Sundaram & Das

Derivatives: Principles and Practice Rangarajan K. Sundaram Stern School of Business New York University New York, NY 10012

Sanjiv R. Das Leavey School of Business Santa Clara University Santa Clara, CA 95053

I McGraw-Hill I Irwin

Contents 3.8 Futures Prices 72 3.9 Exercises 74 Appendix 3A Compounding Frequency 79 Appendix 3B Forward and Futures Prices with Constant Interest Rates 81 Appendix 3C Rolling Over Futures Contracts 83

Author Biographies xv Preface xvi Acknowledgments xxi Chapter 1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6

1

Chapter 4

Forward and Futures Contracts 5 Options 9 Swaps 10 Using Derivatives: Some Comments The Structure of this Book 14 Exercises 15

Pricing Forwards and Futures II: Building on the Foundations 85 11

PART ONE Futures and Forwards Chapter 2 Futures Markets

17

19

2.1 Introduction 19 2.2 The Changing Face of Futures Markets 19 2.3 The Functioning of Futures Exchanges 21 2.4 The Standardization of Futures Contracts 30 2.5 Closing Out Positions 34 2.6 Margin Requirements and Default Risk 36 2.7 Case Studies in Futures Markets 39 2.8 Exercises 53 Appendix 2A Futures Trading and US Regulation: A Brief History 57

Chapter 3 Pricing Forwards and Futures I: The Basic Theory 60 3.1 3.2 3.3 3.4

Introduction 60 Pricing Forwards by Replication 61 Examples 63 Forward Pricing on Currencies and Related Assets 66 3.5 Forward-Rate Agreements 69 3.6 Concept Check 69 3.7 The Marked-to-Market Value of a Forward Contract 70 viii

4.1 Introduction 85 4.2 From Theory to Reality 85 4.3 The Implied Repo Rate 89 4.4 Transactions Costs 92 4.5 Forward Prices and Future Spot Prices 92 4.6 Index Arbitrage 93 4.7 Exercises 97 Appendix 4A Forward Prices with Convenience Yields 100

Chapter 5 Hedging with Futures and Forwards

101

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

Introduction 101 A Guide to the Main Results 103 The Cash Flow from a Hedged Position 104 The Case of No Basis Risk 105 The Minimum-Variance Hedge Ratio 106 Examples 109 Implementation 111 Further Issues in Implementation 112 Index Futures and Changing Equity Risk 114 Fixed-Income Futures and Duration-Based Hedging 115 5.11 Exercises 115 Appendix 5A Derivation of the Optimal Tailed Hedge Ratio h** 120

Chapter 6 Interest-Rate Forwards and Futures 6.1 6.2 6.3 6.4

Introduction 122 Eurodollars and Libor Rates 122 Forward-Rate Agreements 123 Eurodollar Futures 129

122

Contents ix

6.5 Treasury Bond Futures 136 6.6 Treasury Note Futures 139 6.7 Treasury Bill Futures 139 6.8 Duration-Based Hedging 140 6.9 Exercises 143 Appendix 6A Deriving the Arbitrage-Free FRA Rate 147 Appendix 6B PVBP-Based Hedging Using Eurodollar Futures 148 Appendix 6C Calculating the Conversion Factor 149 Appendix 6D Duration as a Sensitivity Measure 150 Appendix 6E The Duration of a Futures Contract 151

Chapter 9 No-Arbitrage Restrictions on Option Prices 196

PART TWO

10.1 10.2 10.3 10.4 10.5

Options

153

Chapter 7 Options Markets

9.5 9.6 9.7 9.8

Introduction 196 Motivating Examples 196 Notation and Other Preliminaries 198 Maximum and Minimum Prices for Options 199 The Insurance Value of an Option 204 Option Prices and Contract Parameters 205 Numerical Examples 208 Exercises 210

Chapter 10 Early Exercise and Put-Call Parity 213 Introduction 213 A Decomposition of Option Prices 213 The Optimality of Early Exercise 216 Put-Call Parity. 220 Exercises 226

155

7.1 7.2 7.3 7.4 7.5

Introduction 155 Definitions and Terminology 155 Options as Financial Insurance 156 Naked Option Positions 158 Options as Views on Market Direction and Volatility 162 7.6 Exercises 165 Appendix 7A Options Markets 167

Chapter 8 Options: Payoffs and Trading Strategies 171 8.1 8.2

9.1 9.2 9.3 9.4

Introduction 171 Trading Strategies I: Covered Calls and Protective Puts 171 8.3 Trading Strategies II: Spreads 174 8.4 Trading Strategies III: Combinations 182 8.5 Trading Strategies IV: Other Strategies 185 8.6 Which Strategies Are the Most Widely Used? 189 8.7 The Barings Case 189 8.8 Exercises 192 Appendix 8A Asymmetric Butterfly Spreads 195

Chapter 11 Option Pricing: An Introduction 11.1 11.2 11.3

228

Overview 228 The Binomial Model 229 Pricing by Replication in a One-Period Binomial Model 231 11.4 Comments 235 11.5 Riskless Hedge Portfolios 237 11.6 Pricing Using Risk-Neutral Probabilities 238 11.7 The One-Period Model in General Notation 242 11.8 The Delta of an Option 242 11.9 An Application: Portfolio Insurance 246 11.10 Exercises 248 Appendix 11A Riskless Hedge Portfolios and Option Pricing 252 Appendix 11B Risk-Neutral Probabilities and Arrow Security Prices 254 Appendix 11C The Risk-Neutral Probability, No-Arbitrage, and Market Completeness 255 Appendix 11D Equivalent Martingale Measures 257

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Contents

Chapter 12 Binomial Option Pricing 259 12.1 12.2 12.3 12.4

Introduction 259 The Two-Period Binomial Tree 261 Pricing Two-Period European Options 262 European Option Pricing in General w-Period Trees 269 12.5 Pricing American Options: Preliminary Comments 269 12.6 American Puts on Non-Dividend-Paying Stocks 270 12.7 Cash Dividends in the Binomial Tree 272 12.8 An Alternative Approach to Cash Dividends 275 12.9 Dividend Yields in Binomial Trees 279 12.10 Exercises 282 Appendix 12A A General Representation of European Option Prices 286

Chapter 13 Implementing the Binomial Model 289 13.1 13.2 13.3

Introduction 289 The Lognormal Distribution 289 Binomial Approximations of the Lognormal 294 13.4 Computer Implementation of the Binomial Model 298 13.5 Exercises 303 Appendix 13A Estimating Historical Volatility 306

Chapter 14 The Black-Scholes Model 308 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9

Introduction 308 Option Pricing in the Black-Scholes Setting 310 Remarks on the Formula 313 Working with the Formulae I: Plotting Option Prices 314 Working with the Formulae II: Algebraic Manipulation 315 Dividends in the Black-Scholes Model 319 Options on Indices, Currencies, and Futures 324 Testing the Black-Scholes Model: Implied Volatility 327 The VIX and Its Derivatives 332

14.10 Exercises 335 Appendix 14A Further Properties of the Black-Scholes Delta 338 Appendix 14B Variance and Volatility Swaps

339

Chapter 15 The Mathematics of Black-Scholes 344 15.1 15.2 15.3 15.4 15.5 15.6

Introduction 344 Geometric Brownian Motion Defined 344 The Black-Scholes Formula via Replication 348 The Black-Scholes Formula via Risk-Neutral Pricing 351 The Black-Scholes Formula via CAPM 353 Exercises 354

Chapter 16 Options Modeling: Beyond Black-Scholes 357 16.1 16.2 16.3 16.4 16.5 16.6

Introduction 357 Jump-Diffusion Models 358 Stochastic Volatility 368 GARCH Models 374 Other Approaches 378 Implied Binomial Trees/Local Volatility Models 379 16.7 Summary 389 16.8 Exercises 389 Appendix 16A Program Code for JumpDiffusions 393 Appendix 16B Program Code for a Stochastic Volatility Model 394 Appendix 16C Heuristic Comments on Option Pricing under Stochastic Volatility 396 Appendix 16D Program Code for Simulating GARCH Stock Prices Distributions 399 Appendix 16E Local Volatility Models: The Fourth Period of the Example 400

Chapter 17 Sensitivity Analysis: The Option "Greeks" 404 17.1 17.2

Introduction 404 Interpreting the Greeks: A Snapshot View 404

Contents xi

17.3 The Option Delta 408 17.4 The Option Gamma 412 17.5 The Option Theta 418 17.6 The Option Vega 423 17.7 The Option Rho 426 17.8 Portfolio Greeks 429 17.9 Exercises 432 Appendix 17A Deriving the Black-Scholes Option Greeks 436

Chapter 18 Exotic Options I: Path-Independent Options 440 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8

Introduction 440 Forward Start Options 442 Binary Options 445 Chooser Options 450 Compound Options 453 Exchange Options 458 Quanta Options 460 Variants on the Exchange Option Theme 462 18.9 Exercises 465

Chapter 19 Exotic Options II: Path-Dependent Options 470 19.1

Path-Dependent Exotic Options 470 _ 19.2 Barrier Options 470 19.3 Asian Options 479 19.4 Lookback Options 485 19.5 Cliquets 488 19.6 Shout Options 490 19.7 Exercises 492 Appendix 19A Barrier Option Pricing Formulae 496

Chapter 20 Value-at-Risk 498 20.1 20.2 20.3 20.4 20.5

Introduction 498 Value-at-Risk 498 Risk Decomposition 505 Coherent Risk Measures 511 Exercises 515

Chapter 21 Convertible Bonds

519

21.1 Introduction 519 21.2 Convertible Bond Terminology 519 21.3 Main Features of Convertible Bonds 520 21.4 Breakeven Analysis 522 21.5 Pricing Convertibles: A First Pass 523 21.6 Incorporating Credit Risk 530 21.7 Convertible Greeks 534 21.8 Convertible Arbitrage 542 21.9 Summary 542 21.10 Exercises 543 Appendix 21A Octave Code for the Blended Discount Rate Valuation Tree 545 Appendix 21B Octave Code for the Simplified Das-Sundaram Model 546

Chapter 22 Real Options 548 22.1 22.2 22.3 22.4 22.5 22.6 22.7

Introduction 548 Preliminary Analysis and Examples 550 A Real Options "Case Study" 554 Creating the State Space 560 Applications of Real Options 563 Summary 564 Exercises 564 Appendix 22A Derivation of Cash-Flow Value in the "Waiting-to-Invest" Example 568

PART THREE Swaps 569 Chapter 23 Interest Rate Swaps and Floating-Rate Products 571 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8

Introduction 571 Floating-Rate Notes 571 Interest Rate Swaps 575 Uses of Swaps 576 Swap Payoffs 579 Valuing and Pricing Swaps 582 Extending the Pricing Arguments 586 Case Study: The Procter & Gamble-Bankers Trust "5/30" Swap 589

xii Contents

23.9

Case Study: A Long-Term Capital Management "Convergence Trade" 593 23.10 Credit Risk and Credit Exposure 596 23.11 Hedging Swaps 597 23.12 Caps, Floors, and Swaptions 599 23.13 The Black Model for Pricing Caps, Floors, and Swaptions 604 23.14 Summary 609 23.15 Exercises 609

Chapter 24 Equity Swaps 613 24.1 24.2 24.3 24.4 24.5 24.6

Introduction 613 Uses of Equity Swaps 614 Payoffs from Equity Swaps 616 Valuation and Pricing of Equity Swaps Summary 628 Exercises 628

622

Chapter 27 Estimating the Yield Curve 667 27.1 Introduction 667 27.2 Bootstrapping 667 27.3 Splines 669 27.4 Polynomial Splines 670 27.5 Exponential Splines 673 27.6 Implementation Issues with Splines 674 27.7 The Nelson-Siegel-Svensson Approach 674 27.8 Summary 676 27.9 Exercises 676 Appendix 27A Bootstrapping by Matrix Inversion 680 Appendix 27B Implementation with Exponential Splines 681

Chapter 28 Modeling Term-Structure Movements 684

Chapter 25 Currency and Commodity Swaps 631

28.1 28.2

25.1 25.2 25.3 25.4 25.5

28.3

Introduction 631 Currency Swaps 631 Commodity Swaps 639 Summary 643 Exercises 644

PART FOUR Interest Rate Modeling 647 Chapter 26 The Term Structure of Interest Rates: Concepts 649 26.1 26.2 26.3 26.4 26.5 26.6 26.7

Introduction 649 The Yield-to-Maturity 649 The Term Structure of Interest Rates 651 Discount Functions 652 Zero-Coupon Rates 653 Forward Rates 654 Yield-to-Maturity, Zero-Coupon Rates, and Forward Rates 656 26.8 Constructing the Yield-to-Maturity Curve: An Empirical Illustration 657 26.9 Summary 661 26.10 Exercises 662 Appendix 26A The Raw YTM Data 664

28.4 28.5 28.6 28.7

Introduction 684 Interest-Rate Modeling versus Equity Modeling 684 Arbitrage Violations: A Simple Example 685 A Gentle Introduction to No-Arbitrage Modeling 687 "No-Arbitrage" and "Equilibrium" Models 693 Summary 697 Exercises 697

Chapter 29 Factor Models of the Term Structure

700

29.1 Overview 700 29.2 The Black-Derman-Toy Model 701 29.3 The Ho-Lee Model 710 29.4 One-Factor Models in Continuous Time 714 29.5 Multifactor Models 720 29.6 Affine Factor Models 722 29.7 Summary 725 29.8 Exercises 726 Appendix 29A Deriving the Fundamental PDE in Factor Models 729

Chapter 30 The Heath-Jarrow-Morton and Libor Market Models 731 30.1 Overview 731

Contents xiii

30.2

The HJM Framework: Preliminary Comments 731 30.3 A One-Factor HJM Model 733 30.4 A Two-Factor HJM Setting 742 30.5 The HJM Risk-Neutral Drifts: An Algebraic Derivation 746 30.6 Libor Market Models 749 30.7 Mathematical Excursion: Marting;ales 750 30.8 Libor Rates: Notation 751 30.9 Risk-Neutral Pricing in the LMM 753 30.10 Simulation of the Market Model 757 30.11 Calibration 757 30.12 Swap Market Models 758 30.13 Swaptions 760 30.14 Summary 761 30.15 Exercises 761 Appendix 30A Risk-Neutral Drifts and Volatilities in HJM 765

PART FIVE Credit Risk 769 Chapter 31 Credit Derivative Products 771 31.1 Introduction 771 31.2 Total Return Swaps 775 31.3 Credit Spread Options/Forwards 779 31.4 Credit Default Swaps / 779 31.5 Credit-Linked Notes ' 788 31.6 Correlation Products 790 31.7 Summary 797 31.8 Exercises 797 Appendix 31A The CDS Big Bang 800

Chapter 32 Structural Models of Default Risk 802 32.1 32.2 32.3 32.4 32.5 32.6

Introduction 802 The Merton (1974) Model 803 Issues in Implementation 812 A Practitioner Model 817 Extensions of the Merton Model 819 Evaluation of the Structural Model Approach 820 32.7 Summary 823 32.8 Exercises 824 Appendix 32A The Delianedis-Geske Model 826

Chapter 33 Reduced-Form Models of Default Risk 829 33.1 33.2 33.3

Introduction 829 Modeling Default I: Intensity Processes \830 Modeling Default II: Recovery Rate Conventions 834 33.4 The Litterman-Iben Model 836 33.5 The Duffie-Singleton Result 841 33.6 Defaultable HJM Models 843 33.7 Ratings-Based Modeling: The JLT Model 845 33.8 An Application of Reduced-Form Models: Pricing CDS 853 33.9 Summary 855 33.10 Exercises 855 Appendix 33A Duffle-Singleton in Discrete Time 859 Appendix 33B Derivation of the Drift-Volatility Relationship 860

Chapter 34 Modeling Correlated Default 34.1 34.2

863

Introduction 863 Examples of Correlated Default Products 863 34.3 Simple Correlated Default Math 865 34.4 Structural Models Based on Asset Values 868 34.5 Reduced-Form Models 874 34.6 Multiperiod Correlated Default 875 34.7 Fast Computation of Credit Portfolio Loss Distributions without Simulation 878 34.8 Copula Functions 881 34.9 Top-Down Modeling of Credit Portfolio Loss 893 34.10 Summary 897 34.11 Exercises 898

Bibliography Index 1-1

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xiv Contents

The following Web chapters are available at www.mhhe.com/sdle: PART SIX Computation 901 Chapter 35 Derivative Pricing with Finite Differencing 903 35.1 35.2 35.3 35.4 35.5 35.6 35.7 35.8

Introduction 903 Solving Differential Equations 904 A First Approach to Pricing Equity Options 907 Implicit Finite Differencing 913 The Crank-Nicholson Scheme 917 Finite Differencing for Term-Structure Models 919 Summary 921 Exercises 922

Chapter 36 Derivative Pricing with Monte Carlo Simulation 923 36.1 36.2 36.3 36.4 36.5 36.6 36.7 36.8 36.9 36.10 36.11 36.12 36.13 36.14

Introduction 923 Simulating Normal Random Variables 924 Bivariate Random Variables 925 Cholesky Decomposition 925 Stochastic Processes for Equity Prices 927 ARCH Models 929 Interest-Rate Processes 930 Estimating Historical Volatility for Equities 932 Estimating Historical Volatility for Interest Rates 932 Path-Dependent Options 933 Variance Reduction 935 Monte Carlo for American Options 938 Summary 942 Exercises 943

Chapter 37 Using Octave 945 37.1 37.2 37.3 37.4 37.5

Some Simple Commands 945 Regression and Integration 948 Reading in Data, Sorting, and Finding Equation Solving 955 Screenshots 955

950