated into dollars,
R ∗F0, ti .
When coupon bonds are swapped, we have to account for the difference in maturity value as well as the coupon payment, which is an annuity. If the dollar bond has a par value of $1, the foreign bond will have a par value of 1/x0, where x0 is the current exchange rate expressed as dollars per unit of the foreign currency. If R ∗ is the coupon rate on the foreign bond and R is the coupon rate on the dollar bond, the present value of the difference in payments on the two bonds is n i=1

RP0, ti − R ∗F0, ti P0, ti /x0 + P0, tn (1 − F0, tn /x0) = 0

255

256

Chapter 8. Swaps

The division by x0 accounts for the fact that a $1 bond is equivalent to 1/x0 bonds with a par value of 1 unit of the foreign currency. The dollar coupon in this case is
R=

n

i=1 P0, ti R F0, ti /x0 + P0, tn (F0, tn /x0 n i=1 P0, ti

− 1)

(8.13)

The fixed payment, R, is the dollar equivalent of the foreign coupon plus the amortized value of the difference in the maturity payments of the two bonds. Problem 8.16 asks you to verify that equation (8.13) gives 6% using the assumptions in Tables 8.5 and 8.6.

Other Currency Swaps
There are other kinds of currency swaps. The preceding examples assumed that all borrowing was fixed rate. Suppose the dollar-based borrower issues a euro-denominated loan with a floating interest rate. In this case there are two future unknowns: the exchange rate at which interest payments are converted, and—because the bond is floating rate—the amount of the interest payment. Swapping this loan to a dollar loan is still straightforward, however; we just require one extra hedging transaction.
We first convert the floating interest rate into a fixed interest rate with a euro interest rate swap. The resulting fixed-rate euro-denominated exposure can then be hedged with currency forwards and converted into dollar interest rate exposure. Given the assumptions in Table 8.6, the euro-denominated loan would swap to a 3.5% floating-rate loan. From that point on, we are in the same position as in the previous example.
In general, we can swap fixed-to-fixed, fixed-to-floating, floating-to-fixed, and floating-to-floating. The analysis is similar in all cases.
One kind of swap that might on its face seem similar is a diff swap, short for differential swap. In this kind of swap, payments are made based on the difference in floating interest rates in two different currencies, with the notional amount in a single currency. For example, we might have a swap with a $10m notional amount, but the swap would pay in dollars, based on the difference in a euro-denominated interest rate and a dollar-denominated interest rate. If the short-term euro interest rate rises from 3.5% to 3.8% with the dollar rate unchanged, the annual swap payment would be 30 basis points on $10m, or $30,000. This is like a standard interest rate swap, only for a diff swap, the reference interest rates are denominated in different currencies.
Standard currency forward contracts cannot be used to hedge a diff swap. The problem is that we can hedge the change in the foreign interest rate, but doing so requires a transaction denominated in the foreign currency. We can’t easily hedge the exchange rate at which the value of the interest rate change is converted because we don’t know in advance how much currency will need to be converted. In effect there is quantity uncertainty regarding the foreign currency to be converted. We have seen this kind of problem before, in our discussion of crop yields in Chapter 4 and in our discussion of dollar-denominated Nikkei index futures in Chapter 5. The diff swap is an example of a quanto swap. We will discuss quantos in Chapter 22.

8.5 SWAPTIONS
An option to enter into a swap is called a swaption. We can see how a swaption works by returning to the two-date oil swap example in Section 8.1. The 2-year oil swap price

8.6 Total Return Swaps

was $110.483. Suppose we are willing to buy oil at $110.483/barrel, but we would like to speculate on the swap price being even lower over the next 3 months.
Consider the following contract: If in 3 months the fixed price for a swap commencing in 9 months (1 year from today) is $110.483 or above, we enter into the swap, agreeing to pay $110.483 and receive the floating price for 2 years. If, on the other hand, the market swap price is below $110.483, we have no obligation. If the swap price in 3 months is $108, for example, we could enter into a swap at that time at the $108 price, or we could elect not to enter any swap.
With this contract we are entering into the swap with $110.483 as the swap price only when the market swap price is greater; hence, this contract will have a premium. In this example, we would have purchased a payer swaption, since we have the right, but not the obligation, to pay a fixed price of $110.483 for 2 years of oil. The counterparty has sold this swaption. When exercised, the swaption commits us to transact at multiple times in the future.
It is possible to exercise the option and then offset the swap with another swap, converting the stream of swap payments into a certain stream with a fixed present value. Thus, the swaption is analogous to an ordinary option, with the present value of the swap obligations
(the price of the prepaid swap) as the underlying asset.
The strike price in this example is $110.483, so we have an at-the-money swaption.
We could make the strike price different from $110.483. For example, we could reduce the swaption premium by setting the strike above $110.483.
Swaptions can be American or European, and the terms of the underlying swap—fixed price, floating index, settlement frequency, and tenor—will be precisely specified.
Example 8.4 Suppose we enter into a European payer oil swaption with a strike price of
$21. The underlying swap commences in 1 year and has two annual settlements. After 3 months, the fixed price on the underlying swap is $21.50. We exercise the option, obligating us to pay $21/barrel for 2 years. If we wish to offset the swap, we can enter into a swap to receive the $21.50 fixed price. In year 1 and year 2 we will then receive $21.50 and pay
$21, for a certain net cash flow each year of $0.50. The floating payments cancel.
A receiver swaption gives you the right to pay the floating price and receive the fixed strike price. Thus, the holder of a receiver swaption would exercise when the fixed swap price is below the strike.
Although we have used a commodity swaption in this example, an interest rate or currency swaption would be analogous, with payer and receiver swaptions giving the right to pay or receive the fixed interest rate.

8.6 TOTAL RETURN SWAPS
A total return swap is a swap in which one party pays the realized total return (dividends plus capital gains) on a reference asset, and the other party pays a floating return such as
LIBOR. The two parties exchange only the difference between these rates. The party paying the return on the reference asset is the total return payer.
As with other swaps, there are multiple settlement dates over the life of the swap. The cumulative effect for the total return payer is of being short the reference asset and long

257

258

Chapter 8. Swaps

FIGURE 8.10
Cash flows for a total return swap. The total return payer pays the per-period total return on the reference asset, receiving the floating rate from the counterparty.

Total Return on
Reference Asset
Total Return Payer

Floating Rate Payer
Floating Rate

an asset paying the floating rate. The cash flows on a total return swap are illustrated in
Figure 8.10.
Example 8.5 ABC Asset Management has a $2 billion investment in the S&P stock index.
However, fund managers have become pessimistic about the market and would like to reduce their exposure to stocks from $2 billion to $1 billion. One way to do this is to sell $1 billion of stocks. However, the fund can retain the stock position but financially transfer the return of the stocks by engaging in a total return swap, obligating the fund to pay the total return
(dividends plus capital gains) on the swapped stocks, while receiving a floating-rate return such as LIBOR on the swapped $1 billion notional amount. This avoids the transaction costs of a sale of physical stock.
Table 8.7 illustrates the payments on such a swap. In year 1, ABC earns 6.5% on the
S&P index. However, on the portion it has swapped, it must pay the 6.5% in exchange for the 7.2% floating rate. The net payment of 0.7% leaves ABC as well off as if it had sold the index and invested in an asset paying the floating rate. In year 2, ABC receives 18%, compensating it for the difference between the 7.5% floating return and the 10.5% loss on the S&P index. Finally, in year 3 the S&P index does well, and ABC pays 16.5% to the counterparty. You might wonder about the economics of a swap like this. The stock index on average earns a higher return than LIBOR. So if the fund swaps the stock index in exchange for
LIBOR, it will on average make payments to the counterparty.
This observation is correct, but notice that the fund is paying the difference between the index return and a short-term interest rate—this difference is the risk premium on the index. In Section 5.3, we had a similar result for a forward contract: On average a short position in a forward contract on a stock index loses money because the risk premium has zero value.
The average loss associated with swapping a stock index for LIBOR is the same as the average loss associated with selling the stock and buying a floating-rate note paying
LIBOR. It is just that the swap makes the loss obvious since it requires a payment.
Some investors have used total return swaps to avoid taxes on foreign stocks. In many cases, countries impose withholding taxes on foreign investors, meaning that if a firm in country A pays a dividend, for example, country A withholds a fraction of that dividend from investors based in country B. A total return swap enables a country-B investor to own country-A stocks without physically holding them, and thus in many cases without having

Chapter Summary

TABLE 8.7

Year
1
2
3

S&P Capital Gain
5%
−12%
22%

Illustration of cash flows on a total return swap with annual settlement for 3 years.

S&P Dividend
1.5%
1.5%
1.5%

Floating Rate
7.2%
7.5%
7.0%

Net Payment to
Total Return Payer
0.7%
18.0%
−16.5%

to pay withholding taxes. For example, a U.S. investor could first swap out of a U.S. stock index and then swap into a European stock index, effectively becoming the counterparty for a European investor wanting to swap out of European stock exposure. Because net swap payments are not always recognized by withholding rules, this transaction can be more tax-efficient than holding the foreign stocks directly.
Another use of total return swaps is the management of credit risk. A fund manager holding corporate debt can swap the return on a particular bond for a floating-rate return.
If the company that issued the bond goes bankrupt, the debt holder receives a payment on the swap compensating for the fact that the bond is worth a fraction of its face value.
If you think about this use of total return swaps, it is a crude tool for managing credit risk specifically. The problem is that bond prices also change due to interest rate changes. A corporate bond holder might wish to retain interest rate risk but not bankruptcy risk. Thus, there are products called default swaps. These are essentially default options, in which the buyer pays a premium, usually amortized over a series of payments. If the reference asset experiences a “credit event” (for example, a failure to make a scheduled payment on a particular bond or class of bonds), then the seller makes a payment to the buyer. Frequently these contracts split the return on the bond into the portion due to interest rate changes (with
Treasury securities used as a reference) and the portion due to credit quality changes, with the swap making payments based only on the latter. We discuss these swaps in Chapter 27.

CHAPTER SUMMARY
A swap is a contract calling for an exchange of payments, on one or more dates, determined by the difference in two prices. A single-payment swap is the same thing as a cash-settled forward contract. In the simplest swaps, a fixed payment is periodically exchanged for a floating payment. A firm can use a swap to lock in a long-term commodity price, a fixed interest rate, or a fixed exchange rate. Considering only the present value of cash flows, the same result is obtained using a strip of forward contracts and swaps. The difference is that hedging with a strip of forward contracts results in net payments that are time-varying. In contrast, hedging with a swap results in net payments that are constant over time. The value of a swap is zero at inception, though as swap payments are made over time, the value of the swap can change in a predictable way.

259

260

Chapter 8. Swaps

TABLE 8.8

Equivalent forms of the swap-rate calculation. For the currency swap, F0, ti is the forward price for the foreign currency and R ∗ is an annuity in the foreign currency. For the commodity swap, F0, ti is the forward price for the commodity. To Obtain Formula for

Substitute in Equation (8.14)

Interest rate swap
Currency swap (annuity)
Commodity swap

f0(ti ) = r0(ti−1, ti ) f0(ti ) = R ∗F0, ti f0(ti ) = F0, ti

The fixed price in a swap is a weighted average of the corresponding forward prices.
The swap formulas in different cases all take the same general form. Let f0(ti ) denote the forward price for the floating payment in the swap. Then the fixed swap payment is
R=

n i=1 P (0, ti )f0 (ti ) n i=1 P (0, ti )

(8.14)

Table 8.8 summarizes the substitutions to make in equation (8.14) to get the various swap formulas shown in the chapter. This formula can be generalized to permit time variation in the notional amount and the swap price, and the swap can start on a deferred basis.
An important characteristic of swaps is that they require only the exchange of net payments, and not the payment of principal. So if a firm enters an interest rate swap, for example, it is required only to make payments based on the difference in interest rates, not on the underlying principal. As a result, swaps have less credit risk than bonds.
Total return swaps involve exchanging the return on an asset for a floating rate such as LIBOR. The term swap is also used to describe agreements like the Procter & Gamble swap (page 253), which required payments based on the difference in interest rates and bond prices, as well as default swaps.

FURTHER READING
The same formulas used to price swaps will appear again in the context of structured notes, which we will encounter in Chapter 15. We will discuss default swaps in Chapter 27.
Litzenberger (1992) provides an overview of the swap market. Turnbull (1987) discusses arguments purporting to show that the use of swaps can have a positive net present value. Default swaps are discussed by Tavakoli (1998). Because of convexity bias (Chapter 7), the market interest rate swap curve is not exactly the same as the swap curve constructed from Eurodollar futures. This is discussed in Burghardt and Hoskins (1995) and Gupta and Subrahmanyam (2000). The SEC complaint against J. P. Morgan Chase is at http://www.sec.gov/litigation/complaints/comp18252.htm.

261

Problems

TABLE 8.9
Quarter

1

2

3

4

5

6

7

8

Oil forward price
21
21.1
20.8
20.5
20.2
20
19.9
19.8
Gas swap price
2.2500 2.4236 2.3503 2.2404 2.2326 2.2753 2.2583 2.2044
Zero-coupon bond price 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763
Euro-denominated
0.9913 0.9825 0.9735 0.9643 0.9551 0.9459 0.9367 0.9274 zero-coupon bond price 0.9056 0.9115 0.9178 0.9244 0.9312 0.9381 0.9452 0.9524
Euro forward price
C
($/= )

PROBLEMS
Some of the problems that follow use Table 8.9. Assume that the current exchange rate is
C
$0.90/= .
8.1 Suppose that 1- and 2-year oil forward prices are $22/barrel and $23/barrel. The 1and 2-year interest rates are 6% and 6.5%. Show that the new 2-year swap price is
$22.483.
8.2 Suppose that oil forward prices for 1 year, 2 years, and 3 years are $20, $21, and $22.
The 1-year effective annual interest rate is 6.0%, the 2-year interest rate is 6.5%, and the 3-year interest rate is 7.0%.
a. What is the 3-year swap price?
b. What is the price of a 2-year swap beginning in one year? (That is, the first swap settlement will be in 2 years and the second in 3 years.)
8.3 Consider the same 3-year oil swap. Suppose a dealer is paying the fixed price and receiving floating. What position in oil forward contracts will hedge oil price risk in this position? Verify that the present value of the locked-in net cash flows is zero.
8.4 Consider the 3-year swap in the previous example. Suppose you are the fixed-rate payer in the swap. How much have you overpaid relative to the forward price after the first swap settlement? What is the cumulative overpayment after the second swap settlement? Verify that the cumulative overpayment is zero after the third payment.
(Be sure to account for interest.)
8.5 Consider the same 3-year swap. Suppose you are a dealer who is paying the fixed oil price and receiving the floating price. Suppose that you enter into the swap and immediately thereafter all interest rates rise 50 basis points (oil forward prices are unchanged). What happens to the value of your swap position? What if interest rates fall 50 basis points? What hedging instrument would have protected you against interest rate risk in this position?

262

Chapter 8. Swaps

8.6 Supposing the effective quarterly interest rate is 1.5%, what are the per-barrel swap prices for 4-quarter and 8-quarter oil swaps? (Use oil forward prices in Table 8.9.)
What is the total cost of prepaid 4- and 8-quarter swaps?
8.7 Using the information about zero-coupon bond prices and oil forward prices in Table
8.9, construct the set of swap prices for oil for 1 through 8 quarters.
8.8 Using the information in Table 8.9, what is the swap price of a 4-quarter oil swap with the first settlement occurring in the third quarter?
8.9 Given an 8-quarter oil swap price of $20.43, construct the implicit loan balance for each quarter over the life of the swap.
8.10 Using the zero-coupon bond prices and oil forward prices in Table 8.9, what is the price of an 8-period swap for which two barrels of oil are delivered in even-numbered quarters and one barrel of oil in odd-numbered quarters?
8.11 Using the zero-coupon bond prices and natural gas swap prices in Table 8.9, what are gas forward prices for each of the 8 quarters?
8.12 Using the zero-coupon bond prices and natural gas swap prices in Table 8.9, what is the implicit loan amount in each quarter in an 8-quarter natural gas swap?
8.13 What is the fixed rate in a 5-quarter interest rate swap with the first settlement in quarter 2?
8.14 Using the zero-coupon bond yields in Table 8.9, what is the fixed rate in a 4-quarter interest rate swap? What is the fixed rate in an 8-quarter interest rate swap?
8.15 What 8-quarter dollar annuity is equivalent to an 8-quarter annuity of = 1?
C
8.16 Using the assumptions in Tables 8.5 and 8.6, verify that equation (8.13) equals 6%.
8.17 Using the information in Table 8.9, what are the euro-denominated fixed rates for
4- and 8-quarter swaps?
8.18 Using the information in Table 8.9, verify that it is possible to derive the 8-quarter dollar interest swap rate from the 8-quarter euro interest swap rate by using equation
(8.13).

PART
Options

I

n earlier chapters we have seen how options work and introduced some of the terminology related to options. In this part of the book we return to options, with the goal of understanding how they are priced.
Forward contracts (and futures and swaps) represent a binding commitment to buy or sell the underlying asset in the future. Because the commitment is binding, but deferred, time value of money is the main economic idea used in determining forward prices. Options, on the other hand, need not be exercised. Intuitively, you would expect the probability distribution of the stock to affect the option price. Consequently, in discussing option pricing we will use some concepts from basic probability. However, it turns out that there is much to say about options without needing to think about the probability distribution of the stock. In Chapter 9 we explore concepts such as

parity in more depth, and discuss some basic intuition about option prices that can be gleaned using only time value of money arguments.
Chapters 10 and 11 introduce the binomial option pricing model. This model assumes that the stock can move only in a very simple way, but provides the intuition underlying more complicated option pricing calculations. Chapter 12 presents the
Black-Scholes option pricing formula, which is one of the most important formulas in finance.
As with forwards, futures, and swaps, option contracts are bought and sold by market-makers who hedge the risk associated with market-making. Chapter 13 looks at how market-makers hedge their option risk and shows the precise sense in which the price of an option reflects the cost of synthetically creating it. Finally, Chapter 14 discusses exotic options, which are variants of the standard options we have been discussing. 9

Parity and Other Option
Relationships

W

ith this chapter we begin to study option pricing. Up to this point we have primarily studied contracts entailing firm commitments, such as forwards, futures, and swaps. These contracts do not permit either party to back away from the agreement. Optionality occurs when it is possible to avoid engaging in unprofitable transactions. The principal question in option pricing is: How do you value the right to back away from a commitment?
Before we delve into pricing models, we devote this chapter to refining our common sense about options. For example, Table 9.1 contains call and put prices for IBM for four different strikes and two different expiration dates. These are American-style options. Here are some observations and questions about these prices:
.

.

.

.

.

What determines the difference between put and call prices at a given strike?
How would the premiums change if these options were European rather than American?
It appears that, for a given strike, the October options are more expensive than the
June options. Is this necessarily true?
Do call premiums always decrease as the strike price increases? Do put premiums always increase as the strike price increases?
Both call and put premiums change by less than the change in the strike price. Does this always happen?

These questions, and others, will be answered in this chapter, but you should take a minute and think about the answers now, drawing on what you have learned in previous chapters. While doing so, pay attention to how you are trying to come up with the answers.
What constitutes a persuasive argument? Along with finding the answers, we want to understand how to think about questions like these.

9.1 PUT-CALL PARITY
Put-call parity is perhaps the single most important relationship among option prices. In
Chapter 2 we argued that synthetic forwards (created by buying the call and selling the put) must be priced consistently with actual forwards. The basic parity relationship for European

265

266

Chapter 9. Parity and Other Option Relationships

TABLE 9.1

IBM option prices, dollars per share, May 6, 2011. The closing price of IBM on that day was $168.89.

Strike

Expiration

160
165
170
175
160
165
170
175

June
June
June
June
October
October
October
October

Calls
Bid ($)
Ask ($)
10.05
6.15
3.20
1.38
14.10
10.85
8.10
5.80

Puts
Bid ($)
Ask ($)

10.15
6.25
3.30
1.43
14.20
11.00
8.20
5.90

1.16
2.26
4.25
7.40
5.70
7.45
9.70
12.40

1.20
2.31
4.35
7.55
5.80
7.60
9.85
12.55

Source: Chicago Board Options Exchange.

options with the same strike price and time to expiration is
Call − put = PV(forward price − strike price)
Equation (3.1) from Chapter 3 expresses this more precisely:
C(K , T ) − P (K , T ) = PV0, T (F0, T − K)
=e

−rT

(9.1)

(F0, T − K)

where C(K , T ) is the price of a European call with strike price K and time to expiration
T , P (K , T ) is the price of a European put, F0, T is the forward price for the underlying asset, K is the strike price, T is the time to expiration of the options, and PV0, T denotes the present value over the life of the options. Note that e−rT F0, T is the prepaid forward price for the asset and e−rT K is the prepaid forward price for the strike, so we can also think of parity in terms of prepaid forward prices.
The intuition for equation (9.1) is that buying a call and selling a put with the strike equal to the forward price (F0, T = K) creates a synthetic forward contract and hence must have a zero price. If we create a synthetic long forward position at a price lower than the forward price, we have to pay PV0, T (F0, T − K) since this is the benefit of buying the asset at the strike price rather than the forward price.
Parity generally fails for American-style options, which may be exercised prior to maturity. Appendix 9.A discusses parity bounds for American options.
We now consider the parity relationship in more detail for different underlying assets.

Options on Stocks
If the underlying asset is a stock and Div is the stream of dividends paid on the stock, then from Chapter 5, e−rT F0, T = S0 − PV0, T (Div). Thus, from equation (9.1), the parity relationship for European options on stocks is

9.1 Put-Call Parity

267

Outright Purchase of Stock

FIGURE 9.1
Cash flows for outright purchase of stock and for synthetic stock created by buying a 40-strike call and selling a 40-strike put.

Day 0

Day 91
Own 1 share

–$40
Buy Call, Sell Put

–$0.79

C(K , T ) = P (K , T ) + [S0 − PV0, T (Div)] − e−rT K

Own 1 share
–$40

(9.2)

where S0 is the current stock price and PV0, T (Div) is the present value of dividends payable over the life of the option. For index options, we know that S0 − PV0, T (Div) = S0e−δT .
Hence, we can write
C(K , T ) = P (K , T ) + S0e−δT − PV0, T (K)
Example 9.1 Suppose that the price of a non-dividend-paying stock is $40, the continuously compounded interest rate is 8%, and options have 3 months to expiration. A 40-strike
European call sells for $2.78 and a 40-strike European put sells for $1.99. This is consistent with equation (9.2):
$2.78 = $1.99 + $40 − $40e−0.08×0.25
Why does the price of an at-the-money call exceed the price of an at-the-money put by $0.79? The answer is that buying a call and selling a put is a leveraged synthetic stock purchase, with different cash flows than an outright purchase. Figure 9.1 compares the synthetic purchase of the stock (buying the call and selling the put) with an outright purchase. Notice that both positions result in the ownership of the stock 3 months from today.
With the outright purchase, we pay $40 today and own the stock for the entire 3 months. With the synthetic purchase, we pay $0.79 today and $40 at expiration. If the price at expiration is above $40, we will own the stock because we exercise the call. If the stock price is below
$40, we will own the stock because we sold a put that will be exercised; as the put-writer, we have to buy the stock.
Although with the synthetic stock purchase we do not own the stock until expiration, we are still exposed to the risk of the stock. A $1 change in the value of the stock will induce a $1 change in the value of the synthetic stock, exactly as if we owned the stock outright.
Thus, by buying the call and selling the put we own the stock economically, but we have deferred the payment of $40 until expiration. To obtain this deferral we must pay 3 months of interest on the $40, the present value of which is $0.79. The option premiums differ by the interest on the deferral of payment for the stock. The difference in the call and put premiums is due to interest on the strike price. This is the reason that an at-the-money
European call on a non-dividend-paying stock always sells for more than an at-the-money
European put with the same expiration.

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Chapter 9. Parity and Other Option Relationships

Note that if we reverse the position by selling the call and buying the put, then we are synthetically short-selling the stock. In 3 months, the options will be exercised and we will receive $40. In this case, the $0.79 compensates us for deferring receipt of the $40.
There are differences between the outright and synthetic positions. First, the stock pays dividends and the synthetic does not. This example assumed that the stock paid no dividends. If it did, the cost of the actual stock would exceed that of the synthetic by the present value of dividends paid over the life of the options. Second, the actual stock has voting rights, unlike the synthetic position.
Example 9.2 Make the same assumptions as in Example 9.1, except suppose that the stock pays a $5 dividend just before expiration. The price of the European call is $0.74 and the price of the European put is $4.85. These prices satisfy parity with dividends, equation
(9.2):
$0.74 − $4.85 = ($40 − $5e−0.08×0.25) − $40e−0.08×0.25
The call price is higher than the put price by interest on the strike ($0.79) and lower by the present value of the dividend ($4.90), for a net difference of $4.11.
In this example, the at-the-money call sells for less than an at-the-money put because dividends on the stock exceed the value of interest on the strike price.
It is worth mentioning a common but erroneous explanation for the higher premium of an at-the-money call compared to an at-the-money put. The profit on a call is potentially unlimited since the stock price can go to infinity, while the profit on a put can be no greater than the strike price. This explanation seems to suggest that the call should be more expensive than the put.1 However, parity shows that the true reason for the call being more expensive (as in Example 9.1) is time value of money.
Synthetic stock. Parity provides a cookbook for the synthetic creation of options, stocks, and T-bills.
The example above shows that buying a call and selling a put is like buying the stock except that the timing of the payment for the stock differs in the two cases. Rewriting equation (9.2) gives us
S0 = C(K , T ) − P (K , T ) + PV0, T (Div) + e−rT K

(9.3)

To match the cash flows for an outright purchase of the stock, in addition to buying the call and selling the put, we have to lend the present value of the strike and dividends to be paid over the life of the option. We then receive the stock in 91 days.
Synthetic T-bills. If we buy the stock, sell the call, and buy the put, we have purchased the stock and short-sold the synthetic stock. This transaction gives us a hedged position that has no risk but requires investment. Parity shows us that
S0 + P (K , T ) − C(K , T ) = PV0, T (K) + PV0, T (Div)
We have thus created a position that costs PV(K) + PV0, T (Div) and that pays K +
F V0, T (Div) at expiration. This is a synthetic Treasury bill.

1. In fact, the argument also seems to suggest that every stock is worth more than its price!

9.1 Put-Call Parity

Since T-bills are taxed differently than stocks, the ability to create a synthetic Treasury bill with the stock and options creates problems for tax and accounting authorities. How should the return on this transaction be taxed—as a stock transaction or as interest income?
Tax rules call for this position to be taxed as interest, but you can imagine taxpayers trying to skirt these rules.
The creation of a synthetic T-bill by buying the stock, buying a put, and selling a call is called a conversion. If we short the stock, buy a call, and sell a put, we have created a synthetic short T-bill position and this is called a reverse conversion.
Synthetic options. Parity tells us that
C(K , T ) = S0 − PV0, T (Div) − PV0, T (K) + P (K , T ) and that
P (K , T ) = C(K , T ) − S0 + PV0, T (K) + PV0, T (Div)
The first relation says that a call is equivalent to a leveraged position on the underlying asset [S0 − PV0, T (Div) − PV(K)], which is insured by the purchase of a put. The second relation says that a put is equivalent to a short position on the stock, insured by the purchase of a call.

Options on Currencies
Suppose we have options to buy euros by paying dollars. From our discussion of currency forward contracts in Chapter 5, we know that the dollar forward price for a euro
(r−r=)T
C is F0, T = x0e
, where x0 is the current exchange rate denominated as $/= , r= is the
C C euro-denominated interest rate, and r is the dollar-denominated interest rate. The parity relationship for options to buy one euro by paying x0 is then
C(K , T ) − P (K , T ) = x0e

−r=T
C

− Ke−rT

(9.4)

Buying a euro call and selling a euro put is equivalent to lending euros and borrowing dollars.
Equation (9.4) tells us that the difference in the call and put premiums simply reflects the difference in the amount borrowed and loaned, in the currency of the country in which the options are denominated.

Options on Bonds
Finally, we can construct the parity relationship for options on bonds. The prepaid forward for a bond differs from the bond price due to coupon payments (which are like dividends).
Thus, if the bond price is B0, we have
C(K , T ) = P (K , T ) + [B0 − PV0, T (Coupons)] − PV0, T (K)

(9.5)

Note that for a pure-discount bond, the parity relationship is exactly like that for a nondividend-paying stock.

Dividend Forward Contracts
Dividends play a critical role in the parity equation. For most of the book we treat either dollar dividends or proportional dividends as known. In practice, however, there is at least

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some uncertainty about future dividend payments and yields. Dividend forward contracts and dividend swaps can be used to hedge or speculate on actual dividend amounts. A dividend forward contract pays FV0, T (D) − F0, T at time T , where F0, T is the dividend forward price and FV0, T (D) is the time T value of dividends paid between time 0 and time
T . It is straightforward to determine the dividend forward price, F0, T (D), using put-call parity. The intuition is that prices of European options reflect expected future ex-dividend stock prices, so that comparing the stock price today with option prices, we can determine the price of a claim on dividends.2
Suppose a stock with price S0 pays uncertain dividends between time 0 and time T.
Consider buying the stock, borrowing Ke−rT , and buying a European put and selling a
European call, both with strike price K expiring at time T . The payoff to this portfolio is the future value of dividends paid between time 0 and time T :3
ST + FV0, T (D) −
Stock plus dividends

K
Borrowing

+ K − ST = FV0, T (D)
Options

The cost of this portfolio is therefore the time 0 price of acquiring dividends paid from 0 to
T , which is the prepaid forward price for a dividend contract. Thus,4
P
F0, T (D) = S0 − Ke−rT + P (K , T ) − C(K , T )

(9.6)

This implies that the dividend forward price is
P
F0, T (D) = erT F0, T (D) = erT S0 + P (K , T ) − C(K , T ) − K

(9.7)

A dividend swap can be constructed from dividend forwards using equation (8.14).

9.2 GENERALIZED PARITY AND EXCHANGE OPTIONS
The preceding section showed how the parity relationship works for different underlying assets. Now we will generalize parity to apply to the case where the strike asset is not necessarily cash but could be any other asset. This version of parity includes all previous versions as special cases.
Suppose we have an option to exchange one asset for another. Let the underlying asset, asset A, have price St , and the strike asset (the asset which, at our discretion, we surrender in exchange for the underlying asset), asset B, have the price Qt . Let FtPT (S)
,
denote the time t price of a prepaid forward on the underlying asset, paying ST at time T , and let FtPT (Q) denote the time t price of a prepaid forward on asset B, paying QT at time
,
T . We use the notation C(St , Qt , T − t) to denote the time t price of an option with T − t periods to expiration, which gives us the right to give up asset B in exchange for asset A.

2. Special and extraordinary dividends are typically excluded from dividend forwards and swaps. This exclusion is consistent with the common practice of adjusting option terms to protect option holders against changes in value due to extraordinary dividends.
3. In practice, dividend forwards and swaps settle annually based on actual dividends, without taking into account the interest from reinvesting dividends.
4. van Binsbergen et al. (forthcoming) study the returns on investment strategies based on equation (9.6). van Binsbergen et al. (2011) show that prices on dividend futures contracts are predictors of growth for
Europe, the U.S., and Japan.

9.2 Generalized Parity and Exchange Options

P (St , Qt , T − t) is defined similarly as the right to give up asset A in exchange for asset
B. Now suppose that the call payoff at time T is
C(ST , QT , 0) = max(0, ST − QT ) and the put payoff is
P (ST , QT , 0) = max(0, QT − ST )
Then for European options we have this form of the parity equation:
C(St , Qt , T − t) − P (St , Qt , T − t) = FtPT (S) − FtPT (Q)
,
,

(9.8)

The use of prepaid forward prices in the parity relationship completely takes into account the dividend and time value of money considerations. This version of parity tells us that there is nothing special about an option having the strike amount designated as cash. In general, options can be designed to exchange any asset for any other asset, and the relative put and call premiums are determined by prices of prepaid forwards on the underlying and strike assets.
To prove equation (9.8) we can use a payoff table in which we buy a call, sell a put, sell a prepaid forward on A, and buy a prepaid forward on B. This transaction is illustrated in Table 9.2.
If the strategy in Table 9.2 does not pay zero at expiration, there is an arbitrage opportunity. Thus, we expect equation (9.8) to hold. All European options satisfy this formula, whatever the underlying asset.

TABLE 9.2

Payoff table demonstrating that there is an arbitrage opportunity unless −C(St , Qt , T − t) + P (St , Qt , T − t)
+ FtPT (S) − FtPT (Q) = 0.
,
,

Transaction

Time 0

Expiration
S T ≤ QT
ST > Q T

Buy call
Sell put

−C(St , Qt , T − t)
P (St , Qt , T − t)

0
ST − QT

ST − Q T
0

Sell prepaid forward on A
FtPT (S)
,

−ST

−ST

Buy prepaid forward on B
−FtPT (Q)
,

QT

QT

0

0

Total

−C(St , Qt , T − t)
+P (St , Qt , T − t)
+FtPT (S) − FtPT (Q)
,
,

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Example 9.3 Suppose that non-dividend-paying stock A has a price of $20, and nondividend-paying stock B has a price of $25. Because neither stock pays dividends, their prepaid forward prices equal their prices. If A is the underlying asset and B is the strike asset, then put-call parity implies that
Call − put = $20 − $25 = −$5
The put is $5 more expensive than the call for any time to expiration of the options.

Options to Exchange Stock
Executive stock options are sometimes constructed so that the strike price of the option is the price of an index, rather than a fixed cash amount. The idea is to have an option that pays off only when the company outperforms competitors, rather than one that pays off simply because all stock prices have gone up. As a hypothetical example of this, suppose that Microsoft executives are given compensation options that payoff only if Microsoft outperforms Google. They will exercise these options if and only if the share price of
Microsoft, SMSFT , exceeds the share price of Google, SGOOG , i.e., SMSFT > SGOOG . From the executives’ perspective, this is a call option, with the payoff max(0, SMSFT − SGOOG )
Now consider the compensation option for Google executives. They receive a compensation option that pays off only if Google outperforms Microsoft, i.e., max(0, SGOOG − SMSFT )
This is a call from the Google executives’ perspective.
Here is the interesting twist: The Google call looks like a Microsoft put! And the
Microsoft call looks like a Google put. Either option can be viewed as a put or call; it is simply a matter of perspective. The distinction between a put and a call in this example depends upon what we label the underlying asset and what we label as the strike asset.

What Are Calls and Puts?
The preceding discussion suggests that labeling an option as a call or put is always a matter of convention. It is an important convention because we use it all the time in talking about options. Nevertheless, in general we can interpret calls as being puts, and vice versa. We can see why by using an analogy.
When you go to the grocery store to obtain bananas, you typically say that you are buying bananas. The actual transaction involves handing cash to the grocer and receiving a banana. This is an exchange of one asset (cash) for another (a banana). We could also describe the transaction by saying that we are selling cash (in exchange for bananas). The point is that an exchange occurs, and we can describe it either as buying the thing we receive, or selling the thing we surrender.
Any transaction is an exchange of one thing for another. Whether we say we are buying or selling is a matter of convention. This insight may not impress your grocer, but it is important for us because it suggests that the labeling we commonly use to distinguish calls and puts is a matter of convention.

9.2 Generalized Parity and Exchange Options

To see how an ordinary call could be considered a put, consider a call option on a stock. This is the right to exchange a given number of dollars, the strike price K, for stock worth S, if the stock is worth more than the dollars. For example, suppose that if S > K, we earn S − K. We can view this as either of two transactions:
.

.

Buying one share of stock by paying K. In this case we exercise when S > K. This is a call option on stock.
Selling K dollars in exchange for one share of stock. Again we exercise when
S > K—i.e., when the dollars we sell are worth less than the stock. This is a put option on dollars, with a share of stock as the strike asset.

Under either interpretation, if S < K we do not exercise the option. If the dollars are worth more than the stock, we would not sell them for the stock.
Now consider a put option on a stock. The put option confers the right to exchange one share of stock for a given number of dollars. Suppose S < K; we earn K − S. We can view this in either of two ways:
.

.

Selling one share of stock at the price K.
Buying K dollars by paying one share of stock. This is a call where we have the right to give up stock to obtain dollars.

If S > K we do not exercise under either interpretation. If the dollars are worth less than the stock, we would not pay the stock to obtain the dollars.

Currency Options
The idea that calls can be relabeled as puts is commonplace in currency markets. A currency transaction involves the exchange of one kind of currency for another. In this context, it is obvious to market participants that referring to a particular currency as having been bought or sold is a matter of convention, depending upon which currency a trader regards as the home currency.
To understand how a call in one currency can be a put in the other, consider how currency options are quoted by dealers. A term sheet for a currency option might specify
“EUR Call USD Put, AMT: EUR 100 million, USD 120 million.” The term sheet thus says explicitly that the option can be viewed either as a call on the euro or a put on the dollar.
C
Exercise of the option will entail an exchange of = 100 million for $120 million.
If we interpret the option as a call on the euro, we have the right to pay $120 million
C
to acquire = 100 million. You can think of this as 100 million calls on the euro, each with a strike of $1.20. If we interpret the option as a put on the dollar, we have the right to sell
C
$120 million in exchange for = 100 million. Because the put is on the dollar, it is natural to think of this as 120 million options on one dollar, each with a strike of 100/120 = 0.8333 euros. We will say that an option is “dollar-denominated” if the strike price and premium are denominated in dollars. An option is “euro-denominated” if the strike price and premium are in euros. It is often helpful to think of a dollar-denominated option as being based on one unit of the foreign currency, and a euro- (or other currency-) denominated option as based on one dollar.

273

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We will now use a numerical example to illustrate how to convert a call in one currency into a put in the other. Suppose the current exchange rate is x0 = $1.25/= , and consider the
C
following two options:5
1. A 1-year dollar-denominated call option on euros with a strike price of $1.20 and
C
premium of $0.06545. In 1 year, the owner of the option has the right to buy = 1 for
$1.20. The payoff on this option, in dollars, is therefore max(0, x1 − 1.20)
1
2. A 1-year euro-denominated put option on dollars with a strike of 1.20 = = 0.833. The
C
= 0.04363. In 1 year the owner of this put has the right to premium of this option is C
C
give up $1 and receive = 0.833; the owner will exercise the put when $1 is worth less
C
than = 0.833. The euro value of $1 in 1 year will be 1/x1. Hence, the payoff of this option is

max 0,

1
1

1.20 x1

1
1
Because x1 > 1.20 exactly when 1.20 > x , the euro-denominated dollar put will be
1
exercised when, and only when, the dollar-denominated euro call is exercised.
Though the two options will be exercised under the same circumstances, they differ in two respects:
.

.

The scale of the two options is different. The dollar-denominated euro call is based on one euro, which has a current dollar value of $1.25, and the euro-denominated dollar
C
put is based on one dollar, which has a current euro value of = 0.8333.
The currency of denomination is different.

We can equate the scale of the two options in one of two ways. First, we can either scale up the dollar-denominated euro calls, holding 1.20 of them, or we can scale down the
1
euro-denominated dollar puts, holding 1.20 of them. To see the equivalence of the euro call and the dollar put, consider the following two transactions, each of which entails exchanging
C
$1.20 for = 1 if the exchange rate is greater than $1.20:
1. Buy one 1-year dollar-denominated euro call option with a strike of $1.20. If we
C
exercise, we will give up $1.20 for = 1. The cost of the option is $0.06545.
2. Buy 1.20 1-year euro-denominated put options on dollars with a strike of = 0.833. If
C
C exercised, these options entail receiving = 1 and giving up $1.20. The cost of this in
C
C dollars is $1.25/= × 1.20 × = 0.04363 = $0.06545.
Table 9.3 compares the payoffs of these two option positions. At exercise, each position results in surrendering $1.20 for = 1 if x1 > 1.20. The two positions must cost the
C
same, or else there is an arbitrage opportunity.
We can summarize this result algebraically. The price of a dollar-denominated foreign currency call with strike K , when the current exchange rate is x0, is C$(x0 , K , T ). The price

5. These are Black-Scholes prices with a current exchange rate of $1.25/= , a dollar-denominated interest
C
rate of 0.005, a euro-denominated interest rate of 0.020, and exchange rate volatility of 10%.

9.3 Comparing Options with Respect to Style, Maturity, and Strike

TABLE 9.3

The equivalence of buying a dollar-denominated euro call and a euro-denominated dollar put. In transaction I, we buy one dollar-denominated call option, permitting us to buy
= 1 for a strike price of $1.20. In transaction II, we buy 1.20
C
euro-denominated puts, each with a premium of = 0.04363,
C
and permitting us to sell $1 for a strike price of = 0.833.
C

Year 0
$

Transaction
I:
II:

Buy 1 euro call
Convert dollars to euros, buy 1.20 dollar puts

=
C

−0.06545
−0.06545


0.05236
−0.05236

of a foreign-currency–denominated dollar put with strike is Pf ( x1 ,
0

1
K,

Year 1 x1 < 1.20 x1 ≥ 1.20
=
=
$
C
$
C

1
K

0

0

−1.20

1

0

0

−1.20

1

, when the exchange rate is

T ). Adjusting for currency and scale differences, the prices are related by
C$(x0, K , T ) = x0 K Pf

1 1
, ,T x0 K

1 x0 ,

(9.9)

This insight—that calls in one currency are the same as puts in the other—is interesting in and of itself. Its generalization to all options provides a fresh perspective for thinking about the difference between calls and puts.

9.3 COMPARING OPTIONS WITH RESPECT TO STYLE,
MATURITY, AND STRIKE
We now examine how option prices change when there are changes in option characteristics, such as exercise style (American or European), the strike price, and time to expiration.
Remarkably, we can say a great deal without a pricing model and without making any assumptions about the distribution of the underlying asset.6 Thus, whatever the particular option model or stock price distribution used for valuing a given option, we can still expect option prices to behave in certain ways.
Here is an example of the kind of questions we will address in this section. Suppose you have three call options, with strikes of $40, $45, and $50. How do the premiums on these options differ? Common sense suggests that, with a call option on any underlying asset, the premium will go down as you raise the strike price; it is less valuable to be able

6. The so-called “theory of rational option pricing,” on which this section is based, was first presented in
1973 by Robert Merton in an astonishing paper (Merton, 1973b). This material is also superbly exposited in Cox and Rubinstein (1985).

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to buy at a higher price.7 Moreover, the decline in the premium cannot be greater than $5.
(The right to buy for a $5 cheaper price cannot be worth more than $5.)
Following this logic, the premium will drop as we increase the strike from $40 to $45, and drop again when we increase the strike further from $45 to $50. Here is a more subtle question: In which case will the premium drop more? It turns out that the decline in the premium from $40 to $45 must be greater than the decline from $45 to $50, or else there is an arbitrage opportunity.
In this section we will explore the following issues for stock options (some of the properties may be different for options on other underlying assets):
.

How prices of otherwise identical American and European options compare.

.

How option prices change as the time to expiration changes.

.

How option prices change as the strike price changes.

A word of warning before we begin this discussion: You may find option price listings reporting prices that seem to permit arbitrage. This can occur if some of the reported option prices are stale, meaning that the comparison is among option prices recorded at different times of the day. Also, an apparent arbitrage opportunity only becomes genuine when bidask spreads (see Table 9.1), commissions, costs of short-selling, and market impact are taken into account.

European Versus American Options
Since an American option can be exercised at any time, it must always be at least as valuable as an otherwise identical European option. (By “otherwise identical” we mean that the two options have the same underlying asset, strike price, and time to expiration.) Any exercise strategy appropriate to a European option can always be duplicated with an American option.
Thus we have
CAmer (S , K , T ) ≥ CEur (S , K , T )

(9.10a)

PAmer (S , K , T ) ≥ PEur (S , K , T )

(9.10b)

We will see that there are times when the right to early-exercise is worthless, and, hence,
American and European options have the same value.

Maximum and Minimum Option Prices
It is often useful to understand just how expensive or inexpensive an option can be. Here are some basic limits.
Calls. The price of a European call option:
.

.

Cannot be negative, because the call need not be exercised.
Cannot exceed the stock price, because the best that can happen with a call is that you end up owning the stock.

7. If you are being fastidious, you will say the option premium cannot increase as the strike goes up. Saying that the option premium will decrease as the strike increases does not account for the possibility that all the premiums are zero, and hence the premium will not go down but will remain unchanged, as the strike price increases.

9.3 Comparing Options with Respect to Style, Maturity, and Strike

.

Must be at least as great as the price implied by parity with a zero put value.

Combining these statements, together with the result about American options never being worth less than European options, gives us
S ≥ CAmer (S , K , T ) ≥ CEur (S , K , T ) ≥ max[0, PV0, T (F0, T ) − PV0, T (K)]

(9.11)

where present values are taken over the life of the option.
Puts. Similarly, a put:
.

.

Cannot be worth more than the undiscounted strike price, since that is the most it can ever be worth (if the stock price drops to zero, the put pays K at some point).
Must be at least as great as the price implied by parity with a zero call value.
Also, an American put is worth at least as much as a European put. This gives us
K ≥ PAmer (S , K , T ) ≥ PEur (S , K , T ) ≥ max[0, PV(K) − PV0, T ]

(9.12)

Early Exercise for American Options
When might we want to exercise an option prior to expiration? An important result is that an American call option on a non-dividend-paying stock should never be exercised prior to expiration. You may, however, rationally exercise an American-style put option prior to expiration. Calls on a non-dividend-paying stock. We can demonstrate in two ways that an
American-style call option on a non-dividend-paying stock should never be exercised prior to expiration. Early exercise is not optimal if the price of an American call prior to expiration satisfies CAmer (St , K , T − t) > St − K
If this inequality holds, you would lose money by early-exercising (receiving St − K) as opposed to selling the option (receiving CAmer (St , K , T − t) > St − K).
We will use put-call parity to demonstrate that early exercise is not rational. If the option expires at T , parity implies that
CEur (St , K , T − t) = St − K

+ PEur (St , K , T − t)

Exercise value

+ K(1 − e

Insurance against ST St − K

(9.13)

Time value of money on K

Since the put price, PEur (St , K , T − t), and the time value of money on the strike, K(1 − e−r(T −t)), are both positive, this equation establishes that the European call option premium on a non-dividend paying stock always is at least as great as St − K. From equation (9.10a), we also know that CAmer ≥ CEur . Thus we have
CAmer ≥ CEur > St − K
Since CAmer , the American option premium, always exceeds S − K, we would lose money exercising an American call prior to expiration, as opposed to selling the option.

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Chapter 9. Parity and Other Option Relationships

Equation (9.13) is useful because it shows us precisely why we would never earlyexercise. Early-exercising has two effects. First, we throw away the implicit put protection should the stock later move below the strike price. Second, we accelerate the payment of the strike price.
A third effect is the possible loss from deferring receipt of the stock. However, when there are no dividends, we lose nothing by waiting to take physical possession of the stock.
We have demonstrated that if a stock pays no dividends, you should never see an option selling for less than St − K. In fact, equation (9.13), like equation (9.11), actually implies the stronger result that you should never see a call on a non-dividend-paying stock sell for less than St − Ke−r(T −t). What happens if you do observe an option selling for too low a price? If C < St − K and the option is American, you can buy the option, exercise it, and earn St − K − CAmer (St , K , T − t) > 0. If the option is European and cannot be exercised early, the arbitrage is: Buy the option, short the stock, and lend the present value of the strike price. Table 9.4 demonstrates the arbitrage in this case. The sources of profit from the arbitrage are the same as those identified in equation (9.13).
It is important to realize that this proposition does not say that you must hold the option until expiration. It says that if you no longer wish to hold the call, you should sell it rather than early-exercising it.8
Exercising calls just prior to a dividend. If the stock pays dividends, the parity relationship is
C(St , K , T − t) = P (St , K , T − t) + St − PVt , T (Div) − PVt , T (K)
Using this expression, we cannot always rule out early exercise as we did above. Early exercise is not optimal at any time where
K − PVt , T (K) > PVt , T (Div)

(9.14)

That is, if interest on the strike price (which induces us to delay exercise) exceeds the present value of dividends (which induces us to exercise), then we will for certain never early-exercise at that time. If inequality (9.14) is violated, this does not tell us that we will exercise, only that we cannot rule it out.
If dividends are sufficiently great, however, early exercise can be optimal. To take an extreme example, consider a 90-strike American call on a stock selling for $100, which is about to pay a dividend of $99.99. If we exercise—paying $90 to acquire the $100 stock—we have a net position worth $10. If we delay past the ex-dividend date, the option is worthless.
If dividends do make early exercise rational, it will be optimal to exercise at the last moment before the ex-dividend date. By exercising earlier than that, we pay the strike price prematurely and thus at a minimum lose interest on the strike price.
Early exercise for puts. It can be optimal to early-exercise a put. To see this, suppose a company is bankrupt and the stock price falls to zero. Then a put not exercised until expiration will be worth PVt , T (K). If we could early-exercise, we would receive K. If the

8. Some options, such as compensation options, cannot be sold. In practice it is common to see executives exercise options prior to expiration and then sell the stock. The discussion in this section demonstrates that such exercise would be irrational if the option could be sold or if the stock could be sold short.

9.3 Comparing Options with Respect to Style, Maturity, and Strike

Demonstration of arbitrage if a call option with price C sells for less than St − Ke−r(T −t) and the stock pays no dividends. Every entry in the row labeled “Total” is nonnegative. TABLE 9.4

Transaction

Time t

Expiration or Exercise, Time T
ST < K
ST > K

Buy call
Short stock
Lend Ke−r(T −t)
Total

−C
St
−Ke−r(T −t)
St − Ke−r(T −t) − C

0
−ST
K
K − ST

ST − K
−ST
K
0

interest rate is positive, K > PV(K). Therefore, early exercise would be optimal in order to receive the strike price earlier.
We can also use a parity argument to understand this. The put will never be exercised as long as P > K − S. Supposing that the stock pays no dividends, parity for the put is
P (St , K , T − t) = C(St , K , T − t) − St + PVt , T (K)
The no-exercise condition, P > K − S, then implies
C(St , K , T − t) − St + PVt , T (K) > K − St or C(St , K , T − t) > K − PVt , T (K)
If the call is sufficiently valueless (as in the above example of a bankrupt company), parity cannot rule out early exercise. This does not mean that we will early-exercise; it simply means that we cannot rule it out.
Early exercise in general. We can summarize this discussion of early exercise. When we exercise an option, we receive something (the stock with a call, the strike price with a put).
A necessary condition for early exercise is that we prefer to receive this something sooner rather than later. For calls, dividends on the stock are a reason to want to receive the stock earlier. For puts, interest on the strike is a reason to want to receive the strike price earlier.
Thus, dividends and interest play similar roles in the two analyses of early exercise. In fact, if we view interest as the dividend on cash, then dividends (broadly defined) become the sole reason to early-exercise an option.
Similarly, dividends on the strike asset become a reason not to early-exercise. In the case of calls, interest is the dividend on the strike asset, and in the case of puts, dividends on the stock are the dividend on the strike asset.
The point of this section has been to make some general statements about when early exercise will not occur, or under what conditions it might occur. Early exercise is a trade-off involving time value of money on the strike price, dividends on the underlying asset, and the value of insurance on the position. In general, figuring out when to exercise requires an option pricing model. We will discuss early exercise further in Chapters 10 and 11.

279

280

Chapter 9. Parity and Other Option Relationships

Time to Expiration
How does an option price change as we increase time to expiration? If the options are
American, the option price can never decline with an increase in time to expiration. If the options are European, the price can go either up or down as we increase time to expiration.
American options. An American call with more time to expiration is at least as valuable as an otherwise identical call with less time to expiration. An American call with 2 years to expiration, for example, can always be turned into an American option with 1 year to expiration by voluntarily exercising it after 1 year. Therefore, the 2-year call is at least as valuable as the 1-year call. For the same reason, a longer-lived American put is always worth at least as much as an otherwise equivalent European put.
European options. A European call on a non-dividend-paying stock will be at least as valuable as an otherwise identical call with a shorter time to expiration. This occurs because, with no dividends, a European call has the same price as an otherwise identical
American call. With dividends, however, longer-lived European calls may be less valuable than shorter-lived European calls.
To see this, imagine a stock that will pay a liquidating dividend 2 weeks from today.9
A European call with 1 week to expiration will have value because it is exercisable prior to the dividend. A European call with 3 weeks to expiration will have no value because the stock will have no value at expiration. This is an example of a longer-lived option being less valuable than a shorter-lived option. Note that if the options were American, we would simply exercise the 3-week option prior to the dividend.
Longer-lived European puts can also be less valuable than shorter-lived European puts. A good example of this is a bankrupt company. The put will be worth the present value of the strike price, with present value calculated until time to expiration. Longer-lived puts will be worth less than shorter-lived puts. If the options were American, they would all be exercised immediately and hence would be worth the strike price.
European options when the strike price grows over time. In discussing the effect of changing time to maturity, we have been keeping the option strike price fixed. The present value of the strike price therefore decreases with time to maturity. Suppose, however, that we keep the present value of the strike constant by setting Kt = Kert . When the strike grows at the interest rate, the premiums on European calls and puts on a non-dividend-paying stock increase with time to maturity.10 We will demonstrate this for puts; the demonstration is identical for calls.
To keep the notation simple, let P (t) denote the time 0 price of a European put maturing at time t, with strike price Kt = Kert . We want to show that P (T ) > P (t) if
T > t. To show this, we will demonstrate an arbitrage if P (T ) ≤ P (t).

9. A liquidating dividend occurs when a firm pays its entire value to shareholders. A firm is worthless after paying a liquidating dividend.
10. Dividends can easily be accommodated in the following way. Suppose that all dividends are reinvested in the stock. Call the resulting position a total return portfolio and let St be the price of this portfolio. Define the options so that the total return portfolio is the underlying asset. The result in this section then obtains for the total return portfolio. The reason is that the total return portfolio is a synthetic non-dividend-paying stock. If the underlying asset does pay dividends, however, then the price of a European call can decrease with time to maturity.

9.3 Comparing Options with Respect to Style, Maturity, and Strike

Demonstration that there is an arbitrage if P (T ) ≤ P (t) with t < T . The strike on the put with maturity t is
Kt = Kert , and the strike on the put with maturity T is KT = Kert . If the option expiring at time t is in-themoney, the payoff, St − Kt , is reinvested until time T .
If P (t) ≥ P (T ), all cash flows in the “total” line are nonnegative. TABLE 9.5

Transaction

Time 0

Payoff at Time T
ST < K T
ST > K T
Payoff at Time t
St < Kt
St > K t
St < Kt
St > Kt

Sell P (t)
Buy P (T )
Total

P (t)
−P (T )
P (t) − P (T )

ST − KT
KT − ST
0

0
KT − ST
KT − ST

ST − KT
0
ST − KT

0
0
0

If the longer-lived put is not more expensive—i.e., if P (T ) ≤ P (t)—buy the put with
T years to expiration and sell the put with t years to expiration. At time t the written put will expire. If St > Kt its value is zero and we can ignore the shorter-lived option from this point on. If St < Kt , the put holder will exercise the short-lived option and our payoff is
St − Kt . Suppose that we keep the stock we receive and borrow to finance the strike price, holding this position until the second option expires at time T . Here is the important step:
Notice that the time-T value of this time-t payoff is ST − Kt er(T −t) = ST − KT .
Table 9.5 summarizes the resulting payoffs. By buying the long-lived put and selling the short-lived put, we are guaranteed not to lose money at time T . Therefore, if P (t) ≥
P (T ) there is an arbitrage opportunity. A practical application of this result is discussed in the box on page 282.

Different Strike Prices
We discussed at the beginning of this section some statements we can make about how option prices vary with the strike price. Here is a more formal statement of these propositions.
Suppose we have three strike prices, K1 < K2 < K3, with corresponding call option prices
C(K1), C(K2), and C(K3) and put option prices P (K1), P (K2), and P (K3). Here are the propositions we discuss in this section:
1. A call with a low strike price is at least as valuable as an otherwise identical call with a higher strike price:
C(K1) ≥ C(K2)

(9.15)

A put with a high strike price is at least as valuable as an otherwise identical put with a low strike price:
P (K2) ≥ P (K1)

(9.16)

281

282

BOX

Chapter 9. Parity and Other Option Relationships

9.1: Portfolio Insurance for the Long Run

Historically, the rate of return from investing in stocks over a long horizon has outperformed that from investing in government bonds in the
United States (see, for example, Siegel, 1998).
This observation has led some to suggest that if held for a sufficiently long period of time, stocks are a safe investment relative to risk-free bonds.
Bodie (1995) suggests using put option premiums to think about the claim that stocks are safe in the long run. Specifically, what would it cost to buy a put option insuring that after T years your stock portfolio would be worth at least as much as if you had instead invested in a zero-coupon bond? If your initial investment was S0, you could

provide this insurance by setting the strike price on the put option equal to KT = S0erT .
Bodie uses the Black-Scholes model to show that the premium on this insurance increases with
T . As Bodie notes, however, this proposition must be true for any valid option pricing model. The payoffs in Table 9.5 demonstrate that the cost of this insurance must increase with T or else there is an arbitrage opportunity. Whatever the historical return statistics appear to say, the cost of portfolio insurance is increasing with the length of time for which you insure the portfolio return. Using the cost of insurance as a measure, stocks are riskier in the long run.

2. The premium difference between otherwise identical calls with different strike prices cannot be greater than the difference in strike prices:
C(K1) − C(K2) ≤ K2 − K1

(9.17)

The premium difference for otherwise identical puts also cannot be greater than the difference in strike prices:
P (K2) − P (K1) ≤ K2 − K1

(9.18)

3. Premiums decline at a decreasing rate as we consider calls with progressively higher strike prices. The same is true for puts as strike prices decline. This is called convexity of the option price with respect to the strike price:
C(K1) − C(K2) C(K2) − C(K3)

K2 − K1
K3 − K 2

(9.19)

P (K2) − P (K1) P (K3) − P (K2)

K2 − K1
K3 − K2

(9.20)

These statements are all true for both European and American options.11 Algebraic demonstrations are in Appendix 9.B. It turns out, however, that these three propositions

11. In fact, if the options are European, the second statement can be strengthened: The difference in option premiums must be less than the present value of the difference in strikes.

9.3 Comparing Options with Respect to Style, Maturity, and Strike

are equivalent to saying that there are no free lunches: If you enter into an option spread, there must be stock prices at which you would lose money on the spread net of your original investment. Otherwise, the spread represents an arbitrage opportunity. These three propositions say that you cannot have a bull spread, a bear spread, or a butterfly spread for which you can never lose money. Specifically:
1. If inequality (9.15) were not true, buy the low-strike call and sell the high-strike call
(this is a call bull spread). If inequality (9.16) were not true, buy the high-strike put and sell the low-strike put (a put bear spread).
2. If inequality (9.17) were not true, sell the low-strike call and buy the high-strike call
(a call bear spread). If inequality (9.18) were not true, buy the low-strike put and sell the high-strike put (a put bull spread).
3. If either of inequalities (9.19) or (9.20) were not true, there is an asymmetric butterfly spread with positive profits at all prices.
We will illustrate these propositions with numerical examples.
Example 9.4 Suppose we observe the call premiums in Panel A of Table 9.6. These values violate the second property for calls, since the difference in strikes is 5 and the difference in the premiums is 6. If we observed these values, we could engage in arbitrage by buying the 55-strike call and selling the 50-strike call, which is a bear spread. Note that we receive
$6 initially and never have to pay more than $5 in the future. This is an arbitrage, whatever the interest rate.
Now consider the third proposition, strike price convexity. There is a different way to write the convexity inequality, equation (9.19). Since K2 is between K1 and K3, we can

TABLE 9.6

Panel A shows call option premiums for which the change in the option premium ($6) exceeds the change in the strike price ($5). Panel B shows how a bear spread can be used to arbitrage these prices. By lending the bear spread proceeds, we have a zero cash flow at time 0; the cash outflow at time
T is always greater than $1.

Panel A
Strike
Premium

50
18

55
12

Panel B
Transaction

Time 0

Expiration or Exercise
ST < 50
50 ≤ ST ≤ 55
ST ≥ 55

Buy 55-strike call
Sell 50-strike call
Total

−12
18
6

0
0
0

0
50 − ST
50 − ST

ST − 55
50 − ST
−5

283

284

Chapter 9. Parity and Other Option Relationships

write it as a weighted average of the other two strikes, i.e.,
K2 = λK1 + (1 − λ)K3 where λ=

K3 − K2
K3 − K1

(9.21)

With this expression for λ, it is possible to rewrite equation (9.19) as
C(K2) ≤ λC(K1) + (1 − λ)C(K3)

(9.22)

Here is an example illustrating convexity.
Example 9.5 If K1 = 50, K2 = 59, and K3 = 65, λ =

65−59
65−50

= 0.4; hence,

59 = 0.4 × 50 + 0.6 × 65
Call prices must then satisfy
C(59) ≤ 0.4 × C(50) + 0.6 × C(65)
Suppose we observe the call premiums in Table 9.7. The change in the option premium per dollar of strike price change from 50 to 59 is 5.1/9 = 0.567, and the change from 59 to 65 is 3.9/6 = 0.65. Thus, prices violate the proposition that the premium decreases at a decreasing rate as the strike price increases.
To arbitrage this mispricing, we engage in an asymmetric butterfly spread: Buy four
50-strike calls, buy six 65-strike calls, and sell ten 59-strike calls.12 By engaging in a butterfly spread, Panel B shows that a profit of at least $3 is earned.
The formula for λ may look imposing, but there is an easy way to figure out what λ is in any situation. In this example, we had the prices 50, 59, and 65. It is possible to express
59 as a weighted average of 50 and 65. The total distance between 50 and 65 is 15, and the distance from 50 to 59 is 9, which is 9/15 = 0.6 of the total distance. Thus, we can write
59 as
59 = (1 − 0.6) × 50 + 0.6 × 65
This is the interpretation of λ in expression (9.22).
Here is an example of convexity with puts.
Example 9.6 See the prices in Panel A of Table 9.8. We have K1 = 50, K2 = 55, and
K3 = 70. λ = 0.75 and 55 = 0.75 × 50 + (1 − 0.75) × 70. Convexity is violated since
P (55) = 8 > 0.75 × 4 + (1 − 0.75) × 16 = 7

12. Note that we get exactly the same arbitrage with any number of calls as long as the ratio at the various strikes remains the same. We could also have bought 0.4 50-strike calls, sold one 59-strike call, and bought
0.6 65-strike calls.

9.3 Comparing Options with Respect to Style, Maturity, and Strike

TABLE 9.7

285

The example in Panel A violates the proposition that the rate of change of the option premium must decrease as the strike price rises. The rate of change from 50 to 59 is 5.1/9, while the rate of change from 59 to 65 is 3.9/6. We can arbitrage this convexity violation with an asymmetric butterfly spread. Panel
B shows that we earn at least $3 plus interest at time T .

Panel A
Strike
Call premium

50
14

59
8.9

65
5

Panel B
Transaction

Time 0

Buy four 50-strike calls
Sell ten 59-strike calls
Buy six 65-strike calls
Lend $3
Total

−56
89
−30
−3
0

0
0
0
3erT
3erT

Expiration or Exercise
50 ≤ ST ≤ 59
59 ≤ ST ≤ 65

ST < 50

TABLE 9.8

4(ST − 50)
0
0
3erT
3erT + 4(ST − 50)

4(ST − 50)
10(59 − ST )
0
3erT
3erT + 6(65 − ST )

ST > 65
4(ST − 50)
10(59 − ST )
6(ST − 65)
3erT
3erT

Arbitrage of mispriced puts using asymmetric butterfly spread.

Panel A
Strike
Put premium

50
4

55
8

70
16

Panel B
Transaction

Time 0

Buy three 50-strike puts
Sell four 55-strike puts
Buy one 70-strike put
Lend $4
Total

−12
32
−16
−4
0

ST < 50
3(50 − ST )
4(ST − 55)
70 − ST
4erT
4erT

Expiration or Exercise
50 ≤ ST ≤ 55
55 ≤ ST ≤ 70
0
4(ST − 55)
70 − ST
4erT
4erT + 3(ST − 50)

0
0
(70 − ST )
4erT
4erT + 70 − ST

ST > 70
0
0
0
4erT
4erT

To arbitrage this mispricing, we engage in an asymmetric butterfly spread: Buy three 50strike puts, buy one 70-strike put, and sell four 55-strike puts. The result is in Panel B of
Table 9.8.
In this case, we always make at least 4. Figure 9.2 illustrates the necessary shape of curves for both calls and puts relating the option premium to the strike price.

286

Chapter 9. Parity and Other Option Relationships

FIGURE 9.2

Option Premium

Illustration of convexity and other strike price properties for calls and puts. For calls the premium is decreasing in the strike, with a slope less than 1 in absolute value.
For puts the premium is increasing in the strike, with a slope less than 1. For both, the graph is convex, i.e., shaped like the cross section of a bowl.

Calls
Puts

50
45
40
35
30
25
20
15
10
5
0

0

20

40

60
Strike Price

80

100

Exercise and Moneyness
If it is optimal to exercise an option, it is also optimal to exercise an otherwise identical option that is more in-the-money. Consider what would have to happen in order for this not to be true.
Suppose a call option on a dividend-paying stock has a strike price of $50, and the stock price is $70. Also suppose that it is optimal to exercise the option. This means that the option must sell for $70 − $50 = $20.
Now what can we say about the premium of a 40-strike option? We know from the discussion above that the change in the premium is no more than the change in the strike price, or else there is an arbitrage opportunity. This means that
C (40) ≤ $20 + ($50 − $40) = $30
Since the 40-strike call is worth $30 if exercised, it must be optimal to exercise it.
Following the same logic, this is also true for puts.

CHAPTER SUMMARY
Put-call parity is one of the most important relations in option pricing. Parity is the observation that buying a European call and selling a European put with the same strike price and time to expiration is equivalent to making a leveraged investment in the underlying asset, less the value of cash payments to the underlying asset over the life of the option. Different versions of parity for different underlying assets appear in Table 9.9. In every case the value

Further Reading

TABLE 9.9

Underlying Asset

Versions of put-call parity. Notation in the table includes the spot currency exchange rate, x0 ; the risk-free interest rate in the foreign currency, rf ; and the current bond price, B0 .

Parity Relationship e−rT F0, T

Futures contract
Stock, no-dividend
S0
Stock, discrete dividend
S0 − PV0, T (Div)
Stock, continuous dividend e−δT S0
Currency
e−rf T x0
Bond
B0 − PV0, T (Coupons)

=
=
=
=
=
=

C(K , T ) − P (K , T ) + e−rT K
C(K , T ) − P (K , T ) + e−rT K
C(K , T ) − P (K , T ) + e−rT K
C(K , T ) − P (K , T ) + e−rT K
C(K , T ) − P (K , T ) + e−rT K
C(K , T ) − P (K , T ) + e−rT K

on the left-hand side of the parity equation is the price of the underlying asset less its cash flows over the life of the option. The parity relationship can be algebraically rearranged so that options and the underlying asset create a synthetic bond, options and a bond create a synthetic stock, and one kind of option together with the stock and bond synthetically create the other kind of option.
The idea of an option can be generalized to permit an asset other than cash to be the strike asset. This insight blurs the distinction between a put and a call. The idea that puts and calls are different ways of looking at the same contract is commonplace in currency markets. Option prices must obey certain restrictions when we vary the strike price, time to maturity, or option exercise style. American options are at least as valuable as European options. American calls and puts become more expensive as time to expiration increases, but
European options need not. European options on a non-dividend-paying stock do become more expensive with increasing time to maturity if the strike price grows at the interest rate.
Dividends are the reason to exercise an American call early, while interest is the reason to exercise an American put early. A call option on a non-dividend-paying stock will always have a price greater than its value if exercised; hence, it should never be exercised early.
There are a number of pricing relationships related to changing strike prices. In particular, as the strike price increases, calls become less expensive with their price decreasing at a decreasing rate. The absolute value of the change in the call price is less than the change in the strike price. As the strike price decreases, puts become less expensive with their price decreasing at a decreasing rate. The change in the put price is less than the change in the strike price.

FURTHER READING
Two of the ideas in this chapter will prove particularly important in later chapters.
The first key idea is put-call parity, which tells us that if we understand calls we also understand puts. This equivalence makes it easier to understand option pricing since the pricing techniques and intuition about one kind of option are directly applicable to the

287

288

Chapter 9. Parity and Other Option Relationships

other. The idea of exchange options—options to exchange one asset for another—also will show up again in later chapters. We will see how to price such options in Chapter 14.
A second key idea that will prove important is the determination of factors influencing early exercise. As a practical matter, it is more work to price an American than a European option, so it is useful to know when this extra work is not necessary. Less obviously, the determinants of early exercise will play a key role in Chapter 17, where we discuss real options. We will see that certain kinds of investment projects are analogous to options, and the investment decision is like exercising an option. Thus, the early-exercise decision can have important consequences beyond the realm of financial options.
Much of the material in this chapter can be traced to Merton (1973b), which contains an exhaustive treatment of option properties that must hold if there is to be no arbitrage.
Cox and Rubinstein (1985) also provides an excellent treatment of this material.

PROBLEMS
9.1 A stock currently sells for $32.00. A 6-month call option with a strike of $35.00 has a premium of $2.27. Assuming a 4% continuously compounded risk-free rate and a
6% continuous dividend yield, what is the price of the associated put option?
9.2 A stock currently sells for $32.00. A 6-month call option with a strike of $30.00 has a premium of $4.29, and a 6-month put with the same strike has a premium of $2.64.
Assume a 4% continuously compounded risk-free rate. What is the present value of dividends payable over the next 6 months?
9.3 Suppose the S&R index is 800, the continuously compounded risk-free rate is 5%, and the dividend yield is 0%. A 1-year 815-strike European call costs $75 and a 1year 815-strike European put costs $45. Consider the strategy of buying the stock, selling the 815-strike call, and buying the 815-strike put.
a. What is the rate of return on this position held until the expiration of the options? b. What is the arbitrage implied by your answer to (a)?
c. What difference between the call and put prices would eliminate arbitrage?
d. What difference between the call and put prices eliminates arbitrage for strike prices of $780, $800, $820, and $840?
C
9.4 Suppose the exchange rate is 0.95 $/= , the euro-denominated continuously compounded interest rate is 4%, the dollar-denominated continuously compounded interest rate is 6%, and the price of a 1-year 0.93-strike European call on the euro is
$0.0571. What is the price of a 0.93-strike European put?
9.5 The premium of a 100-strike yen-denominated put on the euro is ¥8.763. The current exchange rate is 95 ¥/= . What is the strike of the corresponding euro-denominated
C
yen call, and what is its premium?
9.6 The price of a 6-month dollar-denominated call option on the euro with a $0.90 strike is $0.0404. The price of an otherwise equivalent put option is $0.0141. The annual continuously compounded dollar interest rate is 5%.
a. What is the 6-month dollar-euro forward price?

Problems

b. If the euro-denominated annual continuously compounded interest rate is
3.5%, what is the spot exchange rate?
9.7 Suppose the dollar-denominated interest rate is 5%, the yen-denominated interest rate is 1% (both rates are continuously compounded), the spot exchange rate is 0.009
$/¥, and the price of a dollar-denominated European call to buy one yen with 1 year to expiration and a strike price of $0.009 is $0.0006.
a. What is the dollar-denominated European yen put price such that there is no arbitrage opportunity?
b. Suppose that a dollar-denominated European yen put with a strike of $0.009 has a premium of $0.0004. Demonstrate the arbitrage.
c. Now suppose that you are in Tokyo, trading options that are denominated in yen rather than dollars. If the price of a dollar-denominated at-the-money yen call in the United States is $0.0006, what is the price of a yen-denominated at-the-money dollar call—an option giving the right to buy one dollar, denominated in yen—in Tokyo? What is the relationship of this answer to your answer to (a)? What is the price of the at-the-money dollar put?
9.8 Suppose call and put prices are given by
Strike

50

Call premium
Put premium

55

9
7

10
6

What no-arbitrage property is violated? What spread position would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
9.9 Suppose call and put prices are given by
Strike

50

55

Call premium
Put premium

16
7

10
14

What no-arbitrage property is violated? What spread position would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
9.10 Suppose call and put prices are given by
Strike

50

55

60

Call premium
Put premium

18
7

14
10.75

9.50
14.45

Find the convexity violations. What spread would you use to effect arbitrage?
Demonstrate that the spread position is an arbitrage.
9.11 Suppose call and put prices are given by
Strike

80

100

105

Call premium
Put premium

22
4

9
21

5
24.80

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Chapter 9. Parity and Other Option Relationships

Find the convexity violations. What spread would you use to effect arbitrage?
Demonstrate that the spread position is an arbitrage.
9.12 In each case identify the arbitrage and demonstrate how you would make money by creating a table showing your payoff.
a. Consider two European options on the same stock with the same time to expiration. The 90-strike call costs $10 and the 95-strike call costs $4.
b. Now suppose these options have 2 years to expiration and the continuously compounded interest rate is 10%. The 90-strike call costs $10 and the 95strike call costs $5.25. Show again that there is an arbitrage opportunity. (Hint:
It is important in this case that the options are European.)
c. Suppose that a 90-strike European call sells for $15, a 100-strike call sells for
$10, and a 105-strike call sells for $6. Show how you could use an asymmetric butterfly to profit from this arbitrage opportunity.
9.13 Suppose the interest rate is 0% and the stock of XYZ has a positive dividend yield. Is there any circumstance in which you would early-exercise an American XYZ call?
Is there any circumstance in which you would early-exercise an American XYZ put?
Explain.
9.14 In the following, suppose that neither stock pays a dividend.
a. Suppose you have a call option that permits you to receive one share of
Apple by giving up one share of AOL. In what circumstance might you earlyexercise this call?
b. Suppose you have a put option that permits you to give up one share of Apple, receiving one share of AOL. In what circumstance might you early-exercise this put? Would there be a loss from not early-exercising if Apple had a zero stock price?
c. Now suppose that Apple is expected to pay a dividend. Which of the above answers will change? Why?
9.15 The price of a non-dividend-paying stock is $100 and the continuously compounded risk-free rate is 5%. A 1-year European call option with a strike price of $100 ×
1
e0.05×1 = $105.127 has a premium of $11.924. A 1 2 year European call option with a strike price of $100 × e0.05×1.5 = $107.788 has a premium of $11.50. Demonstrate an arbitrage.
9.16 Suppose that to buy either a call or a put option you pay the quoted ask price, denoted
Ca (K , T ) and Pa (K , T ), and to sell an option you receive the bid, Cb (K , T ) and
Pb (K , T ). Similarly, the ask and bid prices for the stock are Sa and Sb . Finally, suppose you can borrow at the rate rH and lend at the rate rL . The stock pays no dividend. Find the bounds between which you cannot profitably perform a parity arbitrage. 9.17 In this problem we consider whether parity is violated by any of the option prices in Table 9.1. Suppose that you buy at the ask and sell at the bid, and that your continuously compounded lending rate is 0.3% and your borrowing rate is 0.4%.
Ignore transaction costs on the stock, for which the price is $168.89. Assume that

9.A Parity Bounds for American Options

IBM is expected to pay a $0.75 dividend on August 8, 2011. Options expire on the third Friday of the expiration month. For each strike and expiration, what is the cost if you:
a. Buy the call, sell the put, short the stock, and lend the present value of the strike price plus dividend (where appropriate)?
b. Sell the call, buy the put, buy the stock, and borrow the present value of the strike price plus dividend (where appropriate)?
9.18 Consider the June 165, 170, and 175 call option prices in Table 9.1.
a. Does convexity hold if you buy a butterfly spread, buying at the ask price and selling at the bid?
b. Does convexity hold if you sell a butterfly spread, buying at the ask price and selling at the bid?
c. Does convexity hold if you are a market-maker either buying or selling a butterfly, paying the bid and receiving the ask?

Appendix 9.A PARITY BOUNDS FOR AMERICAN OPTIONS
The exact parity relationship discussed in Chapter 9 only holds for European options.
However, American options often come close to obeying put-call parity, especially when options have short times to expiration.
With a non-dividend-paying stock, the call will not be exercised early, but the put might be. The effect of early exercise for the put is to accelerate the receipt of the strike price. Since interest on the strike price is small for short times to maturity, parity will come close to holding for short-lived American options on non-dividend-paying stocks.
We now let P and C refer to prices of American options. The American put can be more valuable than the European put, and we have
P ≥ C + PV(K) − S
However, suppose that the put were exercised early. Then it would be worth K − S. For example, if we think of synthetically creating the stock by buying the call and selling the put, there is a chance that we will pay K before expiration, in the event the stock price plummets and the put is early-exercised. Consequently, if we replace the present value of the strike with the undiscounted strike, we have a valid upper bound for the value of the put.
It will be true (and you can verify with a no-arbitrage argument) that
P ≤C+K −S
When there are no dividends, we have C + K − S as an upper bound on the put, and
European parity as a lower bound (since an American put is always worth at least as much as a European put). The parity relationship can be written as a restriction on the put price or on the call price:
C + K − S ≥ P ≥ C + PV(K) − S
P + S − PV(K) ≥ C ≥ P + S − K

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Thus, when there are no dividends, European parity can be violated to the extent of interest on the strike price. Since this will be small for options that are not long-lived, European parity can remain a good approximation for American options.
Dividends add the complication that the call as well as the put may be exercised early. There exists the possibility of a large parity violation because of the following
“whipsaw” scenario: The call is exercised early to capture a large dividend payment, the stock price drops, and the put is then exercised early to capture interest on the strike price.
The possibility that this can happen leads to a wider no-arbitrage band. With dividends, the parity relationship becomes (Cox and Rubinstein, 1985, p. 152)
C + K + PV(D) − S ≥ P ≥ C + PV(K) − S
P + S − PV(K) ≥ C ≥ P + S − PV(D) − K
The upper bound for the call is the same as in European parity, except without dividends. The intuition for the upper bound on the call option (the left-hand side) is that we can avoid the loss of dividends by early-exercising the call; hence, it is the same bound as in the European case with no dividends. The lower bound exists because it may not be optimal to exercise the call to avoid dividends, and it may be optimal to early-exercise the put.
Consider the worst case for the call. Suppose K = $100 and S = $100, and the stock is about to pay a liquidating dividend (i.e., D = $100). We will not exercise the call, since doing so gives us nothing. The put will be exercised after the dividend is paid, once the stock is worthless. So P = $100. The relationship then states
C ≥ P + S − D − K = 100 + 100 − 100 − 100 = 0
And indeed, the call will be worthless in this case.

Appendix 9.B ALGEBRAIC PROOFS OF STRIKE-PRICE RELATIONS
Appendix available at http://www.pearsonhighered.com/mcdonald.

10
I

Binomial Option Pricing:
Basic Concepts

n earlier chapters we discussed how the price of one option is related to the price of another, but we did not explain how to determine the price of an option relative to the price of the underlying asset. In this chapter we discuss the binomial option pricing model, with which we can compute the price of an option, given the characteristics of the stock or other underlying asset.
The binomial option pricing model assumes that, over a period of time, the price of the underlying asset can move up or down only by a specified amount—that is, the asset price follows a binomial distribution. Given this assumption, it is possible to determine a no-arbitrage price for the option. Surprisingly, this approach, which appears at first glance to be overly simplistic, can be used to price options, and it conveys much of the intuition underlying more complex (and seemingly more realistic) option pricing models that we will encounter in later chapters. It is hard to overstate the value of thoroughly understanding the binomial approach to pricing options.
Because of its usefulness, we devote this and the next chapter to binomial option pricing. In this chapter, we will see how the binomial model works and use it to price both
European and American call and put options on stocks, currencies, and futures contracts.
As part of the pricing analysis, we will also see how market-makers can create options synthetically using the underlying asset and risk-free bonds. In the next chapter, we will explore aspects of the model in more depth.

10.1 A ONE-PERIOD BINOMIAL TREE
Binomial pricing achieves its simplicity by making a very strong assumption about the stock price: At any point in time, the stock price can change to either an up value or a down value.
In-between, greater or lesser values are not permitted. The restriction to two possible prices is why the method is called “binomial.” The appeal of binomial pricing is that it displays the logic of option pricing in a simple setting, using only algebra to price options.
The binomial approach to pricing was first used by Sharpe (1978) as an intuitive way to explain option pricing. Binomial pricing was developed more formally by Cox et al. (1979) and Rendleman and Bartter (1979), who showed how to implement the model, demonstrated the link between the binomial model and the Black-Scholes model, and showed that the method provides a tractable way to price options for which early exercise may be optimal.
The binomial model is often referred to as the “Cox-Ross-Rubinstein pricing model.”

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FIGURE 10.1

60

Binomial tree depicting the movement of XYZ stock over 1 year. The current stock price is $41.

41
30

We begin with a simple example. Consider a European call option on the stock of
XYZ, with a $40 strike and 1 year to expiration. XYZ does not pay dividends and its current price is $41. The continuously compounded risk-free interest rate is 8%. We wish to determine the option price.
Since the stock’s return over the next year is uncertain, the option could expire either in-the-money or out-of-the-money, depending upon whether the stock price is more or less than $40. Intuitively, the valuation for the option should take into account both possibilities and assign values in each case. If the option expires out-of-the-money, its value is zero.
If the option expires in-the-money, its value will depend upon how far in-the-money it is.
To price the option, then, we need to characterize the uncertainty about the stock price at expiration. Figure 10.1 represents the evolution of the stock price: Today the price is $41, and in
1 year the price can be either $60 or $30. This depiction of possible stock prices is called a binomial tree. We will see shortly how to construct a binomial tree like that in Figure 10.1.
For now, you should take the tree as given as we work through an option pricing example.
Be aware that if we had started with a different tree, the numbers that follow, including the price, would all be different.

Computing the Option Price
Now we compute the price of our 40-strike 1-year call. Consider two portfolios:
Portfolio A. Buy one call option. The cost of this is the call premium, which is what we are trying to determine.
Portfolio B. Buy 2/3 of a share of XYZ and borrow $18.462 at the risk-free rate.1
This position costs
2/3 × $41 − $18.462 = $8.871
Now we compare the payoffs to the two portfolios 1 year from now. Since the stock can take on only two values, we can easily compute the value of each portfolio at each possible stock price.
For Portfolio A, the time 1 payoff is max[0, S1 − $40]:

1. Of course, it is not possible in practice to buy fractional shares of stock. As an exercise, you can redo this example, multiplying all quantities by 3. You would then compare three call options (Portfolio A) to buying two shares and borrowing $18.462 × 3 = $55.386 (Portfolio B).

10.1 A One-Period Binomial Tree

Stock Price in 1 Year (S1)
$30
Payoff

$60

0

$20

In computing the payoff for Portfolio B, we assume that we sell the shares at the market price and that we repay the borrowed amount, plus interest ($18.462 × e0.08 = $20). Thus we have
Stock Price in 1 Year (S1)
$30
2/3 purchased shares
Repay loan of $18.462
Total payoff

$60

$20
−$20

$40
−$20

0

$20

Note that Portfolios A and B have the same payoff: Zero if the stock price goes down, in which case the option is out-of-the-money, and $20 if the stock price goes up.
Therefore, both portfolios should have the same cost. Since Portfolio B costs $8.871, then given our assumptions, the price of one option must be $8.871. Portfolio B is a synthetic call, mimicking the payoff to a call by buying shares and borrowing.
The idea that positions that have the same payoff should have the same cost is called the law of one price. This example uses the law of one price to determine the option price.
We will see shortly that there is an arbitrage opportunity if the law of one price is violated.
The call option in the example is replicated by holding 2/3 shares, which implies that one option has the risk of 2/3 shares. The value 2/3 is the delta ( ) of the option: the number of shares that replicates the option payoff. Delta is a key concept, and we will say much more about it later.
Finally, we can say something about the expected return on the option. Suppose XYZ has a positive risk premium (i.e., the expected return on XYZ is greater than the risk-free rate). Since we create the synthetic call by borrowing to buy the stock, the call is equivalent to a leveraged position in the stock, and therefore the call will have an expected return greater than that on the stock. The option elasticity, which we will discuss in Chapter 12, measures the amount of leverage implicit in the option.

The Binomial Solution
In the preceding example, how did we know that buying 2/3 of a share of stock and borrowing $18.462 would replicate a call option?
We have two instruments to use in replicating a call option: shares of stock and a position in bonds (i.e., borrowing or lending). To find the replicating portfolio, we need to find a combination of stock and bonds such that the portfolio mimics the option.
To be specific, we wish to find a portfolio consisting of shares of stock and a dollar amount B in lending, such that the portfolio imitates the option whether the stock rises or falls. We will suppose that the stock has a continuous dividend yield of δ, which we reinvest in the stock. Thus, as in Section 5.2, if you buy one share at time t, at time t + h you will have eδh shares. The up and down movements of the stock price reflect the ex-dividend price. Let S be the stock price today. We can write the stock price as uS when the stock goes up and as dS when the price goes down. The stock price tree is then:

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uS
S
dS

In this tree u is interpreted as one plus the rate of capital gain on the stock if it goes up, and d is one plus the rate of capital loss if it goes down. (If there are dividends, the total return is the capital gain or loss, plus the dividend.)
Let Cu and Cd represent the value of the option when the stock goes up or down, respectively. The tree for the stock implies a corresponding tree for the value of the option:
Cu
C
Cd

If the length of a period is h, the interest factor per period is erh . The problem is to solve for and B such that our portfolio of shares and B in lending duplicates the option payoff.
The value of the replicating portfolio at time h, with stock price Sh, is
Sheδh + erhB
At the prices Sh = dS and Sh = uS, a successful replicating portfolio will satisfy2
(

× dS × eδh) + (B × erh) = Cd

(

× uS × eδh) + (B × erh) = Cu and B. Solving for

This is two equations in the two unknowns

Cu − Cd
S(u − d)

(10.1)

uCd − dCu u−d (10.2)

= e−δh

B = e−rh

and B gives

Note that when there are dividends, the formula adjusts the number of shares in the replicating portfolio, , to offset the dividend income.
Given the expressions for and B, we can derive a simple formula for the value of the option. The cost of creating the option is the net cash required to buy the shares and bonds. Thus, the cost of the option is S + B. Using equations (10.1) and (10.2), we have
S + B = e−rh Cu

e(r−δ)h − d u − e(r−δ)h
+ Cd u−d u−d

(10.3)

2. The term eδh arises in the following equations because the owner of the stock receives a proportional dividend that we assume is reinvested in shares.

10.1 A One-Period Binomial Tree

The assumed stock price movements, u and d, should not give rise to arbitrage opportunities. In particular, we require that u > e(r−δ)h > d

(10.4)

To see why this condition must hold, suppose δ = 0. If the condition were violated, we would short the stock to hold bonds (if erh ≥ u), or we would borrow to buy the stock (if d ≥ erh). Either way, we would earn an arbitrage profit. Therefore the assumed process could not be consistent with any possible equilibrium. Problem 10.23 asks you to verify that equation (10.4) must hold when δ > 0.
Note that because is the number of shares in the replicating portfolio, it can also be interpreted as the sensitivity of the option to a change in the stock price. If the stock price changes by $1, then the option price, S + B , changes by . This interpretation will be quite important later.
Example 10.1 Here is the solution for , B, and the option price using the stock price tree depicted in Figure 10.1. There we have u = $60/$41 = 1.4634, d = $30/$41 = 0.7317, and δ = 0. In addition, the call option had a strike price of $40 and 1 year to expiration— hence, h = 1. Thus Cu = $60 − $40 = $20, and Cd = 0. Using equations (10.1) and (10.2), we have
=

$20 − 0
= 2/3
$41 × (1.4634 − 0.7317)

B = e−0.08

1.4634 × $0 − 0.7317 × $20
= −$18.462
1.4634 − 0.7317

Hence, the option price is given by
S + B = 2/3 × $41 − $18.462 = $8.871
Note that if we are interested only in the option price, it is not necessary to solve for and B; that is just an intermediate step. If we want to know only the option price, we can use equation (10.3) directly:
S + B = e−0.08 $20 ×

1.4634 − e0.08 e0.08 − 0.7317
+ $0 ×
1.4634 − 0.7317
1.4634 − 0.7317

= $8.871
Throughout this chapter we will continue to report not only in the price but also in the replicating portfolio.

and B, since we are interested

Arbitraging a Mispriced Option
What if the observed option price differs from the theoretical price? Because we have a way to replicate the option using the stock, it is possible to take advantage of the mispricing and fulfill the dream of every trader—namely, to buy low and sell high.
The following examples illustrate that if the option price is anything other than the theoretical price, arbitrage is possible.

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The Option is Overpriced. Suppose that the market price for the option is $9.00, instead of $8.871. We can sell the option, but this leaves us with the risk that the stock price at expiration will be $60 and we will be required to deliver the stock.
We can address this risk by buying a synthetic option at the same time we sell the actual option. We have already seen how to create the synthetic option by buying 2/3 shares and borrowing $18.462. If we simultaneously sell the actual option and buy the synthetic, the initial cash flow is


$9.00
Receive option premium

2/3 × $41 + $18.462 = $0.129
Cost of shares

Borrowing

We earn $0.129, the amount by which the option is mispriced.
Now we verify that there is no risk at expiration. We have
Stock Price in 1 Year (S1)
$30
$0
$20
−$20

−$20
$40
−$20

$0

Written call
2/3 Purchased shares
Repay loan of $18.462

$60

$0

Total payoff

By hedging the written option, we eliminate risk.
The Option is Underpriced. Now suppose that the market price of the option is $8.25.
We wish to buy the underpriced option. Of course, if we are unhedged and the stock price falls at expiration, we lose our investment. We can hedge by selling a synthetic option. We accomplish this by reversing the position for a synthetic purchased call: We short 2/3 shares and invest $18.462 of the proceeds in Treasury bills. The cash flow is
−$8.25

+

Option premium

2/3 × $41

− $18.462 = $0.621

Short-sale proceeds

Invest in T-bills

At expiration we have
Stock Price in 1 Year (S1)
$30
Purchased call
2/3 Short-sold shares
Sell T-bill
Total payoff

$60

$0
−$20
$20

$20
−$40
$20

$0

$0

We have earned the amount by which the option was mispriced and hedged the risk associated with buying the option.

A Graphical Interpretation of the Binomial Formula
The binomial solution for and B, equations (10.1) and (10.2), is obtained by solving two equations in two unknowns. Letting Ch and Sh be the option and stock value after one binomial period, and supposing δ = 0, the equations for the portfolio describe a line with

10.1 A One-Period Binomial Tree

FIGURE 10.2
The payoff to an expiring call option is the dark heavy line. The payoff to the option at the points dS and uS are
Cd and Cu (at point D).
The portfolio consisting of shares and B bonds has intercept erhB and slope , and by construction goes through both points E and
D. The slope of the line is calculated as Rise between
Run
points E and D, which gives the formula for .

299

Option
Payoff

D

Cu = uS – K

Rise = Cu – Cd

Slope = Δ
Cd = 0

E dS K

uS

Sh (Stock price after one period)

Intercept
A
= e rhB
Run = uS – dS

the formula
Ch =

× Sh + erhB

This is graphed as line AED in Figure 10.2, which shows the option payoff as a function of the stock price at expiration.
We choose and B to yield a portfolio that pays Cd when Sh = dS and Cu when
Sh = uS. Hence, by construction this line runs through points E and D. We can control the slope of a payoff diagram by varying the number of shares, , and its height by varying the number of bonds, B. It is apparent that a line that runs through both E and D must have slope = (Cu − Cd )/(uS − dS). Also, the point A is the value of the portfolio when
Sh = 0, which is the time-h value of the bond position, erhB. Hence, erhB is the y-axis intercept of the line.
You can see by looking at Figure 10.2 that any line replicating a call will have a positive slope ( > 0) and a negative intercept (B < 0). As an exercise, you can verify graphically that a portfolio replicating a put would have negative slope ( < 0) and positive intercept
(B > 0).

Risk-Neutral Pricing
In the preceding option price calculations, we did not make use of (nor did we discuss) the probability that the stock price would move up or down. The strategy of holding shares and B bonds replicates the option payoff whichever way the stock moves, so the probability of an up or down movement is irrelevant for computing S + B, which is the option price.
Although we do not need probabilities to price the option, there is a very important probabilistic interpretation of equation (10.3). Define p∗ =

e(r−δ)h − d u−d (10.5)

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Chapter 10. Binomial Option Pricing: Basic Concepts

We call p ∗ the risk-neutral probability of an increase in the stock price. You can verify that 0 < p ∗ < 1 (this follows from inequality 10.4), so that p ∗ looks like a probability. Using equation (10.5), and the fact that C = S + B, we can rewrite equation (10.3) as
C = e−rh[p ∗Cu + (1 − p ∗)Cd ]

(10.6)

Because p ∗ and 1 − p ∗ are both positive and sum to one, the term p ∗Cu + (1 − p ∗)Cd looks like an expected cash flow computed using the risk-neutral probability. This expected cash flow is then discounted using the risk-free rate. The pricing procedure illustrated in equation
(10.6), in which a risk-neutral expected value is discounted at the risk-free rate, is called risk-neutral valuation.
We can also use equation (10.6) to compute forward and prepaid forward prices. If we substitute next period’s possible stock prices in place of the option prices in equation
(10.6), after some algebra we obtain e−rh p ∗uS + (1 − p ∗)dS = Se−δh

(10.7)

This is the prepaid forward price. If we do not discount the expected payoff, we obtain the forward price, which is a value denominated in time h dollars:3 p ∗uS + (1 − p ∗)dS = Se(r−δ)h = Ft , t+h

(10.8)

You can think about p ∗ as the probability for which the expected stock price is the forward price. When you first encounter risk-neutral valuation, it appears peculiar. The risk-neutral probability is generally not the actual probability of the stock price going up, and the expected cash flow in equation (10.6) is discounted at the risk-free rate even though the option has the risk of a levered investment in the stock. Nevertheless, risk-neutral valuation is one of the most important ideas in the book. We will discuss risk-neutral valuation more in later chapters.

10.2 CONSTRUCTING A BINOMIAL TREE
In this section we explain the construction of the binomial tree. We will model the stock returns u and d using the equations u = e(r−δ)h+σ d = e(r−δ)h−σ




h h (10.9)

where r is the continuously compounded annual interest rate, δ is the continuous dividend yield, σ is the annual volatility, and h is the length of a binomial period in years.
In all likelihood you do not find equation (10.9) to be immediately intuitive. Thus, in this section we explain a number of the concepts underlying equation (10.9). We first review some properties of continuously compounded returns and define volatility. We then explain the construction of u and d, and then discuss the estimation of historical volatility.

3. We derived this expression for the forward price of the stock in Chapter 5, equation (5.6).

10.2 Constructing a Binomial Tree

Continuously Compounded Returns
What follows in this section and the rest of the book relies on calculations based on continuously compounded returns. As we have emphasized in previous chapters, returns can be expressed in a variety of ways. Continuously compounded returns are mathematically convenient and widely used in practice, both in pricing models and when computing volatility. Here we briefly summarize the important properties of continuously compounded returns. You can also consult Appendix B at the end of this book.
.

The logarithmic function computes continuously compounded returns from prices. Let St and St+h be stock prices at times t and t + h. The continuously compounded return between t and t + h, rt , t+h is then rt , t+h = ln(St+h/St )

.

(10.10)

The exponential function computes prices from continuously compounded returns. If we know the continuously compounded return, rt , t+h, we can obtain St+h by exponentiating both sides of equation (10.10). This gives
St+h = St ert , t+h

.

(10.11)

Continuously compounded returns are additive. Suppose we have continuously compounded returns over consecutive periods—for example, rt , t+h, rt+h, t+2h, etc.
The continuously compounded return over a long period is the sum of continuously compounded returns over the shorter periods, i.e., n rt , t+nh =

rt+(i−1)h, t+ih

(10.12)

i=1

Here are some examples illustrating these statements.
Example 10.2 The stock price on four consecutive days is $100, $103, $97, and $98. The daily continuously compounded returns are ln(103/100) = 0.02956;

ln(97/103) = −0.06002;

ln(98/97) = 0.01026

The continuously compounded return from day 1 to day 4 is ln(98/100) = −0.0202. This is also the sum of the daily continuously compounded returns: r1, 2 + r2, 3 + r3, 4 = 0.02956 + (−0.06002) + 0.01026 = −0.0202
Example 10.3 The stock price is $100 at time 0 and $10 at 1 year later. The percentage return is (10 − 100)/100 = −0.9 = −90%. The continuously compounded return is ln(10/100) = −2.30 = −230%. (A continuously compounded return can be less than
100%.)
Example 10.4 The stock price today is $100. Over the next year the continuously compounded return is −500%. Using equation (10.11), the end-of-year price is S1 =
100e−5.00 = $0.6738. The percentage return is 0.6738/100 − 1 = −99.326%.

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Volatility
The volatility of an asset, defined as the standard deviation of continuously compounded returns, is a key input for any option pricing calculation. We can express volatility over different periods. For example, we could compute monthly volatility (the standard deviation of the monthly return) or annual volatility (the standard deviation of the annual return). How are these related?
Suppose that the continuously compounded return over month i is rmonthly, i . From equation (10.12), we can sum continuously compounded returns. Thus, the annual continuously compounded return is
12

rannual =

rmonthly, i i=1 The variance of the annual continuously compounded return is therefore
12

Var(rannual ) = Var

rmonthly, i

(10.13)

i=1

It is common to assume that returns are uncorrelated over time; i.e., the realization of the return in one period does not affect the expected returns in subsequent periods. With this assumption, the variance of a sum is the sum of the variances. Also suppose that each month has the same variance of returns. If we let σ 2 denote the annual variance, then from equation
(10.13) we have
2
σ 2 = 12 × σmonthly

Taking the square root of both sides and rearranging, we can express the monthly standard deviation in terms of the annual standard deviation, σ : σ σmonthly = √
12
To generalize this formula, if we split the year into n periods of length h (so that h = 1/n), the standard deviation over the period of length h, σh, is4

σh = σ h

(10.14)

The standard deviation thus scales with the square root of time. If we know σh, equation
(10.14) implies that σ σ = √h h (10.15)

4. Equation (10.14) assumes that continuously compounded returns are independent and identically distributed. If returns are not independent, volatility estimation becomes more complicated. Commodity prices, for example, may be mean-reverting: If oil supply is temporarily reduced (e.g., due to political upheaval), the price of oil will increase in the short term but will likely revert due to decreased demand and

increased supply in the future. The volatility over T years will be less than σ T , reflecting the tendency of prices to revert from extreme values. Monte Carlo methods (Chapter 19) would be one way to price an option in such a case.

10.2 Constructing a Binomial Tree

Constructing u and d
As a starting point in constructing u and d, we can ask: What if there were no uncertainty about the future stock price? With certainty, the stock price next period must equal the forward price. Recall from Chapter 5 that the formula for the forward price is
Ft , t+h = St e(r−δ)h

(10.16)

Thus, without uncertainty we must have St+h = Ft , t+h; the rate of return on the stock must be the risk-free rate.5
We incorporate uncertainty into the stock return using volatility, which measures how sure we are that the stock rate of return will be close to the expected rate of return. Stocks with a larger σ will have a greater chance of a return far from the expected return. We model the stock price evolution by adding uncertainty to the forward price: uSt = Ft , t+he+σ dSt = Ft , t+he−σ




h h (10.17)

Using equation (10.16), we can rewrite equation (10.17) to obtain equation (10.9). This is the formula we will use to construct binomial trees. Note that if we set volatility equal to zero (i.e., σ = 0), we will have uSt = dSt = Ft , t+h . Thus, with zero volatility, the price will still rise over time, just as with a Treasury bill. Zero volatility does not mean that prices are fixed; it means that prices are known in advance.
We will refer to a tree constructed using equation (10.9) as a “forward tree.” In Section
11.3 we will discuss alternative ways to construct a tree, including the Cox-Ross-Rubinstein tree. Estimating Historical Volatility
In selecting the parameters to use in the binomial model, the most difficult decision usually is choosing the value for σ , which we cannot observe directly. One possibility is to measure σ by computing the standard deviation of continuously compounded historical returns.
Volatility computed from historical stock returns is historical volatility. We will discuss another important volatility measure, implied volatility, in Chapter 12.
Table 10.1 lists 10 weeks of Wednesday closing prices for the S&P 500 composite index and for IBM, along with the standard deviation of the continuously compounded returns, computed using the StDev function in Excel.6 Based on the historical returns in
Table 10.1, the weekly standard deviation of returns was 0.02800 and 0.02486 for the
S&P 500 index and IBM, respectively. These standard deviations measure the variability in weekly returns. We compute annualized standard deviations by using equation (10.15):


Multiply the weekly standard deviations by 52 (because h = 1/ 52), giving annualized

5. With the forward price given by equation (10.16), the total return on the stock would be the stock price increase, which is at the rate r − δ, plus the dividend yield, δ.
6. We use weekly rather than daily data because computing daily statistics is complicated by weekends and holidays. In theory the standard deviation over the 3 days from Friday to Monday should be greater than over the 1 day from Monday to Tuesday. Using weekly data avoids this kind of complication. Further, using Wednesdays avoids most holidays.

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TABLE 10.1

Weekly prices and continuously compounded returns for the S&P 500 index and IBM, from 7/7/2010 to 9/8/2010.

S&P 500
Date
7/7/2010
7/14/2010
7/21/2010
7/28/2010
8/4/2010
8/11/2010
8/18/2010
8/25/2010
9/1/2010
9/8/2010
Standard deviation

Standard deviation × 52

Price
1060.27
1095.17
1069.59
1106.13
1127.24
1089.47
1094.16
1055.33
1080.29
1098.87
0.02800
0.20194

ln(St /St−1)
0.03239
−0.02363
0.03359
0.01890
−0.03408
0.00430
−0.03613
0.02338
0.01705

IBM
Price
127
130.72
125.27
128.43
131.27
129.83
129.39
125.27
125.77
126.08

ln(St /St−1)
0.02887
−0.04259
0.02491
0.02187
−0.01103
−0.00338
−0.03238
0.00398
0.00246

0.02486
0.17926

historical standard deviations of 20.19% for the S&P 500 index and 17.93% for IBM. We can use the estimated annualized standard deviation as σ in constructing a binomial tree.
You should not be misled by the fact that the standard deviations were estimated with

weekly data. Once we annualize the estimate we can then multiply the result by h (as in equation (10.9)) to obtain the appropriate standard deviation for any size binomial step.
You might be wondering about how dividends affect the standard deviation calculation. The returns in Table 10.1 are based on ex-dividend prices, in particular ignoring IBM’s payment of $0.65 with an ex-dividend date of August 6, 2010. Including the dividend in the return calculation in this example changes the estimated annual standard deviation to
0.1778. Most of the time dividends are infrequent and small; the standard deviation will be similar whether you compute a standard deviation accounting for dividends or ignoring them. The more important question is whether the standard deviation calculation should be based on returns that include dividends (the total return volatility). For option pricing it is generally the volatility of the price excluding dividends that matters. For a European option, the payoff clearly depends on the ex-dividend price, so the volatility calculation should exclude dividends. The same is true for American options: If an American call is not exercised before expiration, the payoff depends on the ex-dividend price. If it is exercised before expiration, the option holder exercises prior to a dividend. American puts are exercised ex-dividend. Thus, for standard options, the volatility that matters excludes dividends.
For an option protected against dividends, it would be appropriate to base pricing upon the total return volatility, which includes dividends.

10.2 Constructing a Binomial Tree

FIGURE 10.3
Binomial tree for pricing a European call option; assumes S = $41.00, K = $40.00, σ = 0.30, r =
0.08, T = 1.00 years, δ = 0.00, and h = 1.000. At each node the stock price, option price, , and B are given. Option price in bold italic signifies that exercise is optimal at that node.
$59.954
$19.954
$41.000
$7.839
Δ = 0.738
B = –$22.405
$32.903
$0.000

One-Period Example with a Forward Tree
We began this section by assuming that the stock price followed the binomial tree in Figure
10.1. The up and down stock prices of $30 and $60 were selected to make the example easy to follow. Now we present an example where everything is the same except that we use equation (10.9) to construct the up and down moves.

Suppose volatility is 30%. Since the period is 1 year, we have h = 1, so that σ h =
0.30. We also have S0 = $41, r = 0.08, and δ = 0. Using equation (10.9), we get

1

uS = $41e(0.08−0)×1+0.3×

= $59.954

dS = $41e

= $32.903


(0.08−0)×1−0.3× 1

(10.18)

Because the binomial tree is different than in Figure 10.1, the option price will be different as well.
Using the stock prices given in equation (10.18), we have u = $59.954/$41 = 1.4623 and d = $32.903/$41 = 0.8025. With K = $40, we have Cu = $59.954 − $40 = $19.954, and Cd = 0. Using equations (10.1) and (10.2), we obtain
=

$19.954 − 0
= 0.7376
$41 × (1.4623 − 0.8025)

B = e−0.08

1.4623 × $0 − 0.8025 × $19.954
= −$22.405
1.4623 − 0.8025

Hence, the option price is given by
S + B = 0.7376 × $41 − $22.405 = $7.839
This example is summarized in Figure 10.3.
We have covered a great deal of ground in this section, but there are still many issues remaining. The simple binomial tree seems too simple to provide an accurate option price.
Unanswered questions include how to handle more than one binomial period, how to price put options, and how to price American options. With the basic binomial formula in hand, we can now turn to those questions.

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FIGURE 10.4
Binomial tree for pricing a European call option; assumes S = $41.00, K =
$40.00, σ = 0.30, r = 0.08,
T = 2.00 years, δ = 0.00, and h = 1.000. At each node the stock price, option price,
, and B are given. Option prices in bold italic signify that exercise is optimal at that node.

$87.669 (Suu )
$47.669
$59.954 (Su )
$23.029
Δ = 1.000
B = –$36.925
$41.000 (S )
$10.737
Δ = 0.734
B = –$19.337

$48.114 (Sud = Sdu )
$8.114
$32.903 (Sd )
$3.187
Δ = 0.374
B = –$9.111
$26.405 (Sdd )
$0.000

10.3 TWO OR MORE BINOMIAL PERIODS
We now see how to extend the binomial tree to more than one period. We begin by pricing a 2-year option using a two-period binomial model. Then we will see how to accommodate many periods of arbitrary length.

A Two-Period European Call
We begin first by adding a single period to the tree in Figure 10.3; the result is displayed in
Figure 10.4. We can use that tree to price a 2-year option with a $40 strike when the current stock price is $41, assuming all inputs are the same as before.
Since we are increasing the time to maturity for a call option on a non-dividendpaying stock, then based on the discussion in Section 9.3 we expect the option premium to increase. In this example the two-period tree will give us a price of $10.737, compared to
$7.839 in Figure 10.3.
Constructing the Tree. To see how to construct the tree, suppose that we move up in year
1, to Su = $59.954. If we reach this price, then we can move further up or down according to equation (10.17). We get
Suu = $59.954e0.08+0.3 = $87.669 and Sud = $59.954e0.08−0.3 = $48.114
The subscript uu means that the stock has gone up twice in a row and the subscript ud means that the stock has gone up once and then down.
Similarly, if the price in 1 year is Sd = $32.903, we have
Sdu = $32.903e0.08+0.3 = $48.114

10.3 Two or More Binomial Periods

and
Sdd = $32.903e0.08−0.3 = $26.405
Note that an up move followed by a down move (Sud ) generates the same stock price as a down move followed by an up move (Sdu). This is called a recombining tree. If an up move followed by a down move led to a different price than a down move followed by an up move, we would have a nonrecombining tree.7 A recombining tree has fewer nodes, which means less computation is required to compute an option price. We will see examples of nonrecombining trees in Sections 11.4 and 25.4.
We also could have used equation (10.9) directly to compute the year-2 stock prices.
Recall that u = e0.08+0.3 = 1.462 and d = e0.08−0.3 = 0.803. We have
Sud

Suu = u2 × $41 = e2×(0.08+0.3) × $41 = $87.669
= Sdu = u × d × $41 = e(0.08+0.3) × e(0.08−0.3) × $41 = $48.114
Sdd = d 2 × $41 = e2×(0.08−0.3) × $41 = $26.405

Pricing the Call Option. How do we price the option when we have two binomial periods?
The key insight is that we work backward through the binomial tree. In order to use equation (10.3), we need to know the option prices resulting from up and down moves in the subsequent period. At the outset, the only period where we know the option price is at expiration.
Knowing the price at expiration, we can determine the price in period 1. Having determined that price, we can work back to period 0.
Figure 10.4 exhibits the option price at each node as well as the details of the replicating portfolio at each node. Remember, however, when we use equation (10.3), it is not necessary to compute and B in order to derive the option price.8 Here are details of the solution:
Year 2, Stock Price = $87.669. Since we are at expiration, the option value is max(0, S − K) = $47.669.
Year 2, Stock Price = $48.114. Again we are at expiration, so the option value is
$8.114.
Year 2, Stock Price = $26.405. Since the option is out of the money, the value is 0.
Year 1, Stock Price = $59.954. At this node we use equation (10.3) to compute the option value. (Note that once we are at this node, the “up” stock price, uS, is $87.669, and the “down” stock price, dS, is $48.114.) e−0.08 $47.669 ×

1.462 − e0.08 e0.08 − 0.803
+ $8.114 ×
= $23.029
1.462 − 0.803
1.462 − 0.803

Year 1, Stock Price = $32.903. Again we use equation (10.3) to compute the option value: 7. In cases where the tree recombines, the representation of stock price movements is also (and, some argue, more properly) called a lattice. The term tree would then be reserved for nonrecombining stock movements. 8. As an exercise, you can verify the

and B at each node.

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e−0.08 $8.114 ×

e0.08 − 0.803
1.462 − e0.08
= $3.187
+ $0 ×
1.462 − 0.803
1.462 − 0.803

Year 0, Stock Price = $41. Again using equation (10.3): e−0.08 $23.029 ×

e0.08 − 0.803
1.462 − e0.08
+ $3.187 ×
= $10.737
1.462 − 0.803
1.462 − 0.803

Notice the following:
.

.

.

.

.

The option price is greater for the 2-year than for the 1-year option, as we would expect. We priced the option by working backward through the tree, starting at the end and working back to the first period.
The option’s and B are different at different nodes. In particular, at a given point in time, increases to 1 as we go further into the money.
We priced a European option, so early exercise was not permitted. However, permitting early exercise would have made no difference. At every node prior to expiration, the option price is greater than S − K; hence we would not have exercised even if the option had been American.
Once we understand the two-period option, it is straightforward to value an option using more than two binomial periods. The important principle is to work backward through the tree.

Many Binomial Periods
An obvious objection to the binomial calculations thus far is that the stock can only have two or three different values at expiration. It seems plausible that to increase accuracy we would want to divide the time to expiration into more periods, generating a more realistic tree. Fortunately, the generalization to many binomial periods is straightforward.
To illustrate using more binomial periods, we re-examine the 1-year European call option in Figure 10.3, which has a $40 strike and initial stock price of $41. We use equation
(10.9) to generate the up and down moves. Suppose there are three binomial periods. With a
1
1-year call, the length of a period is h = 3 . We will assume that other inputs stay the same, so r = 0.08 and σ = 0.3. Equation (10.9) then automatically generates a per-period interest

rate of rh = 0.027 and volatility of σ h = 0.1732.
Figure 10.5 depicts the stock price and option price tree for this option. The option price is $7.074, as opposed to $7.839 in Figure 10.3. The difference occurs because the numerical approximation is different; it is quite common to see large changes in a binomial price when the number of periods, n, is changed, particularly when n is small.
Since the length of the binomial period is shorter, u and d are closer to 1 than before
(1.2212 and 0.8637 as opposed to 1.462 and 0.803 with h = 1). Just to be clear about the procedure, here is how the second-period nodes are computed:

Su = $41e0.08×1/3+0.3 1/3 = $50.071

Sd = $41e0.08×1/3−0.3 1/3 = $35.411
The remaining nodes are computed similarly.

10.4 Put Options

FIGURE 10.5
Binomial tree for pricing a European call option; assumes S = $41.00, K =
$40.00, σ = 0.30, r = 0.08,
T = 1.00 years, δ = 0.00, and h = 0.333. At each node the stock price, option price,
, and B are given. Option prices in bold italic signify that exercise is optimal at that node.

$74.678
$34.678
$61.149
$22.202
Δ = 1.000
B = –$38.947
$50.071
$12.889
Δ = 0.922
B = –$33.264
$41.000
$7.074
Δ = 0.706
B = –$21.885

$52.814
$12.814
$43.246
$5.700
Δ = 0.829
B = –$30.139

$35.411
$2.535
Δ = 0.450
B = –$13.405

$37.351
$0.000

$30.585
$0.000
Δ = 0.000
B = $0.000
$26.416
$0.000

The option price is computed by working backward. The risk-neutral probability of the stock price going up in a period is e0.08×1/3 − 0.8637
= 0.4568
1.2212 − 0.8637
The option price at the node where S = $43.246, for example, is then given by e−0.08×1/3 [$12.814 × 0.4568] + [$0 × (1 − 0.4568)] = $5.700
Option prices at the remaining nodes are priced similarly.

10.4 PUT OPTIONS
Thus far we have priced only call options. The binomial method easily accommodates put options also, as well as other derivatives. We compute put option prices using the same stock price tree and in almost the same way as call option prices; the only difference with a
European put option occurs at expiration: Instead of computing the price as max(0, S − K), we use max(0, K − S).
Figure 10.6 shows the binomial tree for a European put option with 1 year to expiration and a strike of $40 when the stock price is $41. This is the same stock price tree as in
Figure 10.5.

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FIGURE 10.6
Binomial tree for pricing a European put option; assumes S = $41.00, K =
$40.00, σ = 0.30, r = 0.08,
T = 1.00 years, δ = 0.00, and h = 0.333. At each node the stock price, option price,
, and B are given. Option prices in bold italic signify that exercise is optimal at that node.

$74.678
$0.000
$61.149
$0.000
Δ = 0.000
B = $0.000
$50.071
$0.741
Δ = –0.078
B = $4.659
$41.000
$2.999
Δ = –0.294
B = $15.039

$52.814
$0.000
$43.246
$1.401
Δ = –0.171
B = $8.809

$35.411
$5.046
Δ = –0.550
B = $24.517

$37.351
$2.649

$30.585
$8.363
Δ = –1.000
B = $38.947
$26.416
$13.584

To illustrate the calculations, consider the option price at the node where the stock price is $35.411. The option price at that node is computed as e−0.08×1/3 $1.401 ×

1.2212 − e0.08×1/3 e0.08×1/3 − 0.8637
+ $8.363 ×
1.2212 − 0.8637
1.2212 − 0.8637

= $5.046

Figure 10.6 does raise one issue that we have not previously had to consider. Notice that at the node where the stock price is $30.585, the option price is $8.363. If this option were American, it would make sense to exercise at that node. The option is worth $8.363 when held until expiration, but it would be worth $40 − $30.585 = $9.415 if exercised at that node. Thus, in this case the American option should be more valuable than the otherwise equivalent European option. We will now see how to use the binomial approach to value
American options.

10.5 AMERICAN OPTIONS
Since it is easy to check at each node whether early exercise is optimal, the binomial method is well-suited to valuing American options. The value of the option if it is left “alive” (i.e., unexercised) is given by the value of holding it for another period, equation (10.3). The value of the option if it is exercised is given by max(0, S − K) if it is a call and max(0, K − S) if it is a put.

10.5 American Options

FIGURE 10.7
Binomial tree for pricing an American put option; assumes S = $41.00, K =
$40.00, σ = 0.30, r = 0.08,
T = 1.00 years, δ = 0.00, and h = 0.333. At each node the stock price, option price,
, and B are given. Option prices in bold italic signify that exercise is optimal at that node.

$74.678
$0.000
$61.149
$0.000
Δ = 0.000
B = $0.000
$50.071
$0.741
Δ = –0.078
B = $4.659
$41.000
$3.293
Δ = –0.332
B = $16.891

$52.814
$0.000
$43.246
$1.401
Δ = –0.171
B = $8.809

$35.411
$5.603
Δ = –0.633
B = $28.018

$37.351
$2.649

$30.585
$9.415
Δ = –1.000
B = $38.947
$26.416
$13.584

Thus, for an American put, the value of the option at a node is given by
P (S , K , t) = max K − S , e−rh P (uS , K , t + h)p ∗ + P (dS , K , t + h)(1 − p ∗)

(10.19)

where, p ∗ is given by equation (10.5), p∗ =

e(r−δ)h − d u−d Figure 10.7 presents the binomial tree for the American version of the put option valued in Figure 10.6. The only difference in the trees occurs at the node where the stock price is
$30.585. The American option at that point is worth $9.415, its early-exercise value.9 We have just seen in the previous section that the value of the option if unexercised is $8.363.
The greater value of the option at that node ripples back through the tree. When the option price is computed at the node where the stock price is $35.411, the value is greater

9. In order to determine whether to exercise the option at the 30.585 node, it is necessary to compute the price of the option if it were and were not exercised, and then compare the two values. The values of and B reported for the 30.585 node refer to the values for the unexercised option.

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in Figure 10.7 than in Figure 10.6; the reason is that the price is greater at the subsequent node Sdd due to early exercise.
The initial option price is $3.293, greater than the value of $2.999 for the European option. This increase in value is due entirely to early exercise at the Sdd node.
In general the valuation of American options proceeds as in this example. At each node we check for early exercise. If the value of the option is greater when exercised, we assign the exercised value to the node. Otherwise, we assign the value of the option unexercised.
We work backward through the tree as usual.

10.6 OPTIONS ON OTHER ASSETS
The binomial model can be modified easily to price options on underlying assets other than non-dividend-paying stocks. In this section we present examples of options on stock indexes, currencies, and futures contracts. In every case the general procedure is the same:
We compute the option price using equation (10.6). The difference for different underlying assets will be the construction of the binomial tree and the risk-neutral probability.
The valuation of an option on a stock that pays discrete dividends is more complex and is covered in Chapter 11.

Option on a Stock Index
Suppose a stock index pays continuous dividends at the rate δ. This type of option has in fact already been covered by our derivation in Section 10.1. The up and down index moves are given by equation (10.9), the replicating portfolio by equations (10.1) and (10.2), and the option price by equation (10.3). The risk-neutral probability is given by equation (10.5).10
Figure 10.8 displays a binomial tree for an American call option on a stock index.
Note that because of dividends, early exercise is optimal at the node where the stock price is $157.101. Given these parameters, we have p ∗ = 0.457; hence, when S = $157.101, the value of the option unexercised is e−0.05×1/3 0.457 × $87.747 + (1 − 0.457) × $32.779 = $56.942
Since 57.101 > 56.942, we exercise the option at that node.

Options on Currencies
With a currency with spot price x0, the forward price is F0, h = x0e(r−rf )h, where rf is the foreign interest rate. Thus, we construct the binomial tree using ux = xe(r−rf )h+σ dx = xe(r−rf )h−σ




h h 10. Intuitively, dividends can be taken into account either by (1) appropriately lowering the nodes on the tree and leaving risk-neutral probabilities unchanged, or (2) by reducing the risk-neutral probability and leaving the tree unchanged. The forward tree adopts the first approach.

10.6 Options on Other Assets

FIGURE 10.8
Binomial tree for pricing an American call option on a stock index; assumes S
= $110.00, K = $100.00, σ = 0.30, r = 0.05, T =
1.00 years, δ = 0.035, and h = 0.333. At each node the stock price, option price,
, and B are given. Option prices in bold italic signify that exercise is optimal at that node.

$187.747
$87.747
$157.101
$57.101
Δ = 0.988
B = –$98.347
$131.458
$33.520
Δ = 0.911
B = –$86.185
$110.000
$18.593
Δ = 0.691
B = –$57.408

$132.779
$32.779
$111.106
$14.726
Δ = 0.833
B = –$77.871

$92.970
$6.616
Δ = 0.447
B = –$34.984

$93.904
$0.000

$78.576
$0.000
Δ = 0.000
B = $0.000
$66.411
$0.000

There is one subtlety in creating the replicating portfolio: Investing in a “currency” means investing in a money-market fund or fixed-income obligation denominated in that currency.
(We encountered this idea previously in Chapter 5.) Taking into account interest on the foreign-currency-denominated obligation, the two equations are
× dxerf h + erh × B = Cd
× uxerf h + erh × B = Cu
The risk-neutral probability of an up move in this case is given by p∗ =

e(r−rf )h − d u−d (10.20)

Notice that if we think of rf as the dividend yield on the foreign currency, these two equations look exactly like those for an index option. In fact, the solution is the same as for an option on an index: Set the dividend yield equal to the foreign risk-free rate and the current value of the index equal to the spot exchange rate.
Figure 10.9 prices a dollar-denominated American put option on the euro. The current
C
C exchange rate is assumed to be $1.05/= and the strike is $1.10/= . The euro-denominated interest rate is 3.1%, and the dollar-denominated rate is 5.5%.

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FIGURE 10.9
Binomial tree for pricing an American put option on a currency; assumes S =
$1.05/= , K = $1.10, σ =
C
0.10, r = 0.055, T = 0.50 years, δ = 0.031, and h =
0.167. At each node the stock price, option price,
, and B are given. Option prices in bold italic signify that exercise is optimal at that node.

$1.201
$0.000
$1.148
$0.000
Δ = 0.000
B = $0.000
$1.098
$0.021
Δ = –0.459
B = $0.525
$1.050
$0.055
Δ = –0.774
B = $0.867

$1.107
$0.000
$1.058
$0.042
Δ = –0.915
B = $1.009

$1.012
$0.088
Δ = –0.995
B = $1.090

$1.020
$0.080

$0.975
$0.125
Δ = –0.995
B = $1.090
$0.940
$0.160

Because volatility is low and the option is in-the-money, early exercise is optimal at three nodes prior to expiration.

Options on Futures Contracts
We now consider options on futures contracts. We assume the forward price is the same as the futures price. Since we build the tree based on the forward price, we simply add up and down movements around the current price. Thus, the nodes are constructed as

h

−σ h

u = eσ d =e

Note that this solution for u and d is exactly what we would get for an option on a stock index if δ, the dividend yield, were equal to the risk-free rate.
In constructing the replicating portfolio, recall that in each period a futures contract pays the change in the futures price, and there is no investment required to enter a futures contract. The problem is to find the number of futures contracts, , and the lending, B , that replicates the option. We have
× (dF − F ) + erh × B = Cd
× (uF − F ) + erh × B = Cu

10.6 Options on Other Assets

Solving gives11
=

Cu − Cd
F (u − d)

B = e−rh Cu

1− d u−1 +Cd u−d u−d

While tells us how many futures contracts to hold to hedge the option, the value of the option in this case is simply B. The reason is that the futures contract requires no investment, so the only investment is that made in the bond. We can again price the option using equation
(10.3).
The risk-neutral probability of an up move is given by p∗ =

1− d u−d (10.21)

Figure 10.10 shows a tree for pricing an American call option on a gold futures contract.
Early exercise is optimal when the price is $336.720. The intuition for early exercise is that when an option on a futures contract is exercised, the option holder pays nothing, is entered into a futures contract, and receives mark-to-market proceeds of the difference between the strike price and the futures price. The motive for exercise is the ability to earn interest on the mark-to-market proceeds.

Options on Commodities
Many options exist on commodity futures contracts. However, it is also possible to have options on the physical commodity. If there is a market for lending and borrowing the commodity, then, in theory, pricing such an option is straightforward.
Recall from Chapter 6 that the lease rate for a commodity is conceptually similar to a dividend yield. If you borrow the commodity, you pay the lease rate. If you buy the commodity and lend it, you receive the lease rate. Thus, from the perspective of someone synthetically creating the option, the commodity is like a stock index, with the lease rate equal to the dividend yield.

11. The interpretation of here is the number of futures contracts in the replicating portfolio. Another interpretation of is the price sensitivity of the option when the price of the underlying asset changes.
These two interpretations usually coincide, but not in the case of options on futures. The reason is that the futures price at time t reflects a price denominated in future dollars. The effect on the option price of a futures price change today is given by e−rh . To see this, consider an option that is one binomial period from expiration and for which uF > dF > K . Then
=

uF − K − (dF − K)
=1
F (u − d)

But we also have
B = e−rh (uF − K)

1− d u−1 + (dF − K) u−d u−d

= e−rh (F − K)
From the second expression, you can see that if the futures price changes by $1, the option price changes by e−rh .

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FIGURE 10.10
Binomial tree for pricing an
American call option on a futures contract; assumes S
= $300.00, K = $300.00, σ = 0.10, r = 0.05, T =
1.00 years, δ = 0.05, and h
= 0.333. At each node the stock price, option price,
, and B are given. Option prices in bold italic signify that exercise is optimal at that node.

$356.733
$56.733
$336.720
$36.720
Δ = 1.000
B = $36.720
$317.830
$21.843
Δ = 0.768
B = $21.843
$300.000
$12.488
Δ = 0.513
B = $12.488

$317.830
$17.830
$300.000
$8.515
Δ = 0.514
B = $8.515

$283.170
$4.066
Δ = 0.260
B = $4.066

$283.170
$0.000

$267.284
$0.000
Δ = 0.000
B = $0.000
$252.290
$0.000

Because this is conceptually the same as the pricing exercise in Figure 10.8 (imagine a commodity with a price of $110, a lease rate of 3.5%, and a volatility of 30%), we do not present a pricing example.
In practice, pricing and hedging an option based on the physical commodity can be problematic. If an appropriate futures contract exists, a market-maker could use it to hedge a commodity option. Otherwise, transactions in physical commodities often have greater transaction costs than for financial assets. Short-selling a commodity may not be possible, for reasons discussed in Chapter 6. Market-making is then difficult.

Options on Bonds
Finally, we will briefly discuss options on bonds. We devote a separate chapter later to discussing fixed-income derivatives, but it is useful to understand at this point some of the issues in pricing options on bonds. As a first approximation we could just say that bonds are like stocks that pay a discrete dividend (a coupon), and price bond options using the binomial model.
However, bonds differ from the assets we have been discussing in two important respects. 1. The volatility of a bond decreases over time as the bond approaches maturity. The prices of 30-day Treasury bills, for example, are much less volatile than the prices of

10.6 Options on Other Assets

30-year Treasury bonds. The reason is that a given change in the interest rate, other things equal, changes the price of a shorter-lived bond by less.
2. We have been assuming in all our calculations that interest rates are the same for all maturities, do not change over time, and are not random. While these assumptions may be good enough for pricing options on stocks, they are logically inconsistent for pricing options on bonds: If interest rates do not change unexpectedly, neither do bond prices.
In some cases, it may be reasonable to price bond options using the simple binomial model in this chapter. For example, consider a 6-month option on a 29-year bond. The underlying asset in this case is a 29.5-year bond. As a practical matter, the volatility difference between a 29.5- and a 29-year bond is likely to be very small. Also, because it is short-lived, this option will not be particularly sensitive to the short-term interest rate, so the correlation of the bond price and the 6-month interest rate will not matter much.
On the other hand, if we have a 3-year option to buy a 5-year bond, these issues might be quite important. Another issue is that bond coupon payments are discrete, so the assumption of a continuous dividend is an approximation.
In general, the conceptual and practical issues with bonds are different enough that bonds warrant a separate treatment. We will return to bonds in Chapter 25.

Summary
Here is the general procedure covering the other assets discussed in this section.
.

Construct the binomial tree for the price of the underlying asset using

h

or

u=


−σ h

or

d=

uSt = Ft , t+he+σ dSt = Ft , t+he

Ft , t+h
St
Ft , t+h
St

e+σ e−σ √ h √

(10.22) h Since different underlying assets will have different forward price formulas, the tree will be different for different underlying assets.
.

The option price at each node, if the option is unexercised, can then be computed as p∗ =
=

Ft , t+h/St − d u−d e(r−δ)h

−d u−d (10.23)

and, as before,
C = e−rh p ∗Cu + (1 − p ∗)Cd

(10.24)

where Cu and Cd are the up and down nodes relative to the current node. For an
American option, at each node take the greater of this value and the value if exercised.
Pricing options with different underlying assets requires adjusting the risk-neutral probability for the borrowing cost or lease rate of the underlying asset. Mechanically, this

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TABLE 10.2
Underlying Asset
Stock index
Currency
Futures contract
Commodity
Coupon bond

Substitutions for pricing options on assets other than a stock index. Interest Rate
Domestic risk-free rate
Domestic risk-free rate
Domestic risk-free rate
Domestic risk-free rate
Domestic risk-free rate

Dividend Yield
Dividend yield
Foreign risk-free rate
Domestic risk-free rate
Commodity lease rate
Yield on bond

means that we can use the formula for pricing an option on a stock index with an appropriate substitution for the dividend yield. Table 10.2 summarizes the substitutions.

CHAPTER SUMMARY
In order to price options, we must make an assumption about the probability distribution of the underlying asset. The binomial distribution provides a particularly simple stock price distribution: At any point in time, the stock price can go from S up to uS or down to dS, where the movement factors u and d are given by equation (10.9).
Given binomial stock price movements, the option can be replicated by holding shares of stock and B bonds. The option price is the cost of this replicating portfolio,
S + B. For a call option, > 0 and B < 0, so the option is replicated by borrowing to buy shares. For a put, < 0 and B > 0. If the option price does not equal this theoretical price, arbitrage is possible. The replicating portfolio is dynamic, changing as the stock price moves up or down. Thus it is unlike the replicating portfolio for a forward contract, which is fixed.
The binomial option pricing formula has an interpretation as a discounted expected value, with the risk-neutral probability (equation (10.5)) used to compute the expected payoff to the option and the risk-free rate used to discount the expected payoff. This is known as risk-neutral pricing.
The up and down binomial asset price movements depend on the asset volatility, which is the annualized standard deviation of the continuously compounded return on the asset. We construct the up and down movements as the forward price multiplied by a term depending on plus or minus the per-period volatility. Historical volatility can be measured using past returns. Option prices, however, should depend on prospective volatility.
The binomial model can be used to price American and European calls and puts on a variety of underlying assets, including stocks, indexes, futures, currencies, commodities, and bonds.

Problems

FURTHER READING
This chapter has focused on the mechanics of binomial option pricing. Some of the underlying concepts will be discussed in more detail in Chapter 11. There we will have more to say about risk-neutral pricing, the link between the binomial tree and the assumed stock price distribution, and how to price options when the stock pays a discrete dividend.
The binomial model provides a foundation for much of what we will do in later chapters. We will see in Chapter 23, for example, that the binomial option pricing formula gives results equivalent to the Black-Scholes formula when h becomes small. Consequently, if you thoroughly understand binomial pricing, you also understand the Black-Scholes formula. In Chapter 23, we will see how to generalize binomial trees to handle two sources of uncertainty.
In addition to the original papers by Cox et al. (1979) and Rendleman and Bartter
(1979), Cox and Rubinstein (1985) provides an excellent exposition of the binomial model.

PROBLEMS
In these problems, n refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
10.1 Let S = $100, K = $105, r = 8%, T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and n =
1.
a. What are the premium,

, and B for a European call?

b. What are the premium,

, and B for a European put?

10.2 Let S = $100, K = $95, r = 8%, T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and n = 1.
a. Verify that the price of a European call is $16.196.
b. Suppose you observe a call price of $17. What is the arbitrage?
c. Suppose you observe a call price of $15.50. What is the arbitrage?
10.3 Let S = $100, K = $95, r = 8%, T = 0.5, and δ = 0. Let u = 1.3, d = 0.8, and n = 1.
a. Verify that the price of a European put is $7.471.
b. Suppose you observe a put price of $8. What is the arbitrage?
c. Suppose you observe a put price of $6. What is the arbitrage?
10.4 Obtain at least 5 years’ worth of daily or weekly stock price data for a stock of your choice. 1. Compute annual volatility using all the data.
2. Compute annual volatility for each calendar year in your data. How does volatility vary over time?
3. Compute annual volatility for the first and second half of each year in your data. How much variation is there in your estimate?
10.5 Obtain at least 5 years of daily data for at least three stocks and, if you can, one currency. Estimate annual volatility for each year for each asset in your data. What do you observe about the pattern of historical volatility over time? Does historical volatility move in tandem for different assets?

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10.6 Let S = $100, K = $95, σ = 30%, r = 8%, T = 1, and δ = 0. Let u = 1.3, d = 0.8, and n = 2. Construct the binomial tree for a call option. At each node provide the premium, , and B.
10.7 Repeat the option price calculation in the previous question for stock prices of $80,
$90, $110, $120, and $130, keeping everything else fixed. What happens to the initial option as the stock price increases?
10.8 Let S = $100, K = $95, σ = 30%, r = 8%, T = 1, and δ = 0. Let u = 1.3, d = 0.8, and n = 2. Construct the binomial tree for a European put option. At each node provide the premium, , and B.
10.9 Repeat the option price calculation in the previous question for stock prices of $80,
$90, $110, $120, and $130, keeping everything else fixed. What happens to the initial put as the stock price increases?
10.10 Let S = $100, K = $95, σ = 30%, r = 8%, T = 1, and δ = 0. Let u = 1.3, d = 0.8, and n = 2. Construct the binomial tree for an American put option. At each node provide the premium, , and B.
10.11 Suppose S0 = $100, K = $50, r = 7.696% (continuously compounded), δ = 0, and T = 1.
a. Suppose that for h = 1, we have u = 1.2 and d = 1.05. What is the binomial option price for a call option that lives one period? Is there any problem with having d > 1?
b. Suppose now that u = 1.4 and d = 0.6. Before computing the option price, what is your guess about how it will change from your previous answer? Does it change? How do you account for the result? Interpret your answer using put-call parity.
c. Now let u = 1.4 and d = 0.4. How do you think the call option price will change from (a)? How does it change? How do you account for this? Use put-call parity to explain your answer.
10.12 Let S = $100, K = $95, r = 8% (continuously compounded), σ = 30%, δ = 0,
T = 1 year, and n = 3.
a. Verify that the binomial option price for an American call option is $18.283.
Verify that there is never early exercise; hence, a European call would have the same price.
b. Show that the binomial option price for a European put option is $5.979.
Verify that put-call parity is satisfied.
c. Verify that the price of an American put is $6.678.
10.13 Repeat the previous problem assuming that the stock pays a continuous dividend of
8% per year (continuously compounded). Calculate the prices of the American and
European puts and calls. Which options are early-exercised?
10.14 Let S = $40, K = $40, r = 8% (continuously compounded), σ = 30%, δ = 0, T =
0.5 year, and n = 2.
a. Construct the binomial tree for the stock. What are u and d?

Problems

b. Show that the call price is $4.110.
c. Compute the prices of American and European puts.
10.15 Use the same data as in the previous problem, only suppose that the call price is $5 instead of $4.110.
a. At time 0, assume you write the option and form the replicating portfolio to offset the written option. What is the replicating portfolio and what are the net cash flows from selling the overpriced call and buying the synthetic equivalent? b. What are the cash flows in the next binomial period (3 months later) if the call at that time is fairly priced and you liquidate the position? What would you do if the option continues to be overpriced the next period?
c. What would you do if the option is underpriced the next period?
C
10.16 Suppose that the exchange rate is $0.92/= . Let r$ = 4%, and r= = 3%, u = 1.2,
C
d = 0.9, T = 0.75, n = 3, and K = $0.85.
a. What is the price of a 9-month European call?
b. What is the price of a 9-month American call?
10.17 Use the same inputs as in the previous problem, except that K = $1.00.
a. What is the price of a 9-month European put?
b. What is the price of a 9-month American put?
10.18 Suppose that the exchange rate is 1 dollar for 120 yen. The dollar interest rate is 5%
(continuously compounded) and the yen rate is 1% (continuously compounded).
Consider an at-the-money American dollar call that is yen-denominated (i.e., the call permits you to buy 1 dollar for 120 yen). The option has 1 year to expiration and the exchange rate volatility is 10%. Let n = 3.
a. What is the price of a European call? An American call?
b. What is the price of a European put? An American put?
c. How do you account for the pattern of early exercise across the two options?
10.19 An option has a gold futures contract as the underlying asset. The current 1-year gold futures price is $300/oz, the strike price is $290, the risk-free rate is 6%, volatility is 10%, and time to expiration is 1 year. Suppose n = 1. What is the price of a call option on gold? What is the replicating portfolio for the call option? Evaluate the statement: “Replicating a call option always entails borrowing to buy the underlying asset.” 10.20 Suppose the S&P 500 futures price is 1000, σ = 30%, r = 5%, δ = 5%, T = 1, and n = 3.
a. What are the prices of European calls and puts for K = $1000? Why do you find the prices to be equal?
b. What are the prices of American calls and puts for K = $1000?
c. What are the time-0 replicating portfolios for the European call and put?

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10.21 For a stock index, S = $100, σ = 30%, r = 5%, δ = 3%, and T = 3. Let n = 3.
a. What is the price of a European call option with a strike of $95?
b. What is the price of a European put option with a strike of $95?
c. Now let S = $95, K = $100, σ = 30%, r = 3%, and δ = 5%. (You have exchanged values for the stock price and strike price and for the interest rate and dividend yield.) Value both options again. What do you notice?
10.22 Repeat the previous problem calculating prices for American options instead of
European. What happens?
10.23 Suppose that u < e(r−δ)h. Show that there is an arbitrage opportunity. Now suppose that d > e(r−δ)h. Show again that there is an arbitrage opportunity.

Appendix 10.A TAXES AND OPTION PRICES
Appendix available at http://www.pearsonhighered.com/mcdonald.

11
C

Binomial Option Pricing:
Selected Topics

hapter 10 introduced binomial option pricing, focusing on how the model can be used to compute European and American option prices for a variety of underlying assets. In this chapter we continue the discussion of binomial pricing, discussing selected topics and delving more deeply into the economics of the model and its underlying assumptions.
First, the binomial model can value options that may be early-exercised. We examine early exercise in more detail, and see that the option pricing calculation reflects the economic determinants of early exercise discussed in Chapter 9.
Second, the binomial option pricing formula can be interpreted as the expected option payoff one period hence, discounted at the risk-free rate. In Chapter 10 we referred to this calculation as risk-neutral pricing. This calculation appears to be inconsistent with standard discounted cash flow valuation, in which expected cash flows are discounted at a riskadjusted rate, not the risk-free rate. We show that, in fact, the binomial pricing formula (and, hence, risk-neutral valuation) is consistent with option valuation using standard discounted cash flow techniques.
Third, we discuss the implicit distributional assumptions in the binomial model, namely that continuously compounded returns are normally distributed in the limit, which implies that prices are lognormally distributed.
Finally, we saw how to price options on stock indices where the dividend is continuous. In this chapter we adapt the binomial model to price options on stocks that pay discrete dividends. 11.1 UNDERSTANDING EARLY EXERCISE
In deciding whether to early-exercise an option, the option holder compares the value of exercising immediately with the value of continuing to hold the option and exercises if immediate exercise is more valuable. This is the comparison we performed at each binomial node when we valued American options in Chapter 10.
We obtain an economic perspective on the early-exercise decision by considering the costs and benefits of early exercise. As discussed in Section 9.3, there are three economic considerations governing the decision to exercise early. By exercising, the option holder:

323

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Chapter 11. Binomial Option Pricing: Selected Topics

.

Receives the stock and therefore receives future dividends.

.

Pays the strike price prior to expiration (this has an interest cost).

.

Loses the insurance implicit in the call. By holding the call instead of exercising, the option holder is protected against the possibility that the stock price will be less than the strike price at expiration. Once the option is exercised, this protection no longer exists. Consider an example where a call option has a strike price of $100, the interest rate is
5%, and the stock pays continuous dividends of 5%. If the stock price is $200, the net effect of dividends and interest encourages early exercise. Annual dividends are approximately
5% of $200, or 0.05 × $200 = $10. The annual interest saved by deferring exercise is approximately 0.05 × $100 = $5. Thus, for a stock price of $200 (indeed, for any stock price above $100) dividends lost by not exercising exceed interest saved by deferring exercise.
The only reason in this case not to exercise early is the implicit insurance the option owner loses by exercising. This implicit insurance arises from the fact that the option holder could exercise and then the stock price could fall below the strike price of $100. Leaving the option unexercised protects against this scenario. The early-exercise calculation for a call therefore implicitly weighs dividends, which encourage early exercise, against interest and insurance, which discourage early exercise.
If volatility is zero, then the value of insurance is zero, and it is simple to find the optimal exercise policy as long as r and δ are constant. For an infinitely lived call, it is optimal to defer exercise as long as interest savings on the strike exceed dividends lost, or rK > δS
It is optimal to exercise when this is not true, or
S>

rK δ In the special case when r = δ and σ = 0, any in-the-money option should be exercised immediately. If δ = 0.5r, then we exercise when the stock price is twice the exercise price.
The decision to exercise is more complicated when volatility is positive. In this case the implicit insurance has value that varies with time to expiration. Figure 11.1 displays the price above which early exercise is optimal for a 5-year option with K = $100, r = 5%, and δ = 5%, for three different volatilities, computed using 500 binomial steps. Recall from
Chapter 9 that if it is optimal to exercise a call at a given stock price, then it is optimal to exercise at all higher stock prices. Figure 11.1 thus shows the lowest stock price at which exercise is optimal. The oscillation in this lowest price, which is evident in the figure, is due to the up and down binomial movements that approximate the behavior of the stock; with an infinite number of binomial steps, the early-exercise schedule would be smooth and continuously decreasing. Comparing the three lines, we observe a significant volatility effect. A 5-year option with a volatility of 50% should only be exercised if the stock price exceeds about $360. If volatility is 10%, the boundary drops to $130. This volatility effect stems from the fact that the insurance value lost by early-exercising is greater when volatility is greater.
Figure 11.2 performs the same experiment for put options with the same inputs. The picture is similar, as is the logic: The advantage of early exercise is receiving the strike price

11.1 Understanding Early Exercise

FIGURE 11.1
Early-exercise boundaries for volatilities of 10%,
30%, and 50% for a 5-year
American call option. In all cases, K = $100, r = 5%, and δ = 5%.

Exercise Boundary
400
σ = 0.50

350
300
250

σ = 0.30
200
150
100

FIGURE 11.2
Early-exercise boundaries for volatilities of 10%,
30%, and 50% for a 5-year
American put option. In all cases, K = $100, r = 5%, and δ = 5%.

σ = 0.10
0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Time in Years

Exercise Boundary
100
90

σ = 0.10

80
70
60

σ = 0.30

50 σ = 0.50

40
30
20

0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Time in Years

sooner rather than later. The disadvantages are the dividends lost by giving up the stock, and the loss of insurance against the stock price exceeding the strike price.
Figures 11.1 and 11.2 also show that, other things equal, early-exercise criteria become less stringent closer to expiration. This occurs because the value of insurance diminishes as the options approach expiration.
While these pictures are constructed for the special case where δ = r, the overall conclusion holds generally.

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11.2 UNDERSTANDING RISK-NEUTRAL PRICING
In Chapter 10, we saw that the binomial option pricing formula can be written
C = e−rh[p ∗Cu + (1 − p ∗)Cd ]

(11.1)

where p∗ =

e(r−δ)h − d u−d (11.2)

We labeled p ∗ the risk-neutral probability that the stock will go up. Equation (11.1) has the appearance of a discounted expected value, where the expected value calculation uses p ∗ and discounting is done at the risk-free rate.
In this section we explain why p ∗ is called the risk-neutral probability and show that option valuation is consistent with standard discounted cash flow calculations.

The Risk-Neutral Probability
The idea that an option price is the result of a present value calculation is reassuring, but at the same time equation (11.1) is puzzling. A standard discounted cash flow calculation would require computing an expected value using the true probability that the stock price would go up. Discounting would then be done using the expected return on an asset of equivalent risk, not the risk-free rate.
It is common in finance to emphasize that investors are risk averse. To see what risk aversion means, suppose you are offered either (a) $1000, or (b) $2000 with probability
0.5, and $0 with probability 0.5. A risk-averse investor prefers (a), since alternative (b) is risky and has the same expected value as (a). This kind of investor will require a premium to bear risk when expected values are equal.
A risk-neutral investor is indifferent between a sure thing and a risky bet with an expected payoff equal to the value of the sure thing. A risk-neutral investor, for example, will be equally happy with alternative (a) or (b).
Let’s consider what an imaginary world populated by risk-neutral investors would be like. In such a world, investors care only about expected returns and not about riskiness.
Assets would have no risk premium since investors would be willing to hold assets with an expected return equal to the risk-free rate.
In this hypothetical risk-neutral world, we can solve for the probability of the stock going up, p ∗, such that the stock is expected to earn the risk-free rate. In the binomial model, the probability that the stock will go up, p ∗, must satisfy p ∗uSeδh + (1 − p ∗)dSeδh = erhS
Solving for p ∗ gives us equation (11.2), which is why we refer to p ∗ as the risk-neutral probability that the stock price will go up. It is the probability that the stock price would increase in a risk-neutral world.
Not only would the risk-neutral probability, equation (11.2), be used in a risk-neutral world, but also all discounting would take place at the risk-free rate. Thus, the option pricing formula, equation (11.1), can be said to price options as if investors are riskneutral. ous, It is important to note that we are not assuming that investors are actually

11.2 Understanding Risk-Neutral Pricing

risk-neutral, and we are not assuming that risky assets are actually expected to earn the riskfree rate of return. Rather, risk-neutral pricing is an interpretation of the formulas above.
Those formulas in turn arise from finding the cost of the portfolio that replicates the option payoff. This interpretation of the option-pricing procedure has great practical importance; risk-neutral pricing can sometimes be used where other pricing methods are too difficult.
We will see in Chapter 19 that risk-neutral pricing is the basis for Monte Carlo valuation, in which asset prices are simulated under the assumption that assets earn the risk-free rate, and these simulated prices are used to value the option.

Pricing an Option Using Real Probabilities
We are left with this question: Is option pricing consistent with standard discounted cash flow calculations? The answer is yes. We can use the true distribution for the future stock price in computing the expected payoff to the option. This expected payoff can then be discounted with a rate based on the stock’s required return.
Discounted cash flow is not used in practice to price options because there is no reason to do so: However, we present two examples of valuing an option using real probabilities to see the difficulty in using real probabilities, and also to understand how to determine the risk of an option.
Suppose that the continuously compounded expected return on the stock is α and that the stock does not pay dividends. Then if p is the true probability of the stock going up, p must be consistent with u, d, and α: puS + (1 − p)dS = eαhS

(11.3)

Solving for p gives us p= eαh − d u−d (11.4)

For probabilities to be between 0 and 1, we must have u > eαh > d. Using p, the actual expected payoff to the option one period hence is pCu + (1 − p)Cd =

eαh − d u − eαh
Cu +
Cd
u−d u−d (11.5)

Now we face the problem with using real as opposed to risk-neutral probabilities: At what rate do we discount this expected payoff? It is not correct to discount the option at the expected return on the stock, α, because the option is equivalent to a leveraged investment in the stock and, hence, is riskier than the stock.
Denote the appropriate per-period discount rate for the option as γ . To compute γ , we can use the fact that the required return on any portfolio is the weighted average of the returns on the assets in the portfolio.1 In Chapter 10, we saw that an option is equivalent to

1. See, for example, Brealey et al. (2011, ch. 9).

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shares of stock and B bonds. The expected return on

holding a portfolio consisting of this portfolio is eγ h =

S
S

+B

eαh +

S

B erh +B

(11.6)

We can now compute the option price as the expected option payoff, equation (11.5), discounted at the appropriate discount rate, given by equation (11.6). This gives e−γ h

eαh − d u − eαh
Cu +
Cd
u−d u−d (11.7)

It turns out that this gives us the same option price as performing the risk-neutral calculation.
Appendix 11.A demonstrates algebraically that equation (11.7) is equivalent to the riskneutral calculation, equation (11.1).
The calculations leading to equation (11.7) started with the assumption that the expected return on the stock is α. We then derived a consistent probability, p, and discount rate for the option, γ . You may be wondering if it matters whether we have the “correct” value of α to start with. The answer is that it does not matter: Any consistent pair of α and γ will give the same option price. Risk-neutral pricing is valuable because setting α = r results in the simplest pricing procedure.
A One-Period Example. To see how to value an option using true probabilities, we will compute two examples. First, consider the one-period binomial example in Figure 11.3.
Suppose that the continuously compounded expected return on XYZ is α = 15%. Then the true probability of the stock going up, from equation (11.4), is p= e0.15 − 0.8025
= 0.5446
1.4623 − 0.8025

The expected payoff to the option in one period, from equation (11.5), is
0.5446 × $19.954 + (1 − 0.5446) × $0 = $10.867
The replicating portfolio, and B, does not depend on p or α. In this example, and B = −$22.405. The discount rate, γ , from equation (11.6) is given by
0.738 × $41
−$22.405
e0.15 + e0.08 0.738 × $41 − $22.405
0.738 × $41 − $22.405
= 1.386

eγ h =

$59.954
$19.954

FIGURE 11.3
Binomial tree for pricing a European call option; assumes S = $41.00, K =
$40.00, σ = 0.30, r = 0.08,
T = 1.00 years, δ = 0.00, and h = 1.000. This is the same as Figure 10.3.

$41.000
$7.839
Δ = 0.738
B = –$22.405
$32.903
$0.000

= 0.738

11.2 Understanding Risk-Neutral Pricing

Thus, γ = ln(1.386) = 32.64%. The option price is then given by equation (11.7): e−0.3264 × $10.867 = $7.839
This is exactly the price we obtained before.
Notice that in order to compute the discount rate, we first had to compute and B.
But once we have computed and B, we can simply compute the option price as S + B.
There is no need for further computations. It can be helpful to know the actual expected return on an option, but for valuation it is pointless.
A Multi-Period Example. To demonstrate that this method of valuation works over multiple periods, Figure 11.4 presents the same binomial tree as Figure 10.5, with the addition that the true discount rate for the option, γ , is reported at each node. Given the
15% continuously compounded discount rate, the true probability of an up move in Figure
11.4 is e0.15×1/3 − 0.8637
= 0.5247
1.2212 − 0.8637
To compute the price at the node where the stock price is $61.149, we discount the expected option price the next period at 26.9%. This gives e−0.269×1/3 0.5247 × $34.678 + (1 − 0.5247) × $12.814 = $22.202
When the stock price is $43.246, the discount rate is 49.5%, and the option price is

$74.678
$34.678
γ = N/A

FIGURE 11.4
Binomial tree for pricing an American call option; assumes S = $41.00, K
= $40.00, σ = 0.30, r =
0.08, T = 1.00 years, δ =
0.00, and h = 0.333. The continuously compounded true expected return on the stock, α, is 15%. At each node the stock price, option price, and continuously compounded true discount rate for the option, γ , are given. Option price in bold italic signify that exercise is optimal at that node.

$61.149
$22.202
γ = 0.269
$50.071
$12.889 γ = 0.323

$52.814
$12.814
γ = N/A
$43.246
$5.700 γ = 0.495

$41.000
$7.074
γ = 0.357
$35.411
$2.535 γ = 0.495

$37.351
$0.000
γ = N/A
$30.585
$0.000 γ = N/A
$26.416
$0.000 γ = N/A

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e−0.495×1/3 0.5247 × $12.814 + (1 − 0.5247) × $0 = $5.700
These are both the same option prices as in Figure 10.5, where we used risk-neutral pricing.
We continue by working back through the tree. To compute the price at the node where the stock price is $50.071, we discount the expected option price the next period at 32.3%.
Thus,
e−0.323×1/3 0.5247 × $22.202 + (1 − 0.5247) × $5.700 = $12.889
Again, this is the same price at this node as in Figure 10.5.
The actual discount rate for the option changes as we move down the tree at a point in time and also over time. The required return on the option is less when the stock price is
$61.149 (26.9%) than when it is $43.246 (49.5%). The discount rate increases as the stock price decreases because the option is equivalent to a leveraged position in the stock, and the degree of leverage increases as the option moves out of the money.
These examples illustrate that it is possible to obtain option prices using standard discounted-cash-flow techniques. Generally, however, there is no reason to do so. Moreover, the fact that risk-neutral pricing works means that it is not necessary to estimate α, the expected return on the stock, when pricing an option. Since expected returns are hard to estimate precisely, this makes option pricing a great deal easier.
Appendix 11.B goes into more detail about risk-neutral pricing.

11.3 THE BINOMIAL TREE AND LOGNORMALITY
The usefulness of the binomial pricing model hinges on the binomial tree providing a reasonable representation of the stock price distribution. In this section we discuss the motivation for and plausibility of the binomial tree. We will define a lognormal distribution and see that the binomial tree approximates this distribution.

The Random Walk Model
It is sometimes said that stock prices follow a random walk. More precisely, a random walk provides a foundation for modeling the prices of stocks and other assets. In this section, we will explain what a random walk is. In the next section, we use the random walk model to build a model of stock prices.
To understand a random walk, imagine that we flip a coin repeatedly. Let the random variable Y denote the outcome of the flip. If the coin lands displaying a head, Y = 1. If the coin lands displaying a tail, Y = −1. If the probability of a head is 50%, we say the coin is fair. After n flips, with the ith flip denoted Yi , the cumulative total, Zn, is n Zn =

Yi

(11.8)

i=1

It turns out that the more times we flip, on average, the farther we will move from where we start. We can understand intuitively why with more flips the average distance from the starting point increases. Think about the first flip and imagine you get a head. You move to
+1, and as far as the remaining flips are concerned, this is your new starting point. After the second flip, you will either be at 0 or +2. If you are at zero, it is as if you started over;

11.3 The Binomial Tree and Lognormality

however, if you are at +2, you are starting at +2. Continuing in this way, your average distance from the starting point increases with the number of flips.2
Another way to represent the process followed by Zn is in terms of the change in Zn:
Zn − Zn−1 = Yn
We can rewrite this more explicitly as
Heads:

Zn − Zn−1 = +1

(11.9)

Tails:

Zn − Zn−1 = −1

(11.10)

With heads, the change in Z is 1, and with tails, the change in Z is −1. This random walk is illustrated in Figure 11.5.
The idea that asset prices should follow a random walk was articulated in Samuelson
(1965). In efficient markets, an asset price should reflect all available information. By definition, new information is a surprise. In response to new information the price is equally likely to move up or down, as with the coin flip. The price after a period of time is the initial price plus the cumulative up and down movements due to informational surprises.

Modeling Stock Prices as a Random Walk
The idea that stock prices move up or down randomly makes sense; however, the description of a random walk in the previous section is not a satisfactory description of stock price movements. Suppose we take the random walk model in Figure 11.5 literally. Assume the beginning stock price is $100, and the stock price will move up or down $1 each time we flip the coin. There are at least three problems with this model:
1. If by chance we get enough cumulative down movements, the stock price will become negative. Because stockholders have limited liability (they can walk away from a bankrupt firm), a stock price will never be negative.
2. The magnitude of the move ($1) should depend upon how quickly the coin flips occur and the level of the stock price. If we flip coins once a second, $1 moves are excessive; in real life, a $100 stock will not typically have 60 $1 up or down movements in 1 minute. Also, if a $1 move is appropriate for a $100 stock, it likely isn’t appropriate for a $5 stock.
3. The stock on average should have a positive return. The random walk model taken literally does not permit this.

2. After n flips, the average squared distance from the starting point will be n. Conditional on the first flip being a head, your average squared distance is 0.5 × 0 + 0.5 × 22 = 2. If your first flip had been a tail, your average squared distance after two moves would also be 2. Thus, the unconditional average squared
2
distance is 2 after 2 flips. If Dn represents your squared distance from the starting point, then
2
2
Dn = 0.5 × (Dn−1 + 1)2 + 0.5 × (Dn−1 − 1)2 = Dn−1 + 1
2
2
Since D0 = 0, this implies that Dn = n. This idea that with a random walk you drift increasingly farther from the starting point is an important concept later in the book.

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Cumulative Z
100

FIGURE 11.5
In the top panel is an illustration of a random walk, where the counter,
Z, increases by 1 when a fair coin flip comes up heads, and decreases by 1 with tails. In the bottom panel is a particular path through a 10,000-step binomial tree, where the up and down moves are the same as in the top panel. Assumes S0 = $100, r = 6%, σ = 30%, T = 10 years, and h = 0.0001.

0
–100
–200
–300
–400

0

20,000

40,000
60,000
Number of Coin Flips

80,000

100,000

8

10

Stock Price ($)
120
100
80
60
40
0

2

4

6
Time (years)

It turns out that the binomial model is a variant of the random walk model that solves all of these problems at once. The binomial model assumes that continuously compounded returns are a random walk with drift.

The Binomial Model
The binomial stock price is
St+h = St e(r−δ)h±σ

√ h Taking logs, we obtain

ln(St+h/St ) = (r − δ)h ± σ h

(11.11)

Since ln(St+h/St ) is the continuously compounded return from t to t + h, rt , t+h, the binomial model is simply a particular way to model the continuously compounded return.
That return has two parts, one of which is certain [(r − δ)h], and the other of which is

uncertain and generates the up and down stock price moves (±σ h).
Let’s see how equation (11.11) solves the three problems in the random walk discussed earlier: 11.3 The Binomial Tree and Lognormality

1. The stock price cannot become negative. Even if we move down the binomial tree many times in a row, the resulting large, negative, continuously compounded return will give us a positive price.
2. As stock price moves occur more frequently, h gets smaller, therefore up and down moves get smaller. By construction, annual volatility is the same no matter how many binomial periods there are. Since returns follow a random walk, the percentage price change is the same whether the stock price is $100 or $5.
3. There is a (r − δ)h term, and we can choose the probability of an up move, so we can guarantee that the expected change in the stock price is positive.
The bottom panel of Figure 11.5 illustrates the stock price that results when the continuously compounded return follows equation (11.11). The figure is one particular path through a 10,000-step binomial tree, with the path generated by the same sequence of coin flips as in the top panel of Figure 11.5.

Lognormality and the Binomial Model
The binomial tree approximates a lognormal distribution, which is commonly used to model stock prices. The lognormal distribution is the probability distribution that arises from the assumption that continuously compounded returns on the stock are normally distributed.
When we traverse the binomial tree, we are implicitly adding up binomial random return

components of (r − δ)h ± σ h. In the limit (as n → ∞ or, the same thing, h → 0), the sum of binomial random variables is normally distributed. Thus, continuously compounded returns in a binomial tree are (approximately) normally distributed, which means that the stock is lognormally distributed. We defer a more complete discussion of this to Chapters
18 and 20, but we can see with an example how it works.
The binomial model implicitly assigns probabilities to the various nodes. Figure 11.6 depicts the construction of a tree for three binomial periods, along with the risk-neutral probability of reaching each final period node. There is only one path—sequence of up

Probability

FIGURE 11.6
Construction of a binomial tree depicting stock price paths, along with riskneutral probabilities of reaching the various terminal prices. S0u 3

p*3

S0u 2d

3p*2(1 – p* )

S0 u 2
S0 u
S0

S0du
S0 d 2 u 3p*(1 – p* )2

S0 d
S0 d 2

S0 d 3

(1 – p* )3

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Chapter 11. Binomial Option Pricing: Selected Topics

FIGURE 11.7

Lognormal Probability

Comparison of lognormal distribution with threeperiod binomial approximation.

Binomial Probability

0.014

0.45

0.012

0.40
0.35

0.010

0.30

0.008

0.25

0.006

0.20
0.15

0.004

0.10

0.002
0

0 50

0.05
0
100 150 200 250 300 350 400
Lognormal

Binomial

and down moves—reaching the top or bottom node (uuu or ddd), but there are three paths reaching each intermediate node. For example, the first node below the top (S0u2d) can be reached by the sequences uud, udu, or duu. Thus, there are more paths that reach the intermediate nodes than the extreme nodes.
We can take the probabilities and outcomes from the binomial tree and plot them against a lognormal distribution with the same parameters. Figure 11.7 compares a threeperiod binomial approximation with a lognormal distribution assuming that the initial stock price is $100, volatility is 30%, the expected return on the stock is 10%, and the time horizon is 1 year. Because we need different scales for the discrete and continuous distributions, lognormal probabilities are graphed on the left vertical axis and binomial probabilities on the right vertical axis.
Suppose that a binomial tree has n periods and the risk-neutral probability of an up move is p ∗. To reach the top node, we must go up n times in a row, which occurs with a probability of (p ∗)n. The price at the top node is Sun. There is only one path through the tree by which we can reach the top node. To reach the first node below the top node, we must go up n − 1 times and down once, for a probability of (p ∗)n−1 × (1 − p ∗). The price at that node is Sun−1d. Since the single down move can occur in any of the n periods, there are n ways this can happen. The probability of reaching the ith node below the top is
(p ∗)n−i × (1 − p ∗)i . The price at this node is Sun−i d i . The number of ways to reach this node is
Number of ways to reach ith node =

n n! = i (n − i)! i!

where n! = n × (n − 1) × . . . × 1.3
We can construct the implied probability distribution in the binomial tree by plotting the stock price at each final period node, Sun−i d i , against the probability of reaching that

3. The expression

n i can be computed in Excel using the combinatorial function, Combin(n, i).

11.3 The Binomial Tree and Lognormality

Lognormal Probability

FIGURE 11.8
Comparison of lognormal distribution with 25-period binomial approximation.

Binomial Probability

0.014

0.18

0.012

0.16
0.14

0.010

0.12

0.008

0.10

0.006

0.08
0.06

0.004

0.04

0.002
0

0

100

200
Lognormal

300

400

0.02
0
500

Binomial

node. The probability of reaching any given node is the probability of one path reaching that node times the number of paths reaching that node:
Probability of reaching ith node = p ∗n−i (1 − p ∗)i

n!
(n − i)! i!

(11.12)

Figure 11.8 compares the probability distribution for a 25-period binomial tree with the corresponding lognormal distribution. The two distributions appear close; as a practical matter, a 25-period approximation works fairly well for an option expiring in a few months.
Figures 11.7 and 11.8 show you what the lognormal distribution for the stock price looks like. The stock price is positive, and the distribution is skewed to the right; that is, there is a chance that extremely high stock prices will occur.

Alternative Binomial Trees
There are other ways besides equation (11.11) to construct a binomial tree that approximates a lognormal distribution. An acceptable tree must match the standard deviation of the continuously compounded return on the asset and must generate an appropriate distribution as the length of the binomial period, h, goes to 0. Different methods of constructing the binomial tree will result in different u and d stock movements. No matter how we construct the tree, however, we use equation (10.5) to determine the risk-neutral probability and equation (10.6) to determine the option value.
The Cox-Ross-Rubinstein Binomial Tree. The best-known way to construct a binomial tree is that in Cox et al. (1979), in which the tree is constructed as

h

−σ h

u = eσ d =e

(11.13)

The Cox-Ross-Rubinstein approach is often used in practice. A problem with this approach,

however, is that if h is large or σ is small, it is possible that erh > eσ h, in which case the

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binomial tree violates the restriction in equation (10.4). In real applications h would be small, so this problem does not occur. In any event, the tree based on the forward price never violates equation (10.4).
The Lognormal Tree. Another alternative is to construct the tree using

h

2 )h−σ h
(r−δ−0.5σ

u = e(r−δ−0.5σ d =e

2 )h+σ

(11.14)

This procedure for generating a tree was proposed by Jarrow and Rudd (1983) and is sometimes called the Jarrow-Rudd binomial model. It has a very natural motivation that you will understand after we discuss lognormality in Chapter 18. You will find when using equation (10.5) that the risk-neutral probability of an up-move is generally close to 0.5.
All three methods of constructing a binomial tree yield different option prices for finite n, but they approach the same price as n → ∞. Also, while the different binomial trees all have different up and down movements, all have the same ratio of u to d:

√ u = e2σ h or ln(u/d) = 2σ h d This is the sense in which, however the tree is constructed, the proportional distance between u and d measures volatility.

Is the Binomial Model Realistic?
Any option pricing model relies on an assumption about the behavior of stock prices. As we have seen in this section, the binomial model is a form of the random walk model, adapted to modeling stock prices. The lognormal random walk model in this section assumes, among other things, that volatility is constant, that “large” stock price movements do not occur, and that returns are independent over time. All of these assumptions appear to be violated in the data.
We will discuss the behavior of volatility in Chapter 24. However, there is ample evidence that volatility changes over time (see Bollerslev et al., 1994). It also appears that on occasion stocks move by a large amount. The binomial model has the property that stock price movements become smaller as the period length, h, becomes smaller. Occasional large price movements—“jumps”—are therefore a feature of the data inconsistent with the binomial model. We will also discuss such moves in Chapters 19 and 21. Finally, there is some evidence that stock returns are correlated across time, with positive correlations at the short to medium term and negative correlation at long horizons (see Campbell et al., 1997, ch. 2).
The random walk model is a useful starting point for thinking about stock price behavior, and it is widely used because of its elegant simplicity. However, it is not sacrosanct.

11.4 STOCKS PAYING DISCRETE DIVIDENDS
Although it may be reasonable to assume that a stock index pays dividends continuously, individual stocks pay dividends in discrete lumps, quarterly or annually. In addition, over short horizons it is frequently possible to predict the amount of the dividend. How should we price an option when the stock will pay a known dollar dividend during the life of the option?

11.4 Stocks Paying Discrete Dividends

The procedure we have already developed for creating a binomial tree can accommodate this case. However, we will also discuss a preferable alternative due to Schroder (1988).

Modeling Discrete Dividends
When no dividend will be paid between time t and t + h, we create the binomial tree as in Chapter 10. Suppose that a dividend will be paid between times t and t + h and that its future value at time t + h is D. The time t forward price for delivery at t + h is then
Ft , t+h = St erh − D
Since the stock price at time t + h will be ex-dividend, we create the up and down moves based on the ex-dividend stock price:
Stu = St erh − D eσ



Std = St erh − D e−σ

h

(11.15)

√ h How does option replication work when a dividend is imminent? When a dividend is paid, we have to account for the fact that the stock earns the dividend. Thus, we have
Stu + D

+ erhB = Cu

Std + D

+ erhB = Cd

The solution is
=

Cu − Cd
Stu − Std

B = e−rh

StuCd − Std Cu
Stu − Std



De−rh

Because the dividend is known, we decrease the bond position by the present value of the certain dividend. (When the dividend is proportional to the stock price, as with a stock index, we reduce the stock position, equation (10.1).) The expression for the option price is given by equation (10.24).

Problems with the Discrete Dividend Tree
The practical problem with this procedure is that the tree does not completely recombine after a discrete dividend. In all previous cases we have examined, we reached the same price after a given number of up and down movements, regardless of the order of the movements.
Figure 11.9, in which a dividend with a period-2 value of $5 is paid between periods
1 and 2, demonstrates that with a discrete dividend, the order of up and down movements affects the price. In the third binomial period there are six rather than four possible stock prices. To see how the tree is constructed, period-1 prices are

337

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Chapter 11. Binomial Option Pricing: Selected Topics

FIGURE 11.9

Binomial Period:
Dividend:

0

The tree depicts a discrete $5 dividend paid between the first and second binomial periods. There are eight discrete terminal nodes (six of them distinct) rather than four. Assumes S = $41, σ =
0.3, r = 0.08, t = 1 year, and h = 0.333.

1
0

2
5

3
0
$67.417

$55.203
$47.679
$47.679

$50.071
$39.041

$33.720
$41.000

$45.553
$37.300
$35.411
$32.216
$32.216
$26.380

$22.784


$41e0.08×1/3+0.3× 1/3 = $50.071

$41e0.08×1/3−0.3× 1/3 = $35.411
The period-2 prices from the $50.071 node are

($50.071e0.08×1/3 − 5) × e0.3× 1/3 = $55.203

($50.071e0.08×1/3 − 5) × e−0.3× 1/3 = $39.041
Repeating this procedure for the node S = $35.411 gives prices of $37.300 and $26.380.
You can see that there are now four prices instead of three after two binomial steps: The ud and du nodes do not recombine. There are six distinct prices in the final period as each set of ex-dividend prices generates a distinct tree (three prices arise from the top two prices in period 2 and three prices arise from the bottom two prices in period 2). Each discrete dividend causes the tree to bifurcate.
There is also a conceptual problem with equation (11.15). Since the amount of the dividend is fixed, the stock price could in principle become negative if there have been large downward moves in the stock prior to the dividend.
This example demonstrates that handling fixed dividends requires care. We now turn to a method that is computationally easier than constructing a tree using equation (11.15) and that will not generate negative stock prices.

11.4 Stocks Paying Discrete Dividends

A Binomial Tree Using the Prepaid Forward
Schroder (1988) presents an elegant method of constructing a tree for a dividend-paying stock that solves both problems encountered with the method in Figure 11.9. The key insight is that if we know for certain that a stock will pay a fixed dividend, then we can view the stock price as being the sum of two components: the dividend and the present value of the exdividend value of the stock—in other words, the prepaid forward price. With the dividend known, all volatility is attributed to the prepaid forward component of the stock price.
Suppose we know that a stock will pay a dividend D at time TD < T , where T is the expiration date of the option. For t < TD , the stock price is the sum of the prepaid forward price and the present value of the dividend:
St = FtPT + De−r(TD −t)
,

(11.16)

where FtPT = St − De−r(TD −t). We construct the binomial tree by attributing all uncertainty
,
to the prepaid forward price. As before, we have up and down movements of u = erh+σ

√ h d = erh−σ



h

The observed stock price at time t + h < TD is then
St+h = FtP erh±σ

√ h + De−r(TD −(t+h))

We measure σ by observing movements in St , but σ is used in this equation to characterize movements in FtP . We want the total dollar volatility of the prepaid forward to equal that of the stock, so we assign a volatility to the prepaid forward using the ad hoc adjustment σ F = σs ×

S
FP

Figure 11.10 shows the construction of the binomial tree for a specific example, using the same initial inputs as Figure 11.9. Both the observed stock price and the stock price less the present value of dividends (the prepaid forward price) are included in the figure.
Assuming that the dividend is paid just before the second period, the initial prepaid forward
P
price is F0 = 41 − 5e−0.08×2/3 = 36.26. The volatility for the prepaid forward is therefore
$41
0.3 × $36.26 = 0.3392.
To understand Figure 11.10, note first that u = 1.2492. Look at the node where the stock price is $61.584. This is a cum-dividend price, just before the dividend is paid. The nodes in the last period are constructed based on the ex-dividend price, for example,
($61.584 − $5) × 1.2492 = $70.686
As a final point, we obtain risk-neutral probabilities for the tree in the same way as in the absence of dividends. Because up and down movements are based on the prepaid forward, which pays no dividends, the risk-neutral probability of an up move in the prepaid forward price is given by equation (10.5), as for a non-dividend paying stock.

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Chapter 11. Binomial Option Pricing: Selected Topics

FIGURE 11.10
Binomial tree for pricing an American call option on a stock paying a discrete dividend of $5 in 8 months; assumes S = $41.00, K =
$40.00, σ = 0.3392, r =
0.08, T = 1.00 years, δ =
0.00, and h = 0.333. At each node the stock price, prepaid forward price, and option price are given. Option prices in bold italic signify that exercise is optimal at that node.

$70.686
F P = 70.686
$30.686
$61.584
F P = 56.584
$21.584
$50.164
F P = 45.296
$11.308

$47.777
F P = 47.777
$7.777
$43.246
F P = 38.246
$3.417

$41.000
F P = 36.260
$5.770
$35.485
F P = 30.616
$1.501

$32.293
F P = 32.293
$0.000
$30.851
F P = 25.851
$0.000
$21.827
F P = 21.827
$0.000

CHAPTER SUMMARY
Both call and put options may be rationally exercised prior to expiration. The early-exercise decision weighs three considerations: dividends on the underlying asset, interest on the strike price, and the insurance value of keeping the option alive. Calls will be early-exercised in order to capture dividends on the underlying stock; interest and insurance weigh against early exercise. Puts will be early-exercised in order to capture interest on the strike price; dividends and insurance weigh against early exercise. For both calls and puts, the earlyexercise criterion becomes less stringent as the option has less time to maturity.
Risk-neutral option valuation is consistent with valuation using more traditional discounted cash flow methods. With risk-neutral pricing it is not necessary to estimate the expected return on the stock in order to price an option. With traditional discounted cash flow methods, the correct discount rate for the option varies along the binomial tree; thus, valuation is considerably more complicated than with risk-neutral pricing.
The binomial model, which approximates the lognormal distribution, is a random walk model adapted to modeling stock prices. The model assumes that the continuously compounded return on the stock is normally distributed, which implies that the stock price is lognormally distributed.
The binomial model can be adapted to price options on a stock that pays discrete dividends. Discrete dividends can lead to a nonrecombining binomial tree. If we assume that the prepaid forward price follows a binomial process instead of the stock price, the tree becomes recombining.

Problems

FURTHER READING
The binomial model can be used to derive the Black-Scholes model, which we discuss in
Chapter 12. The practical importance of risk-neutral pricing will become evident in Chapter
19, when we see that Monte Carlo valuation hinges upon risk-neutral pricing. In that chapter we will also reexamine Figure 11.4 and show how the option price may be computed as an expected value using only stock prices in the final period.
The issue of how the stock price is distributed will also arise frequently in later chapters. Chapter 18 discusses lognormality in more detail and presents evidence that stock prices are not exactly lognormally distributed. Chapter 20 will examine in more detail the question of how the stock price moves, in particular what happens when h gets very small in the binomial model.
We will return to the determinants of early exercise in Chapter 17, when we discuss real options.
The literature on risk-neutral pricing is fairly technical. Cox and Ross (1976) was the first paper to use risk-neutral pricing, and Harrison and Kreps (1979) studied the economic underpinnings. Two good treatments of this topic are Huang and Litzenberger (1988, ch.
8)—their treatment inspired Appendix 11.B—and Baxter and Rennie (1996). We study risk-neutral pricing in more detail in Chapter 22.
Campbell et al. (1997) and Cochrane (2001) summarize evidence on the distribution of stock prices. The original Samuelson work on asset prices following a random walk
(Samuelson, 1965) remains a classic, modern empirical evidence notwithstanding.
Broadie and Detemple (1996) discuss the computation of American option prices, and also discuss alternative binomial approaches and their relative numerical efficiency.

PROBLEMS
Many (but not all) of these questions can be answered with the help of the BinomCall and
BinomPut functions available on the spreadsheets accompanying this book.
11.1 Consider a one-period binomial model with h = 1, where S = $100, r = 0, σ = 30%, and δ = 0.08. Compute American call option prices for K = $70, $80, $90, and
$100.
a. At which strike(s) does early exercise occur?
b. Use put-call parity to explain why early exercise does not occur at the higher strikes. c. Use put-call parity to explain why early exercise is sure to occur for all lower strikes than that in your answer to (a).
11.2 Repeat Problem 11.1, only assume that r = 0.08. What is the greatest strike price at which early exercise will occur? What condition related to put-call parity is satisfied at this strike price?
11.3 Repeat Problem 11.1, only assume that r = 0.08 and δ = 0. Will early exercise ever occur? Why?
11.4 Consider a one-period binomial model with h = 1, where S = $100, r = 0.08, σ = 30%, and δ = 0. Compute American put option prices for K = $100, $110,
$120, and $130.

341

342

Chapter 11. Binomial Option Pricing: Selected Topics

a. At which strike(s) does early exercise occur?
b. Use put-call parity to explain why early exercise does not occur at the other strikes. c. Use put-call parity to explain why early exercise is sure to occur for all strikes greater than that in your answer to (a).
11.5 Repeat Problem 11.4, only set δ = 0.08. What is the lowest strike price at which early exercise will occur? What condition related to put-call parity is satisfied at this strike price?
11.6 Repeat Problem 11.4, only set r = 0 and δ = 0.08. What is the lowest strike price
(if there is one) at which early exercise will occur? If early exercise never occurs, explain why not.
For the following problems, note that the BinomCall and BinomPut functions are array functions that return the option delta ( ) as well as the price. If you know , you can compute B as C − S .
11.7 Let S = $100, K = $100, σ = 30%, r = 0.08, t = 1, and δ = 0. Let n = 10. Suppose the stock has an expected return of 15%.
a. What is the expected return on a European call option? A European put option? b. What happens to the expected return if you increase the volatility to 50%?
11.8 Let S = $100, σ = 30%, r = 0.08, t = 1, and δ = 0. Suppose the true expected return on the stock is 15%. Set n = 10. Compute European call prices, , and B for strikes of $70, $80, $90, $100, $110, $120, and $130. For each strike, compute the expected return on the option. What effect does the strike have on the option’s expected return?
11.9 Repeat the previous problem, except that for each strike price, compute the expected return on the option for times to expiration of 3 months, 6 months, 1 year, and 2 years.
What effect does time to maturity have on the option’s expected return?
11.10 Let S = $100, σ = 30%, r = 0.08, t = 1, and δ = 0. Suppose the true expected return on the stock is 15%. Set n = 10. Compute European put prices, , and B for strikes of $70, $80, $90, $100, $110, $120, and $130. For each strike, compute the expected return on the option. What effect does the strike have on the option’s expected return?
11.11 Repeat the previous problem, except that for each strike price, compute the expected return on the option for times to expiration of 3 months, 6 months, 1 year, and 2 years.
What effect does time to maturity have on the option’s expected return?
11.12 Let S = $100, σ = 0.30, r = 0.08, t = 1, and δ = 0. Using equation (11.12) to compute the probability of reaching a terminal node and Sui d n−i to compute the price at that node, plot the risk-neutral distribution of year-1 stock prices as in Figures
11.7 and 11.8 for n = 3 and n = 10.
11.13 Repeat the previous problem for n = 50. What is the risk-neutral probability that
S1 < $80? S1 > $120?

11.A Pricing Options with True Probabilities

11.14 We saw in Section 10.1 that the undiscounted risk-neutral expected stock price equals the forward price. We will verify this using the binomial tree in Figure 11.4.
a. Using S = $100, r = 0.08, and δ = 0, what are the 4-month, 8-month, and
1-year forward prices?
b. Verify your answers in (a) by computing the risk-neutral expected stock price in the first, second, and third binomial period. Use equation (11.12) to determine the probability of reaching each node.
11.15 Compute the 1-year forward price using the 50-step binomial tree in Problem 11.13.
11.16 Suppose S = $100, K = $95, r = 8% (continuously compounded), t = 1, σ = 30%, and δ = 5%. Explicitly construct an eight-period binomial tree using the Cox-RossRubinstein expressions for u and d: u = eσ

√ h d = e−σ

√ h Compute the prices of European and American calls and puts.
11.17 Suppose S = $100, K = $95, r = 8% (continuously compounded), t = 1, σ = 30%, and δ = 5%. Explicitly construct an eight-period binomial tree using the lognormal expressions for u and d: u = e(r−δ−.5σ

2 )h+σ



h

d = e(r−δ−.5σ

2 )h−σ



h

Compute the prices of European and American calls and puts.
11.18 Suppose that S = $50, K = $45, σ = 0.30, r = 0.08, and t = 1. The stock will pay a $4 dividend in exactly 3 months. Compute the price of European and American call options using a four-step binomial tree.

Appendix 11.A PRICING OPTIONS WITH TRUE PROBABILITIES
In this appendix we demonstrate algebraically that computing the option price in a consistent way using α as the expected return on the stock gives the correct option price. Using the definition of γ , equation (11.6), we can rewrite equation (11.7) as
( S + B)

eαh

1
S + erhB

erh − d u − erh eαh − erh
Cu +
Cd +
(Cu − Cd ) u−d u−d u−d (11.17)

Since S + B is the call price, we need only show that the expression in large parentheses is equal to one. From the definitions of and B we have erh − d u − erh
Cu +
Cd =erh( S + B) u−d u−d
We can rewrite equation (11.17) as
( S + B)

1 erh( S + B) + (eαh − erh) S eαh S + erhB

This follows since the expression in large parentheses equals one.

= S+B

343

344

Chapter 11. Binomial Option Pricing: Selected Topics

Appendix 11.B WHY DOES RISK-NEUTRAL PRICING WORK?
In this appendix we use the binomial model to explain the economic underpinnings of riskneutral pricing. Chapter 22 contains a fuller explanation of risk-neutral pricing.

Utility-Based Valuation
The starting point is that the well-being of investors is not measured in dollars, but in utility, which is a measure of satisfaction. Economists typically assume that investors exhibit declining marginal utility: Starting from a given level of wealth, the utility gained if we add
$1 to wealth is less than the utility lost if we remove $1 from wealth. Thus, we expect that more dollars will make an investor happier, but that if we keep adding dollars, each additional dollar will make the investor less happy than the previous dollars.
Declining marginal utility implies that investors are risk-averse, which means that an investor will prefer a safer investment to a riskier investment that has the same expected return. Since losses are more costly than gains are beneficial, a risk-averse investor will avoid a fair bet, which by definition has equal expected gains and losses.4
To illustrate risk-neutral pricing, we imagine a world where there are two assets, a risky stock and a risk-free bond. Investors are risk-averse. Suppose the economy in one period will be in one of two states, a high state and a low state. How do we value assets in such a world? We need to know three things:
1. What utility value, expressed in terms of dollars today, does an investor attach to the marginal dollar received in each state in the future? Denote the values today of $1 received in the high and low states as χH and χL, respectively.5 Because the investor is risk-averse, $1 received in the high state is worth less than $1 received in the low state; hence, χH < χL.
2. How many dollars will an asset pay in each state? The bond pays $1 in each state, while the risky stock pays SH in the high state and SL in the low state.
3. What is the probability of each state occurring? Denote the probability of the high state as p. Another name for p is the physical probability of an up move.
We begin by defining a state price as the price of a security that pays $1 only when a particular state occurs. Let QH be the price of a security that pays $1 when the high state occurs, and QL the price of a security paying $1 when the low state occurs.6 Since χH and χL are the value today of $1 in each state, the price we would pay is just the value times the

4. This is an example of Jensen’s Inequality (see Appendix C at the end of this book). A risk-averse investor has a concave utility function, which implies that
E[U (x)] < U [E(x)]
The expected utility associated with a gamble, E[U (x)], is less than the utility from receiving the expected value of the gamble for sure, U [E(x)].
5. Technically χH and χL are ratios of marginal utilities, discounted by the rate of time preference. However, you can think of them as simply converting future dollars in a particular state into dollars today.
6. These are often called “Arrow-Debreu” securities, named after Nobel Prize–winning economists Kenneth
Arrow and Gerard Debreu.

11.B Why Does Risk-Neutral Pricing Work?

probability that state is reached:
QH = p × χ H
QL = (1 − p) × χL

(11.18)

Since there are only two possible states, we can value any future cash flow using these state prices. The price of the risky stock, S0, is
S0 = (QH × SH ) + (QL × SL)

(11.19)

The risk-free bond, with price B0, pays $1 in each state. Thus, we have
B0 = (QH × 1) + (QL × 1)

(11.20)

We can calculate rates of return by dividing expected cash flows by the price. Thus, the risk-free rate is
1+ r =
=

1
QH + QL
1
B0

(11.21)

The expected return on the stock is
1+ α =

p × SH + (1 − p)SL
QH × SH + QL × SL

p × SH + (1 − p)SL
=
S0

(11.22)

Standard Discounted Cash Flow
The standard discounted cash flow calculation entails computing the security price by discounting the expected cash flow at the expected rate of return. In the case of the stock, this gives us pSH + (1 − p)SL
= S0
1+ α
This is simply a rewriting of equation (11.22); hence, it is obviously correct. Similarly, the bond price is
1
= Price of bond
1+ r

Risk-Neutral Pricing
The point of risk-neutral pricing is to sidestep the utility calculations above. We want probabilities we can use to compute expected cash flows without explicit utility adjustments, and for which discounting that expectation at the risk-free rate will provide the correct answer. The trick is the following: Instead of utility-weighting the cash flows and computing expectations, we utility-weight the probabilities, creating new “risk-neutral” probabilities.

345

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Chapter 11. Binomial Option Pricing: Selected Topics

Use the state prices in equation (11.18) to define the risk-neutral probability of the high state, p ∗, as p∗ =

p × χH
QH
= p × χH + (1 − p) × χL QH + QL

Now we compute the stock price by using the risk-neutral probabilities to compute expected cash flow, and then discounting at the risk-free rate. We have p ∗SH + (1 − p ∗)SL
=
1+ r

QH
QH +QL SH

+

QL
QH +QL SL

1+ r
QH SH + QLSL
=
QH + QL (1 + r)
= Q H SH + Q L SL

which is the price of stock, from equation (11.19). This shows that we can construct riskneutral probabilities and use them to price risky assets.

Physical vs. Risk-Neutral Probabilities
The difference between the actual probability of an event and the risk-neutral probability can be expressed in terms of the risk premium on the asset. As before, we have
S0 =

pSH + (1 − p)SL
1+ α

(11.23)

Because 1 + α = 1 + (α − r) + r, we can rewrite equation (11.23) as pSH + (1 − p)SL − S0(α − r) = (1 + r)S0
Notice that the right-hand side of this equation is the forward price of the stock, F0, 1. We can rewrite this equation to obtain p− S0(α − r)
SH − S L

(SH − SL) + SL = (1 + r)S0

Rewriting and dividing by 1 + r, we have pSH + (1 − p)SL
ˆ
ˆ
= S0
1+ r

(11.24)

where the new probability p is defined as
ˆ
p=p−
ˆ

S0(α − r)
SH − SL

(11.25)

Equation (11.24) is the risk-neutral valuation equation for the stock. Thus, p equals p ∗, the
ˆ
risk-neutral probability of the high state.
Equation (11.25) shows that the actual probability, p, and the risk-neutral probability, p, differ by a term that is proportional to the dollar risk premium on the stock, S0(α − r).
ˆ
The actual probability, p, is reduced by the dollar risk premium per dollar of risk, SH − SL.
Thus, the real and risk-neutral probabilities differ by an amount that is determined by investor attitudes toward risk.

11.B Why Does Risk-Neutral Pricing Work?

TABLE 11.1

Probabilities, utility weights, and equity cash flows in high and low states of the economy.

High State
Cash flow to risk-free bond
Cash flow to stock
Probability
Value of $1

Low State

BH = $1
SH = $180 p = 0.52 χH = $0.87

BL = $1
SL = $30 p = 0.48 χL = $0.98

Example
Table 11.1 contains assumptions for a numerical example.
State Prices. Using equation (11.18), the state prices are QH = 0.52 × $0.87 = $0.4524, and QL = 0.48 × $0.98 = $0.4704.
Valuing the Risk-Free Bond. The risk-free bond pays $1 in each state. Thus, using equation (11.20), the risk-free bond price, B0 , is
B0 = QH + QL = $0.4524 + $0.4704 = $0.9228

(11.26)

The risk-free rate is r= 1
− 1 = 8.366%
0.9228

Valuing the Risky Stock Using Real Probabilities. Using equation (11.19), the price of the stock is
S0 = 0.4524 × $180 + 0.4704 × $30 = $95.544

(11.27)

The expected cash flow on the stock in one period is
E(S1) = 0.52 × $180 + 0.48 × $30 = $108
The expected return on the stock is therefore α= $108
− 1 = 13.037%
$95.544

By definition, if we discount E(S1) at the rate 13.037%, we will obtain the price
$95.544.
Risk-Neutral Valuation of the Stock. The risk-neutral probability is
$0.4524
$0.4524 + $0.4704
= 49.025%

p∗ =

Now we can value the stock using p ∗ instead of the true probabilities and discount at the risk-free rate:

347

348

Chapter 11. Binomial Option Pricing: Selected Topics

0.49025 × $180 + (1 − 0.49025) × $30
1.08366
= $95.544

S0 =

We will discuss risk-neutral pricing in much more detail in Chapter 22. However, at this point we can see that risk-neutral pricing requires that investors agree about the equilibrium risk premium for assets. They will agree if each investor values and invests in both the stock and risk-free bond. In equilibrium, investors will be pleased with their investment amounts. As long as this is true, investors agree about the risk premium associated with each asset and therefore agree on the transformation that creates risk-neutral probabilities.
When would risk-neutral pricing not work? Suppose you have an asset that cannot be traded or hedged, or you have a nontradable asset with cash flows that cannot be replicated by the cash flows of traded assets. If you cannot trade or offset the risk of the asset, then there is no guarantee that the risk premium that you as an individual use to value payoffs from this asset in a given state will be the same as that used by other investors. If risk premia differ across investors, there will be disagreement about valuation and the calculations in this appendix are not possible. Valuing the nontradable stream of cash flows then requires computing the utility value of the payoffs.

12
I

The Black-Scholes
Formula

n 1973 Fischer Black and Myron Scholes (Black and Scholes, 1973) published a formula— the Black-Scholes formula—for computing the theoretical price of a European call option on a stock. Their paper, coupled with closely related work by Robert Merton, revolutionized both the theory and practice of finance. The history of the Black-Scholes formula is discussed in the box on page 350.
In this chapter we present the Black-Scholes formula for pricing European options, explain how it is used for different underlying assets, and discuss the so-called option
Greeks—delta, gamma, theta, vega, rho, and psi—which measure the behavior of the option price when inputs to the formula change. We also show how observed option prices can be used to infer the market’s estimate of volatility. Finally, while there is in general no simple formula comparable to Black-Scholes for valuing options that may be exercised early, perpetual options are an exception. We present the pricing formulas for perpetual
American calls and puts.

12.1 INTRODUCTION TO THE BLACK-SCHOLES FORMULA
To introduce the Black-Scholes formula, we first return to the binomial model, discussed in Chapters 10 and 11. When computing a binomial option price, we can vary the number of binomial steps, holding fixed the time to expiration. Table 12.1 computes binomial call option prices, using the same inputs as in Figure 10.3, and increases the number of steps,
n. Changing the number of steps changes the option price, but once the number of steps becomes great enough we appear to approach a limiting value for the price. The last row reports the call option price if we were to use an infinite number of steps. We can’t literally have an infinity of steps in a binomial tree, but it is possible to show that as the number of steps approaches infinity, the option price is given by the Black-Scholes formula. Thus, the
Black-Scholes formula is a limiting case of the binomial formula for the price of a European option. Call Options
The Black-Scholes formula for a European call option on a stock that pays dividends at the continuous rate δ is
C(S , K , σ , r , T , δ) = Se−δT N (d1) − Ke−rT N (d2)

(12.1)

349

350

BOX

Chapter 12. The Black-Scholes Formula

12.1: The History of the Black-Scholes Formula

T

he Black-Scholes formula was first published in the May/June 1973 issue of the Journal of Political Economy (JPE) (see Black and Scholes,
1973). By coincidence, the Chicago Board Options Exchange (CBOE) opened at almost the same time, on April 26, 1973. Initially, the exchange traded call options on just 16 stocks. Puts did not trade until 1977. In 2000, by contrast, the
CBOE traded both calls and puts on over 1200 stocks. Fischer Black told the story of the formula in Black (1989). He and Myron Scholes started working on the option-pricing problem in 1969, when Black was an independent consultant in
Boston and Scholes an assistant professor at
MIT. While working on the problem, they had extensive discussions with Robert Merton of MIT, who was also working on option pricing.
The first version of their paper was dated October 1970 and was rejected for publication by the

TABLE 12.1

JPE and subsequently by another prominent journal. However, in 1971, Eugene Fama and Merton
Miller of the University of Chicago, recognizing the importance of their work, interceded on their behalf with the editors of the JPE. Later in 1973
Robert Merton published an important and wideranging follow-up paper (Merton, 1973b), which, among other contributions, established the standard no-arbitrage restrictions on option prices discussed in Chapter 9, significantly generalized the Black-Scholes formula and their derivation of the model, and provided formulas for pricing perpetual American puts and down-and-out calls. In 1997, Robert Merton and Myron Scholes won the Nobel Prize in Economics for their work on option pricing. Fischer Black was ineligible for the prize, having died in 1995 at the age of 57.

Binomial option prices for different numbers of binomial steps. As in Figure 10.3, all calculations assume that the stock price S = $41, the strike price K = $40, volatility σ
= 0.30, risk-free rate r = 0.08, time to expiration T = 1, and dividend yield δ = 0.

Number of Steps (n)

Binomial Call Price ($)

1
4
10
50
100
500


7.839
7.160
7.065
6.969
6.966
6.960
6.961

12.1 Introduction to the Black-Scholes Formula

where
1
ln(S/K) + (r − δ + 2 σ 2)T

σ T

d2 = d1 − σ T

d1 =

(12.2a)
(12.2b)

As with the binomial model, there are six inputs to the Black-Scholes formula: S, the current price of the stock; K, the strike price of the option; σ , the volatility of the stock; r, the continuously compounded risk-free interest rate; T , the time to expiration; and δ, the dividend yield on the stock.
The function N (x) in the Black-Scholes formula is the cumulative normal distribution function, which is the probability that a number randomly drawn from a standard normal distribution (i.e., a normal distribution with mean 0 and variance 1) will be less than x.
Most spreadsheets have a built-in function for computing N (x). In Excel, the function is
“NormSDist.” The normal and cumulative normal distributions are illustrated in Figure 18.2 on page 547.
Two of the inputs (K and T ) describe characteristics of the option contract. The others describe the stock (S, σ , and δ) and the discount rate for a risk-free investment (r). All of the inputs are self-explanatory with the exception of volatility, which we discussed in
Section 10.2. Volatility is the standard deviation of the rate of return on the stock—a measure of the uncertainty about the future return on the stock.
It is important to be clear about units in which inputs are expressed. Several of the inputs in equation (12.1) are expressed per unit time: The interest rate, volatility, and dividend yield are typically expressed on an annual basis. In equation (12.1), these inputs are all multiplied by time: The interest rate, dividend, and volatility appear as r × T , δ × T ,

and σ 2 × T (or equivalently, σ × T ). Thus, when we enter inputs into the formula, the specific time unit we use is arbitrary as long as we are consistent. If time is measured in years, then r, δ, and σ should be annual. If time is measured in days, then we need to use the daily equivalent of r, σ , and δ, and so forth. We will always assume inputs are per year unless we state otherwise.
Example 12.1 Let S = $41, K = $40, σ = 0.3, r = 8%, T = 0.25 (3 months), and δ = 0.
Computing the Black-Scholes call price, we obtain1


2
41 ln( 40 ) + (0.08 − 0 + 0.3 ) × 0.25
2

$41 × e−0×0.25 × N ⎝

0.3 0.25


2
41 ln( 40 ) + (0.08 − 0 − 0.3 ) × 0.25
2
⎠ = $3.399
− $40 × e−0.08×0.25 × N ⎝

0.3 0.25
There is one input that does not appear in the Black-Scholes formula—namely, the expected return on the stock. You might guess that stocks with a high beta would have a

1. The call price here can be computed using the Black-Scholes formula call spreadsheet formula, BSCall:
BSCall(S , K , σ , r , t , δ) = BSCall(41, 40, 0.3, 0.08, 0.25, 0) = $3.399

351

352

Chapter 12. The Black-Scholes Formula

higher expected return; hence, options on these stocks would have a higher probability of settlement in-the-money. The higher expected return would seem to imply a higher option price. However, as we saw in Section 11.2, a high stock beta implies a high option beta, so the discount rate for the expected payoff to such an option is correspondingly greater.
The net result—one of the key insights from the Black-Scholes analysis—is that beta is irrelevant: The larger average payoff to options on high beta stocks is exactly offset by the larger discount rate.

Put Options
The pricing formula for a European put follows from put-call parity:
P (S , K , σ , r , T , δ) = C(S , K , σ , r , T , δ) + Ke−rT − Se−δT

(12.3)

Using the Black-Scholes equation for the call, equation (12.1), we can also write the put formula as
P (S , K , σ , r , T , δ) = Ke−rT N (−d2) − Se−δT N (−d1)

(12.4)

where d1 and d2 are given by equations (12.2a) and (12.2b).
Equation (12.4) follows from equations (12.1) and (12.3), together with the fact that for any x, 1 − N (x) = N (−x) (i.e., the probability that a random draw from the standard normal distribution is above x, 1 − N (x), equals the probability that a draw is below −x,
N (−x)).
Example 12.2 Using the same inputs as in Example 12.1, the put price is $1.607. We can compute the put price in two ways. First, computing it using equation (12.4), we obtain2


2
41 ln( 40 ) + (0.08 − 0 − 0.3 )0.25
2

$40e−0.08×0.25N ⎝−

0.3 0.25


2
41 ln( 40 ) + (0.08 − 0 + 0.3 )0.25
2
⎠ = $1.607
− $41e−0×0.25N ⎝−

0.3 0.25
Computing the price using put-call parity, equation (12.3), we have
P (41, 40, 0.3, 0.08, 0.25, 0) = 3.339 + 40e−0.08×0.25 − 41
= $1.607

When Is the Black-Scholes Formula Valid?
The derivation of the Black-Scholes formula makes a number of assumptions that can be sorted into two groups: assumptions about how the stock price is distributed, and

2. The put price here can be computed using the Black-Scholes put spreadsheet formula, BSPut:
BSPut(S , K , σ , r , t , δ) = BSPut(41, 40, 0.3, 0.08, 0.25, 0) = $1.607

12.2 Applying the Formula to Other Assets

assumptions about the economic environment. For the version of the formula we have presented, assumptions about the distribution of the stock price include the following:
.

Continuously compounded returns on the stock are normally distributed and independent over time. (As discussed in Chapter 11, we assume there are no “jumps” in the stock price.)

.

The volatility of continuously compounded returns is known and constant.

.

Future dividends are known, either as a dollar amount or as a fixed dividend yield.

Assumptions about the economic environment include these:
.

The risk-free rate is known and constant.

.

There are no transaction costs or taxes.

.

It is possible to short-sell costlessly and to borrow at the risk-free rate.

Many of these assumptions can easily be relaxed. For example, with a small change in the formula, we can permit the volatility and interest rate to vary over time in a known way. In
Appendix 10.A we discussed why, even though there are taxes, tax rates do not appear in the binomial formula; the same argument applies to the Black-Scholes formula.
As a practical matter, the first set of assumptions—those about the stock price distribution—are the most crucial. Most academic and practitioner research on option pricing concentrates on relaxing these assumptions. They will also be our focus when we discuss empirical evidence. You should keep in mind that almost any valuation procedure, including ordinary discounted cash flow, is based on assumptions that appear strong; the interesting question is how well the procedure works in practice.

12.2 APPLYING THE FORMULA TO OTHER ASSETS
The Black-Scholes formula is often thought of as a formula for pricing European options on stocks. Specifically, equations (12.1) and (12.4) provide the price of a call and put option, respectively, on a stock paying continuous dividends. In practice, we also want to be able to price European options on stocks paying discrete dividends, options on futures, and options on currencies. We have already seen in Chapter 10, Table 10.2, that the binomial model can be adapted to different underlying assets by adjusting the dividend yield. The same adjustments work in the Black-Scholes formula.
We can rewrite d1 in the Black-Scholes formula, equation (12.2a), as d1 =

1 ln(Se−δT /Ke−rT ) + 2 σ 2T

σ T

When d1 is rewritten in this way, it is apparent that the dividend yield enters the formula only to discount the stock price, as Se−δT , and the interest rate enters the formula only to discount the strike price, as Ke−rT . Notice also that volatility enters only as σ 2T .
P
The prepaid forward prices for the stock and strike asset are F0, T (S) = Se−δT and
P
F0, T (K) = Ke−rT . Then we can write the Black-Scholes formula, equation (12.1), entirely

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in terms of prepaid forward prices and σ T : 3
P
P
P
P
C(F0, T (S), F0, T (K), σ , T ) = F0, T (S)N (d1) − F0, T (K)N (d2)

(12.5)

P
P
1 ln[F0, T (S)/F0, T (K)] + 2 σ 2T

σ T

d2 = d1 − σ T

d1 =

This version of the formula is interesting because the dividend yield and the interest rate do not appear explicitly; they are implicitly incorporated into the prepaid forward prices.
To price options on underlying assets other than stocks, we can use equation (12.5) in conjunction with the forward price formulas from Chapters 5 and 6. For all of the examples
P
in this chapter, we will have a strike price denominated in cash, so that F0, T (K) = Ke−rT .

Options on Stocks with Discrete Dividends
When a stock makes discrete dividend payments, the prepaid forward price is
P
F0, T (S) = S0 − PV0, T (Div)

where PV0, T (Div) is the present value of dividends payable over the life of the option.
Thus, using equation (12.5), we can price a European option with discrete dividends by subtracting the present value of dividends from the stock price and entering the result into the formula in place of the stock price. The use of the prepaid forward price here should remind you of the approach to pricing options on dividend-paying stocks in Section 11.4.
Example 12.3 Suppose S = $41, K = $40, σ = 0.3, r = 8%, and T = 0.25 (3 months).
The stock will pay a $3 dividend in 1 month, but makes no other payouts over the life of the option (hence, δ = 0). The present value of the dividend is
PV(Div) = $3e−0.08×1/12 = $2.98
Setting the stock price in the Black-Scholes formula equal to $41 − $2.98 = $38.02, the
Black-Scholes call price is $1.763.
Compared to the $3.399 price computed in Example 12.1, the dividend reduces the option price by about $1.64, or over half the amount of the dividend. Note that this is the price of a European option. An American option might be exercised just prior to the dividend, and hence would have a greater price.

Options on Currencies
We can price an option on a currency by replacing the dividend yield with the foreign interest rate. If the spot exchange rate is x (expressed as domestic currency per unit of

3. We can also let V (T ) = σ T represent total volatility—uncertainty about the relative time-T values of the underlying and strike assets—over the life of the option. The option price can then be written solely in
P
P terms of F0, T (S), F0, T (K), and V (T ). This gives us a minimalist version of the Black-Scholes formula:
To price an option, you need to know the prepaid forward prices of the underlying asset and the strike asset, and the relative volatility of the two.

12.2 Applying the Formula to Other Assets

foreign currency), and the foreign currency interest rate is rf , the prepaid forward price for the currency is
P
F0, T (x) = x0e−rf T

Using equation (12.5), the Black-Scholes formula becomes
C(x , K , σ , r , T , rf ) = xe−rf T N (d1) − Ke−rT N (d2)

(12.6)

1 ln(x/K) + (r − rf + 2 σ 2)T

σ T

d2 = d1 − σ T

d1 =

This formula for the price of a European call on currencies is called the Garman-Kohlhagen model, after Garman and Kohlhagen (1983).
The price of a European currency put is obtained using parity:
P (x , K , σ , r , T , rf ) = C(x , K , σ , r , T , rf ) + Ke−rT − xe−rf T
Example 12.4 Suppose the spot exchange rate is x = $1.25/= , K = $1.20, σ = 0.10, r =
C
1% (the dollar interest rate), T = 1, and rf = 3% (the euro-denominated interest rate). The price of a dollar-denominated euro call is $0.061407, and the price of a dollar-denominated euro put is $0.03641.

Options on Futures
The prepaid forward price for a futures contract is just the present value of the futures price.
Thus, we price a European option on a futures contract by using the futures price as the stock price and setting the dividend yield equal to the risk-free rate. The resulting formula is also known as the Black formula:
C(F , K , σ , r , T , r) = F e−rT N (d1) − Ke−rT N (d2)

(12.7)

1 ln(F /K) + 2 σ 2T

σ T

d2 = d1 − σ T

d1 =

The put price is obtained using the parity relationship for options on futures:
P (F , K , σ , r , T , r) = C(F , K , σ , r , T , r) + Ke−rT − F e−rT
Example 12.5 Suppose the 1-year futures price for natural gas is $6.50/MMBtu and the volatility is 0.25. We have F = $6.50, K = $6.50, σ = 0.25, r = 0.02, T = 1, and δ = 0.02
(the dividend yield is set to equal the interest rate). The Black-Scholes call price and put price are both $0.63379.

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12.3 OPTION GREEKS
Option Greeks are formulas that express the change in the option price when an input to the formula changes, taking as fixed all the other inputs.4 One important use of Greek measures is to assess risk exposure. For example, a market-making bank with a portfolio of options would want to understand its exposure to changes in stock prices, interest rates, volatility, etc. An options investor would like to know how interest rate changes and volatility changes affect profit and loss.
Keep in mind that the Greek measures by assumption change only one input at a time.
In real life, we would expect interest rates and stock prices, for example, to change together.
The Greeks answer the question, What happens when one and only one input changes?
The formulas for the Greeks appear in Appendix 12.B. Greek measures can be computed for options on any kind of underlying asset, but we will focus here on stock options. Definition of the Greeks
The units in which changes are measured are a matter of convention. Thus, when we define a Greek measure, we will also provide the assumed unit of change.
.

Delta ( ) measures the option price change when the stock price increases by $1.

.

Gamma ( ) measures the change in delta when the stock price increases by $1.

.

.

.

.

Vega measures the change in the option price when there is an increase in volatility of 1 percentage point.5
Theta (θ ) measures the change in the option price when there is a decrease in the time to maturity of 1 day.
Rho (ρ) measures the change in the option price when there is an increase in the interest rate of 1 percentage point (100 basis points).
Psi ( ) measures the change in the option price when there is an increase in the continuous dividend yield of 1 per