derivatives markets 2nd by mc donald

derivatives markets 2nd b derivatives markets 2nd b derivatives markets 2nd b derivatives markets 2nd by mc donald derivatives markets 2nd by mc donald derivatives markets 2nd by mc donald derivatives markets 2nd by mc donald

The Addison-Wesley Series in Finance

of Financial Derivatives McDonald

Derivatives Markets Megginson

Cmporate Finance Themy Melvin

International Money and Finance Mishkin/Eakins

Financial Markets and Institutions Moffett

Cases in International Finance Moffett/Stonehill!Eiteman Fundamentals of

Multinational Finance Rejda

Principles of Risk Management and Insurance

Solnik!McLeavey International Investments

Derivatives Markets

Second Edition

Northwestern University Kellogg School of Management

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Library of Congress Cataloging-in-Publication Data

McDonald, Robert L (Robert Lynch),

1954-Derivatives markets, 2e I Robert L McDonald

Fo1· Irene, Claire, David, and Hemy

1.2 The Role of Financial Markets 4

Financial Markets and the Averages 4

Risk-Sharing 5

1.3 Derivatives in Practice 6

Growth in Derivatives Trading 7

How Are Derivatives Used? 10

1.4 Buying and Short-Selling Financial

Assets 11

Buying an Asset 11

Short-Selling 12

The Lease Rate of an Asset 14

Risk and Scarcity in Short-Selling 15

Chapter Sumn!al)' 16

Further Reading 16

Problems 17

HEDGING, AND SIMPLE STRATEGIES

19

and Options 21

2.1 Forward Contracts 21

The Payoff on a Forward Contract 23

Graphing the Payoff on a Forward

Payoff and Profit for a Written Call Option 3 7

The "Moneyness" of an Option 4 3

2.4 Summary of Forward and Option Positions 43

Long Positions 44 Short Positions 44

2.5 Options Are Insurance 45

Homeowner's Insurance Is a Put Option 45 But I Thought Insurance Is Prudent and Put Options Are Risky 47

Call Options Are Also Insurance 4 7

Dividends 5 6

vii

3.1 Basic Insurance Strategies 59

Insuring a Long Position:

3.3- Spreads and Collars 70

Bull and Bear Spread s 7 1

Asymmetric Butterfly Spread s 82

3.5 Example: Another Equity-Linked

Hed ging with a Forward Contract 9 2

Insurance: Guaranteeing a Minimum Price

with a Put Option 9 3

Insuring by Selling a Call 95

Ad justing the Amount of Insurance 96

4.2 Basic Risk Management: The Buyer's

Perspective 98

Hed ging with a Forward Contract 9 8

Insurance: Guaranteeing a Maximum Price with a Call Option 9 9

4.3 Why Do Firms Manage Risk? 100

An Example Where Hed ging Ad d s Value 10 1

Reasons to Hed ge 10 3 Reasons Not to Hed ge 106 Empirical Evidence on Hed ging 106

4.4 Golddiggers Revisited 108

Selling the Gain: Collars 10 8 Other Collar Strategies 1 12 Paylater Strategies 1 13

4.5 Selecting the Hedge Ratio 113

Cross-Hed ging 114 Quantity Uncertainty 116

Chapter Stmti11G1)' 119 Further Reading 120 Problems 120

FUTURES, AND SWAPS 125

Futures 127 5.1 Alternative Ways to Buy a Stock 12 7 5.2 Prepaid Forward Contracts on

5.3 Forward Contracts on Stock 133

Creating a Synthetic Forward Contract 135

Synthetic Forward s in Market-Making and Arbitrage 136 ·

No-Arbitrage Bounds with Transaction

The S&P 500 Futures Contract 14 3

Margins and Marking to Market 144

Comparing Futures and Forward

Prices 14 6

Arbitrage in Practice: S&P 500 Index

Arbitrage 14 7

Quanta Index Contracts 149

5.5 Uses of Index Futures 150

Pencils Have a Positive Lease Rate 17 6

6.5 The Commodity Lease Rate 178

The Lease Market for a Commodity 178 Forward Prices and the Lease Rate 17 9

6.6 Carry Markets 181

Storage Costs and Forward Prices 18 1 Storage Costs and the Lease Rate 18 2 The Convenience Yield 18 2

6.7 Gold Futures 184

Gold Investments 18 7 Evaluation of Gold Production 18 7

6.8 Seasonality: The Corn Forward Market 188

6.9 Natural Gas 191 6.10 Oil 194

6.11 Commodity Spreads 195 6.12 Hedging Strategies 196

Basis Risk 197 Hedging Jet Fuel with Crude Oil 19 9 Weather Derivatives 19 9

Chapter SummGI)' 200 Further Reading 201 Problems 201

Futures 205 7.1 Bond Basics 205

Zero-Coupon Bonds 20 6 Implied Forward Rates 208 Coupon Bonds 210 Zeros from Coupons · 211 Interpreting the Coupon Rate 212 Continuously Compounded Yields 213

7.2 Forward Rate Agreements, Eurodollars, and Hedging 214

Forward Rate Agreements 214 Synthetic FRAs 216

Eurodollar Futures 218 Interest Rate Strips and Stacks 223

Physical Versus Financial Settlement 248

Why Is the Swap Price Not $20 50 ? 250

The Swap Counterpart}' 250

The Market Value of a Swap 25 3

8.2 Interest Rate Swaps 254

A Simple Interest Rate Swap 254

Pricing and the Swap Counterpart}' 255

Computing the Swap Rate in General 257

The Swap Curve 258

The Swap's Implicit Loan Balance 260

Deferred Swaps 26 1

Why Swap Interest Rates? 26 2

Am ortizing and Accreting Swaps 26 3

8.3 Currency Swaps 264

Currency Swap Formulas 267

Other Currency Swaps 267

8.4 Commodity Swaps 268

The Commodity Swap Price 26 8

Swaps with Variable Quantity and

Relationships 281 9.1 Put-Call Parity 281

Options on Stocks 28 3 Options on Currencies 28 6 Options on Bonds 28 6

9.2 Generalized Parity and Exchange Options 287

Options to Exchange Stock 28 8 What Are Calls and Puts? 28 9 Currency Options 290

9.3 Comparing Options with Respect to Style, Maturity, and Strike 292

European Versus American Options 29 3 Maximum and Minimum Option Prices 29 3

Early Exercise for American Options 294 Time to Expiration 297

Different Strike Prices 29 9 Exercise and Moneyness 3 0 4

Chapter Szmmzary• 305 Further Reading 306 Problems 306 Appendix 9.A: Parity Bounds for American Options 310

Appendix 9.B: Algebraic Proofs of Strike-Price Relations 311

Pricing: I 313 10.1 A One-Period Binomial Tree 313

Computing the Option Price 3 14 The Binomial Solution 3 15 Arbitraging a Mispriced Option 3 18

Constructing a Binomial Tree 3 21

Another One-Period Example 3 22

Summary 3 22

Two or More Binomial Periods 323

A Two-Period European Call 3 23

Many Binomial Periods 3 26

Put Options 328

American Options 329

Options on Other Assets 330

Option on a Srock Index 3 3 0

The Risk-Neutral Probability 3 4 6

Pricing an Option Using Real

Probabilities 3 47

11.3 The Binomial Tree and

Lognormality 351

The Random Walk Model 3 5 1

Modeling Stock Prices as a Random

Walk 3 5 2

Continuously Compounded Returns 353

The Standard Deviation of Returns 3 54

The Binomial Model 3 55

Lognormality and the Binomial

Model 3 55

11.4 11.5

CO NTENTS � xi

Alternative Binomial Trees 3 5 8

Is the Binomial Model Realistic? 3 59

Estimating Volatility 360 Stocks Paying Discrete Dividends 361

Modeling Discrete Dividends 3 6 1 Problems with the Discrete Dividend Tree 3 6 2

A Binomial Tree Using the Prepaid Forward 3 6 3

Chapter Summary 365 Further Reading 366 Problems 366 Appendix ll.A: Pricing Options with True

Probabilities 369 Appendix 11.B: Why Does Risk-Neutral Pricing Work? 369

Utility-Based Valuation 3 6 9 Standard Discounted Cash Flow 3 7 1 Risk-Neutral Pricing 3 7 1

Example 3 7 2 Why Risk-Neutral Pricing Works 3 73

Formula 375 12.1 Introduction to the Black-Scholes Formula 375

Call Options 3 7 5 Put Options 378 When Is the Black-Scholes Formula Valid? 3 7 9

12.2 Applying the Formula to Other Assets 379

Options on Srocks with Discrete Dividends 3 8 0

Options on Currencies 3 8 1 Options on Futures 3 8 1

12.3 Option Greeks 382

Definition of the Greeks 3 8 2 Greek Measures for Portfolios 3 88 Option Elasticity 3 8 9

Xii � C O NT E NTS

12.4 Profit Diagrams Before Maturity 395

Purchased Call Option 3 96

Calendar Spreads 3 97

12.5 Implied Volatility 400

Computing Implied Volatility 400

Using Implied Volatility 40 2

12.6 Perpetual American Options 403

Barrier Present Values 403

Interpreting the Profit Calculation 418

Delta-Hedging for Several Days 4 20

A Self-Financing Portfolio: The Stock

.Moves One a 422

13.4 The Mathematics of

Delta-Hedging 422

Using Gamma to Better Approximate the

Change in the Option Price 4 23

Delta-Gamma Approximations 4 24 Theta: Accounting for Time 425 Understanding the Market-Maker's Profit 427

13.5 The Black-Scholes Analysis 429

The Black-Scholes Argument 4 29 Delta-Hedging of American Options 430 What Is the Advantage to Frequent Re-Hedging? 43 1

Delta-Hedging in Practice 432 Gamma-Neutrality 433

13.6 Market-Making as Insurance 436

Insurance 436 Market-Makers 437

Chapter Swnmmy 438 Further Reading 438 Problems 438 Appendix 13.A: Taylor Series Approximations 441 Appendix 13.B: Greeks in the Binomial Model 441

14.1 Introduction 443 14.2 Asian Options 444

XYZ's Hedging Problem 445 Options on the Average 446 Comparing Asian Options 447

An Asian Solution for XYZ 448

Options on Dividend-Paying Stocks 455 Currency Hedging with Compound Options 456

Gap Options 457 Exchange Options 459

European Exchange Options 459

Chapter Summary 461 Further Reading 462

Infinitely Lived Exchange Option 46 8

ENGINEERING AND APPLICATIONS

471

Security Design 473

15.1 The Modigliani-Miller Theorem 473

15.2 Pricing and Designing Structured

15.3 Bonds with Embedded Options 482

Options in Coupon Bonds 482

Options in Equity-Linked Notes 483

Valuing and Structuring an Equity-Linked

Notes with Embedded Options 48 8

15.5 Strategies Motivated by Tax and

Debt and Equity as Options 503 Multiple Debt Issues 511 Warrants 512

Convertible Bonds 513 Callable Bonds 516 Bond Valuation Based on the Stock Price

520 Other Bond Features S20 Put Warrants 522

16.2 Compensation Options 523

Whose Valuation? 525 Valuation Inputs 527

An Alternative Approach to Expensing Option Grants 528

Repricing of Compensation Options 531 Reload Options 532

17.1 Investment and the NPV Rule 548

Static NPV 548 The Correct Use of NPV 549 The Project as an Option 550

17.2 Investment under Uncertainty 551

A Simple DCF Problem 551 Valuing Derivatives on the Cash Flow

552 Evaluating a Project with a 2 -Year Investment Horizon 554 Evaluating the Project with an Infinite Investment Horizon 558

XiV � CO NTENTS

17.3 Real Options in Practice 55 8

Peak-Load Electricity Generation 559

Research and Development 563

Valuing an Infinite Oil Reserve 570

17.5 Commodity Extraction with

Shut-Down and Restart Options 572

Permanent Shutting Down 574

Investment When Shutdown Is

Appendix 17.A: Calculation of Optimal

Time to Drill an Oil Well 583

Appendix 17.B: The Solution with

Shutting Down and Restarting 583

Distribution 587

18.1 The Normal Distribution 587

Converting a-Normal Random Variable to

Standard Normal 590

Sums of Normal Random Variables 591

18.2 The Lognormal Distribution 593

18.3 A Lognormal Model of Stock

18.5 Estimating the Parameters of a Lognormal Distribution 605 18.6 How Are Asset Prices

Distributed? 608

Histograms 6 0 8 Normal Probability Plots 609

Chapter Summary 613 Further Reading 613 Problems 614 Appendix 18.A: The Expectation of a Lognomzal Variable 615

Appendix 18.B: Constructing a Normal Probability Plot 616

19.1 Computing the Option Price as a Discounted Expected Value 617

Valuation with Risk-Neutral Probabilities 618

Valuation with True Probabilities 6 19

19.2 Computing Random Numbers 621

Using Sums of Uniformly Distributed Random Variables 6 22

Using the Inverse Cumulative Normal Distribution 6 22

19.3 Simulating Lognormal Stock Prices 623

Simulating a Sequence of Stock Prices 623

19.4 Monte Carlo Valuation 624

Monte Carlo Valuation of a European Call 6 25

Accuracy of Monte Carlo 6 26 Arithmetic Asian Option 6 27

19.5 Efficient Monte Carlo Valuation 630

Control Variate Method 630 Other Monte Carlo Methods 63 2

19.6 Valuation of American Options 633

19.7 The Poisson Distribution 636

19.8 Simulating Jumps with the Poisson

Definition of Brownian Motion 650

Properties of Brownian Motion 6 5 2

Arithmetic Brownian Motion 6 5 3

The Ornstein-Uhlenbeck Process 654

20.3 Geometric Brownian Motion 655

20.4 The Sharpe Ratio 659

20.5 The Risk-Neutral Process 660

20.6 Ito's Lemma 663

Functions of an Ito Process 66 3

Multivariate Ito's Lemma 665

20.7 Valuing a Claim on sa 666

The Process Followed by S" 667

Proving the Proposition 66 8

Equation 679 21.1 Differential Equations and Valuation under Certainty 679

The Valuation Equation 6 8 0 Bonds 6 8 0

Dividend-Paying Stocks 6 8 1 The General Structure 6 8 1

21.2 The Black-Scholes Equation 681

Verif ying the Formula f or a Derivative 6 8 3

The Black-Scholes Equation and Equilibrium Returns 686 What If the Underlying Asset Is Not an Investment Asset? 6 8 8

21.4 Changing the Numeraire 693 21.5 Option Pricing When the Stock Price CanJump 696

Merton's Solution f or Diversifiable Jumps

697

Chapter Summa/)' 698 Further Reading 698 Problems 699 Appendix 21.A: Multivariate Black-Scholes Analysis 700 Appendix 21.B: Proof of Proposition 21.1

701

22.1 Ali-or-Nothing Options 703

Terminology 70 3 Cash-or-Nothing Options 704 Asset-or-Nothing Options 706

Ali-or-Nothing Barrier Options 710

Cash-or-Nothing Barrier Options 710

Asset-or-Nothing Barrier Options 715

Rebate Options 716

Barrier Options 717

Quanto� 718

The Yen Perspective 7 20

The Dollar Perspective 7 21

A Binomial Model for the

Time-Varying Volatility: ARCH 747

The GARCH Model 751

Realized Quadratic Variation 755

23.3 Hedging and Pricing Volatility 757

Variance and Volatility Swaps 758 Pricing Volatility 759

23.4 Extending the Black-Scholes Model

763

Jump Risk and Implied Volatility 764 Constant Elasticity of Variance 766 The Heston Model 76 8

Evidence 771

Chapter Summa/)' 773 Further Reading 773 Problems 774 Appendix 23.A 777

24.1 Market-Making and Bond Pricing 779

The Behavior of Bonds and Interest Rates 780

An Impossible Bond Pricing Model 7 8 0

A n Equilibrium Equation for Bonds 7 8 1 Delta-Gamma Approximations for Bonds 784

24.2 Equilibrium Short-Rate Bond Price Models 785

The Rendelman-Bartter Model 7 8 5 The Vasicek Model 7 8 6

The Cox-Ingersoll-Ross Model 7 8 7 Comparing Vasicek and CIR 7 8 8

24.3 Bond Options, Caps, and the Black Model 790

24.4 A Binomial Interest Rate Model 793

Zero-Coupon Bond Prices 794 Yields and Expected Interest Rates 796 Option Pricing 797

24.5 The Black-Derman-Toy Model 798

Verifying Yields 8 0 2 Verifying Volatilities 8 03 Constructing a Black-Derman-Toy Tree 8 04

Pricing Examples 8 0 5

Value at Risk f or One Stock 815

VaR f or Two or More Stocks 817

VaR f or Nonlinear Portf olios 81 9

VaR f or Bonds 8 26

Estimating Volatility 830

Bootstrapping Return Distributions 83 1

25.2 Issues with VaR 832

Alternative Risk Measures 83 2

VaR and the Risk-Neutral Distribution

26.1 Default Concepts and Terminology

841

26.2 The Merton Default Model 843

Def ault at Maturity 843

Collateralized Debt Obligations 853

Credit Def ault Swaps and Related

Compounding 875 B.1 The Language of Interest Rates 875 B.2 The Logarithmic and Exponential Functions 876

Changing Interest Rates 877 Symm etry f or Increases and Decreases

878

Problems 878

C.1 Example: The Exponential Function

881 C.2 Example: The Price of a Call 882 C.3 Proof of Jensen's Inequality 884 Problems 884

Visual Basic for Applications 885 D.1 Calculations without VBA 885 D.2 How to Learn VBA 886 D.3 Calculations with VBA 886

Creating a Simple Function 886

A Simple Example of a Subroutine 888 Creating a Button to Invoke a Subroutine

888 Functions Can Call Functions 88 9 Illegal Function Names 88 9 Dif f erences between Functions and Subroutines 890

XViii � CO NTENTS

D.4 Storing and Retrieving Variables in a

Worksheet 890

Using a Named Range to Read and Write

Numbers f rom a Spreadsheet 8 91

Reading and Writing to Cells That Are Not

Named 8 9 2

Using the Cells Functions to Read and

Write to Cells 8 9 2

Reading f rom within a Function 8 93

D.S Using Excel Functions from within

A Simple for Loop 8 9 9

Creating a Binomial Tree 900 Other Kind s of Loops 9 01

D.9 Reading and Writing Arrays 901

Arrays as Outputs 901 Arrays as Inputs 903

D.lO Miscellany 904

Getting Excel to Generate Macros f or You

904 Using Multiple Modules 905 Recalculation Speed 905 Debugging 9 06

Creating an Add-In 906

Deriva6v;, have moved to the cent« of modem coq>nmte finance, ;nve"men", and the management of financial institutions They have also had a profound impact on other management functions such as business strategy, operations management, and marketing A major drawback, however, to making the power of derivatives accessible

to students and practitioners alike has been the relatively high degree of mathematical sophistication required for understanding the underlying concepts and tools

that is a wonderful blend of the economics and mathematics of derivatives pricing and easily accessible to MBA students and advanced undergraduates It is a special pleasure for me to introduce this new edition, since I have long had the highest regard for the author's professional achievements and personal qualities

The book' s orientation is neither overly sophisticated nor watered down, but rather

a mix of intuition and rigor that creates an inherent flexibility for the structuring of a derivatives course The author begins with an introduction to forwards and futures and

ment He looks in detail at forwards and futures on stocks, stock indices, currencies, interest rates, and swaps His treatment of options then follows logically from con­cepts developed in the earlier chapters The heart of the text-an extensive treatment

of the binomial.option model and the Black-Scholes equation-showcases the author' s crystal-clear writing and logical development of concepts Excellent chapters on finan­cial engineering, security design, corporate applications, and real options follow and shed light on how the concepts can be applied to actual problems

The last third of the text provides an advanced treatment of the most important concepts of derivatives discussed earlier This part can be used by itself in an advanced derivatives course, or as a useful reference in introductory courses A rigorous de­velopment of the Black-Scholes equation, exotic options, and interest rate models are presented using Brownian Motion and Ito's Lemma Monte Carlo simulation methods are also discussed in detail New chapters on volatility and credit risk provide a clear discussion of these fast-developing areas

Derivatives concepts are now required for every advanced finance topic There­fore, it is essential to introduce these concepts at an early stage of MBA and under­graduate business or economics programs, and in a fashion that most students can un­derstand This text achieves this goal in such an appealing, inviting way that students will actually enjoy their journey toward an understanding of derivatives

EDUARDO 5 SCHWARTZ

xix

ThiTiy yea.-' ago the Blaok-Scho!o' focmula wa, new, and derivative< wa, an e<oteric and

modern finance For example, corporations routinely hedge and insure using deriva­tives, finance activities with structured products, and use derivatives models in capital budgeting This book will help you to understand the derivative instruments that exist, how they are used, who sells them, how they are priced, and how the tools and concepts are useful more broadly in finance

Derivatives is 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 Most of the book should thus be accessible to anyone who has studied elementary finance For those who want to understand the subject at a deeper level, the last part of the book

standard mathematical tools used in option pricing, such as Ito's Lemma There are also chapters dealing with applications: corporate applications, financial engineering, and real options

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 1 8

Most o f the calculations i n this book can b e replicated using Excel spreadsheets on the CD-ROM that comes with the book 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 easily 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 built-in Excel functions are also mentioned throughout the book

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

xxi

xxii � P R E FA C E

to the first two-thirds and last third of the book The first course i s 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 as fully as possible No one expects that a 1 0-week MBA-level course will produce rocket scien­tists, but mathematics is the language of derivatives and it would be cheating students

to pretend otherwise

You may want to cover the material in a different order than it occurs in the book,

wrote the book expecting that the chapters on lognormality and Monte Carlo simulation might be used in a first derivatives course

blocks o{ 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 The main question is: What is the pre-specified price, and how is it determined? Chapter

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 1 1 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

covers financial engineering, which is the creation of new financial products from the derivatives building blocks in earlier chapters Debt and equity pricing, compensation

application of derivatives models to the valuation and management of physical invest­ments

explains in more detail the structure and assumptions underlying the standard derivatives models Chapter 1 8 covers the lognormal model and shows how the Black-Scholes formula is an expected value Chapter 19 discusses Monte Carlo valuation, a powerful and commonly used pricing technique Chapter 20 explains what it means to say that

WHAT Is N EW I N TH E S ECO N D E D ITI O N � xxiii

stock prices follow a diffusion process, and also covers Ito's Lemma, which is a key result in the study of derivatives (At this point you will discover that Ito's Lemma has already been developed intuitively in Chapter 13, using a simple numerical example.) Chapter 21 derives the Black-Scholes partial differential equation (PDE) Although the Black-Scholes formula is famous, the Black-Scholes equation, discussed in this chapter, is the more profound result Chapter 22 covers exotic options in more detail than Chapter 14, including digital barrier options and quantos Chapter 23 discusses volatility_ estimation and stochastic volatility pricing models Chapter 24 shows how the Black-Scholes and binomial analysis apply to bonds and interest rate derivatives Chapter 25 covers value-at-risk, and Chapter 26 discusses the burgeoning market in credit products

WHAT IS NEW IN THE SECOND EDITION

There are two new chapters in this edition, covering volatility and credit risk:

• Chapter 23 covers empirical volatility models, such as GARCH and realized volatility; financial instruments that can be used to hedge volatility, such as vari­ ance swaps; and pricing models that incorporate j umps and stochastic volatility, such as the Heston model

• Chapter 26 covers structural models of bankruptcy risk (the Merton model); tranched structures such as collateralized debt obligations; credit default swaps and credit indexes

There are numerous changes and new examples throughout the book Among the more important changes are the following:

o An expanded discussion of bond convexity

o An expanded treatment of computing hedge ratios

o An expanded treatment of convertible and callable bonds

• Discussion of the new option expensing rules in FAS 123R and the B ulow-Shoven expensing proposal

• Discussion of a variable prepaid forward on Disney stock issued by Roy Disney

• In-depth discussion of a mandatorily convertible bond issued by Marshall & Ilsley, including pricing and structuring

• The use of simulation to price American options

• Additional discussion of implied volatility

• Enhanced discussion of the link between discounted cash flow valuation and risk­ neutral valuation

:xxiv � P R E FACE

• A n expanded discussion o f value-at-risk

• New spreadsheet functions for pricing options with fixed dividends, CEV option pricing, the Merton jump model, and others

NAVIGATING THE MATERIAL

There are potentially many ways to cover the material in this book The material is generally presented in order of increasing mathematical difficulty, which means that related material is sometimes split across distant chapters For example, fixed income is covered in Chapters 7 and 24, and exotic options in Chapters 14 and 22 Each of these chapters is at the level of the neighboring chapters As an illustration of one way to use the book, here is the material I cover in the courses I teach (within the chapters I skip some specific topics due to time constraints):

Many of the numerical examples in this book display intermediate steps to assist you

in following the calculations In most cases it will also be possible for you to cre­ ate a spreadsheet and compute the same answers starting from the basic assumptions However, numbers displayed in the text are generally 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 The displayed calculations will help you follow the logic of a calculation, but a spreadsheet will be helpful in reproducing the final result

SUPPLEMENTS

A robust package of ancillary materials for both instructors and students accompanies the text

S U P PL E M E NTS � XXV

Possible chapters for different courses Chapters marked with a

"y" are strongly recommended, those marked with a "*" are recommended, and those with a "t" 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 backgroundfor all introductory courses

Introductory

Risk General Futures Options Manageme ? t Advanced

3 Insurance, Collars, and Other Strategies y y y y

9 Parity and Other Option Relationships * t y t

II Binomial Option Pricing: II * :::

XXVi � P R EFAC E

An online Instructor's Solutions Manual by Mark Cassano, University of Cal­ gary, and Rudiger Fahlenbrach, Ohio State University, contains complete solutions to all end-of-chapter problems in 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 both print and electronic formats, including Windows

or Macintosh TestGen files and Microsoft Word files The TestGen and Test Bank are available online at http://www.aw-bc.com/irc

Online PowerPoint slides, developed by Charles Cao, Pennsylvania State Uni­ versity; Ufuk Ince, University of Washington; and Ekaterina Emm, Georgia State Uni­ versity, provide lecture outlines and selected art from the book Copies of the slides can

be downsized and distributed to students to facilitate note taking during class

The Instructors Resource Disk contains the computerized Test Bank files (Test­ Gen), the Instructor Manual files (Word), the Test B ank files (Word) and PowerPoint files

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 Web site

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 the second edition, I received invaluable assistance from Rudiger Fahlenbrach, Ohio State University, who read much of the new material with a critical eye, and who both caught mistakes and offered valuable suggestions Numerous other students, colleagues, and readers provided comments on the first edition Colleagues in the Kellogg finance department who were generous with their time include Torben An­ dersen, Kathleen Hagerty, Ravi Jagannathan, Deborah Lucas, Mitchell Petersen, Ernst Schaumburg, Costis Skiadas, and David Stowell Many Kellogg MBA and Ph.D stu­ dents helped, but I want to especially thank Arne Staal, Caroline Sasseville, and Alex

AC K N OW LE D G M E NTS � XXVU

Wolf Others who reviewed new material include David Bates, University of Iowa; Luca Benzoni, University of Minnesota; Mikhail Chernov, Columbia University; and Darrell Duffie, Stanford University Mark Schroder, Michigan State University, kindly provided code to calculate the non-central chi-squared distribution, and Kellogg student Scott Freemon implemented this code in VBA

A special note of thanks goes to David Hait, president of OptionMetrics, for per­ mission to include options data on the CD-ROM

I also received help and comments from George Allayanis, University of Vir­ ginia; Jeremy Bulow, Stanford University; Raul Guerrero, Dynamic Decisions; Darrell Karolyi, Compensation Strategies, Inc.; C F Lee, Rutgers University; David Nachman, University of Georgia; Ani! Shivdasani, University of North Carolina;- and Nicholas Wonder, Western Washington University

I would like to particularly thank those who provided valuable feedback for the second edition, including Turan B ali, Baruch College, City University of New York; Philip Bond, Wharton School, University of Pennsylvania; Michael Brandt, Duke Uni­ versity; Charles Cao, Pennsylvania State University; B ruce Grundy, Melbourne Business School, Australia; Shantaram Hegde, University of Connecticut; Frank Leiber, B ell At­ lantic ; Ehud Ronn, University of Texas, Austin; Nejat Seyhun, University of Michigan; John Stansfield, University of Missouri, Columbia; Christopher Stivers, University of Georgia; Joel Vanden, Dartmouth College; and Guofu Zhou, Washington University, St Louis

I would be remiss not to acknowledge those who assisted with the first edition, including Tom Arnold, Louisiana State University; David Bates, University of Iowa; Luca Benzoni, University of Minnesota; Mark B roadie, Columbia University; Mark A Cassano, University of Calgary; George M Constantinides, University of Chicago; Kent Daniel, Northwestern University ; Jan Eberly, Northwestern University; Virginia France, University of Illinois; Steven Freund, Suffolk University; Rob Gertner, University of Chicago; Kathleen Hagerty, Northwestern University; David Haushalter, University of Oregon; James E Hodder, University of Wisconsin-Madison; Ravi Jagannathan, North­ western University; Avraham Kamara, University of Washington; Kenneth Kavajecz, Whartori School, University of Pennsylvania; Arvind Krishnamurthy, Northwestern Uni­ versity; Dennis Lasser, State University of New York at B inghamton; Camelis A Los, Kent State University; Deborah Lucas, Northwestern University; Alan Marcus, B oston College; Mitchell Petersen, Northwestern University; Todd Pulvino, Northwestern Uni­ versity; Ernst Schaumburg, Northwestern University ; Eduardo Schwartz, University of California-Los Angeles; David Shimko, Risk Capital Management Partners, Inc.; Ani! Shivdasani, University of North Carolina-Chapel Hill ; Costis Skiadas, Northwestern University; Donald S mith, Boston University; David Stowell, Northwestern University; Alex Triantis, University of Maryland; and Zhenyu Wang, Yale University The follow­ ing served as software reviewers: James B ennett, University of Massachusetts-Boston; Gordon H Dash, University of Rhode Island; Adam Schwartz, University of Mississippi; and Robert E Whaley, Duke University

Special thanks are due to George Constantinides, Jennie France, Kathleen Hagerty, Ken Kavajecz, Alan Marcus, Costis Skiadas, and Alex Triantis for their willingness to

XXViii � P R E FA C E

read and comment upon some of the material multiple times an d for class-testing Mark Broadie generously provided his pricing software, which I used both to compute the Heston model and to double-check my own calculations

I thank Rudiger Fahlenbrach, Mark Cassano, Matt Will, and Charles Cao for their excellent work on the ancillary materials for this book In addition, Rudiger Fahlenbrach, Paskalis Glabadanidis, Jeremy Graveline, Dmitry Novikov, and Krish­namurthy Subramanian served as accuracy checkers for the book and Andy Kaplin provided programming assistance

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 Every derivatives book is deeply in his debt, and this one is no exception His classic papers from the 1970s are as essential today as they were 3o" 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 199 1 at the age of 35 was

a great loss to the profession, as well as to me personally

The editorial and production team at Addison-Wesley made it clear from the outset that their goal was to produce a high-quality book I was lucky to have the project over­seen by Addison Wesley's talented and tireless Finance Editor, Donna Battista Project Manager Mary Clare McEwing expertly kept track of myriad details and offered ex- cellent advice when I needed a sounding· board Development Editor Mmjorie Singer Anderson offered innumerable suggestions, improving the manuscript significantly Pro­duction Supervisor, Nancy Fenton marshalled forces to tum manuscript into a physical book Among those forces were the excellent teams at Elm Street Publishing Services and Techsetters I received numerous compliments on the design of the first edition, which has been carried through ably into the second Kudos are due to Gina Kolenda and Rebecca Light for their creativity in text and cover design

The Addison-Wesley 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.aw-bc.com/mcd onald Please let us know if you do find errors so we can update the list

I produced the original manuscript and revision using Gnu Emacs and MikTeX, extraordinarily powerful and robust tools for authors I am deeply grateful to the world­wide community that produces and supports this software

My deepest and most heartfelt thanks go to my family Through both 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

AC K N OWLE D G M E NTS � xxix Robert L McDonald is E1win P Nemmers Distinguished Professor of Finance at North­ western University 's Kellogg School of Management, where he has taught since 1984

He is co-Editor of the Review of Financial Studies and has been Associate Editor of

Carolina at Chapel Hill and a Ph.D in Economics from MIT

last 30 years Simple stocks and bonds now seem almost quaint alongside the dazzling, fast-paced, and seemingly arcane world of futures, options, swaps, and other "new" financial products (The word "new" is in quotes because it turns out that some of these products have been around for hundreds of years )

Frequently this world pops up in the popular press: Procter & Gamble lost

$ 150 million in 1994, B arings bank lost $ 1 3 billion in 1995, Long-Term Capital Man­ agement lost $3.5 billion in 1998 and (according to some press accounts) almost brought the world financial system to its knees 1 What is not in the headlines is that, most of the time, for most companies and most users, these financial products are an 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 j ust a few examples

1 1 WHAT IS A DERIVATIVE ?

Options, futures, and swaps are examples of derivatives A derivative is a financial instrument (or more simply, an agreement between two people) that has a value deter­ mined by the price of something else For example, a bushel of com is not a derivative;

it is a commodity with a value determined by the price of corn However, you could enter into an agreement with a friend that says: If the price of a bushel of com in one year is greater than $3, you will pay the friend $ 1 If the price of com 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 (com, in this case)

You might think: "That's not a derivative; that's just a bet on the price of com."

So it is: Derivatives can be thought of as bets on the price of something But don ' t

1 A readable summary o f these and other infamous derivatives-related losses i s in Jorion (200 1 )

1

� I NTRO D U CT I O N TO D E R I VATI VES

automatically think the term "bet" is 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 could also use this kind of contract simply to speculate on the price of

contrac._r itself, but how it is used, and who uses it, that determines whether or not it is risk-reducing Context is everything

Although we' ve just defined a derivative, if you are new to the subject the impli­cations of the definition will probably 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

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 The corn example above illustrates this in a simple way: The farmer-a seller

of com-enters into a contract which makes a payment when -the price of corn is low

It is common to think of derivatives· as forbiddingly complex, but many derivatives are simple and familiar Every form of insurance is a derivative, for example Automobile

tree, your insurance is valuable; if the car remains intact, 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 1 300 and 1400 one year from today, derivatives can be constructed to let you do that

Reduced transaction costs Sometimes derivatives provide a lower-cost way to effect

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 of holding

W H AT I S A D E R I VATI V E? � 3

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

The economic observer Finally, we can look at the use of derivatives, the activities of the market-makers, the organization of the markets, 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, but as a general point, in the early chapters the book emphasizes the end-user perspective In the late chapters, the book emphasizes the market-maker perspective At all times, however, the economic observer is interested in making sense of everything

Financial Engineering and Security Design

possible to create a given payoff in multiple ways The construction of a given financial

� I NTRO D U CTI O N TO D E R I VAT I V ES

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 market­maker 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

A theme running throughout the book is that derivative products can generally be constructed from other products

1.2 THE ROLE OF FINANCIAL MARKETS

We take for granted headlines saying that the Dow Jones Industrial Average has gone up

1 00 points, the dollar has fallen against the yen, and interest rates have risen B ut why

do we care about these things? Is the rise and fall of a particular financial index (such

as the Dow Jones Industrial Average) simply a way to keep score, to track winners and losers in the economy? Is watching the stock market like watching sports, where we root for certain players and teams-a tale told by journalists, full of sound and fury, but signifying nothing?

Financial markets in fact have an enormous, often underappreciated, impact on everyday life To help us understand the role of financial markets we will consider the Average family, living in Anytown Joe and Sarah Average have 2.3 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 What is left over goes toward savings earmarked for their children's college tuition and their own retirement What role do global financial markets and derivatives play in the lives of the Averages?

Financial Markets and the Averages

The Averages are largely unaware of the ways in which financial markets affect their lives Here are a few:

operations and investments It is not dependent on the local bank for funds because

it can raise the money it needs by issuing stocks and bonds in global markets

TH E RO LE O F F I N A N C I A L M A R K ETS � 5

casualty insurance for its buildings, it uses global derivatives markets to protect itself against adverse currency, interest rate, and commodity price changes By

likely to throw the Averages into unemployment

than if they tried to achieve comparable diversification by buying individual stocks

times they will lose their jobs The mutual funds in which they invest own stocks

in a broad array of companies, ensuring that the failure of any one company will not wipe out their savings

their insurance company were completely local, it could not offer tornado insurance because one disaster would leave it unable to pay claims By selling tornado risk

in global markets, the insurance company can in effect pool Anytown tornado risk with Japan earthquake risk and Florida hurricane risk This pooling makes insurance available at lower rates

sold the mortgage to other investors, freeing itself from interest rate and default risk associated with the mortgage, leaving that to others Because the risk of their mortgage is borne by those willing to pay the highest price for it, the Averages get the lowest possible mortgage rate

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 An insurance company does not bear all the risk of a disaster Risk-sharing is one of the most important functions of financial markets

In the face of this risk, it seems natural to have arrangements where the lucky share with the unlucky Risk-sharing occurs informally in families and communities The insurance market makes formal risk-sharing possible 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 bum down The lucky, meanwhile, did not need insurance and have lost their premium The market makes it possible for the lucky to help the unlucky

6 � I NTRO D U CTI O N TO D E R I VATI V ES

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

Reinsurers pool different kinds of risks, thereby enabling insurance risks to become more widely held

that the issuer need not repay if there is a specified event, such as a large earthquake, causing large insu(ance 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

Although there are mechanisms for sharing many kinds of risks, some have argued that significantly more risk-sharing is possible and desirable The economist Robert Shiller (Shiller, 2003) envisions the creation of entirely new markets for risk-sharing, including home equity insurance (to trade risks associated with house prices), income­

occupation), and macro insurance (contracts with payments linked to national incomes) While these markets do not yet exist, there is a trend toward more inclusive markets for risk transfer For example, Goldman Sachs and Deutsche Bank have recently created an

"economic derivatives" market in which it is possible to buy claims with payouts based

on economic statistics The box on page 7 discusses this market

You might be wondering what this discussion has to do with the notions of diver­

risk if H is unrelated to other risks The risk that a lightning strike will cause a factory

share a small piece of this risk, it has no significant effect on anyone Risk that does not

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 mm*ets discussed in this book is that the existence of risk-sharing mechanisms benefits evei)'Oile

1.3 DERIVATIVES IN PRACTICE

Derivatives use and the variety of derivatives have grown over the last 30 years

Economic Derivatives

periodically announce economic statistics,

such as the number of jobs in the economy,

production in different sectors, and the level

of sales These statistics provide information

about the performance of the economy

and-since policy makers rely upon

them can provide clues to future government

policy Consequently, money managers,

dealers, and other market participants pay

close attention to these statistics; their release

often results in a flurry of trading activity and

changes in stock and bond prices

Because of the importance of these

statistics for financial markets, Goldman

Sachs and Deutsche Bank created a market in

economic derivatives, in which it is possible

to trade claims with payoffs based on these

statistics Specifically, beginning in late 2002,

it became possible to trade claims based on

employment (U.S nonfarm payrolls),

industrial production (the Purchasing

Growth in Derivatives Trading

Whereas it is possible to trade stocks and bonds on any business day, the market for most economic derivatives is open only briefly before the government releases the statistic Specifically, if the nonfarm payroll number is to be released on a Friday, on the day before there will be one hour during which participants can submit orders to buy or sell various derivatives based on the nonfarm payroll number At the end of the hour, buyers are matched against sellers and prices are determined using a procedure known as a Dutch auction.2 This market permits the trading of many different kinds of claims that

we will discuss in later chapters, including forwards, calls, puts, spreads, straddles, strangles, and digital options

The introduction of derivatives in a market often coincides with an increase in price risk in that market Currencies were officially permitted to float in 1 97 1 when the gold standard was officially abandoned OPEC's 1 973 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 1 970s The market for natural gas has been deregulated

of electricity began during the 1990s Figures 1 1 , 1 2, and 1 3 show the changes for oil prices, exchange rates, and interest rates The link between price variability and the development of derivatives markets is natural-there is no need to manage risk when

2 In a Dutch auction there is a single price that is paid by buyers who are willing to pay that price or

more, and that is received by sellers who are willing to accept that price or less

8 � I NT RO D U CTI O N TO D E R I VATI V ES

Monthly percentage

change in the producer

price index for oil,

c:: u

co

"'

I:

u Ill

3It is sometimes argued that the existence of derivatives markets can increase the price variability of

the underlying asset or commodity Without some price risk in the first place, however, the derivatives

market is unlikely to exist

traded annually at the

Chicago Board of Trade

Source: CRB Commodity Yearbook

A futures exchange is an organized and regulated marketplace for trading futures contracts, a kind of derivative Figure 1 4 depicts futures contract volume for the three largest U.S futures exchanges over the last 30 years Clearly, the usage of futures con­tracts has grown significantly Exchanges in other countries have generally experienced comparable or greater growth In 2002, Eurex, an electronic exchange headquartered

in Frankfurt, Germany, traded 528 million contracts, the largest trading volume of any futures exchange in the world