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 95053I McGraw-Hill I Irwin
Contents
Author Biographies xv
Preface xvi
Acknowledgments xxi
Chapter 1
Introduction 1
1.1 Forward and Futures Contracts 5 1.2 Options 9
-1.3 Swaps 10
1.4 Using Derivatives: Some Comments
1.5 The Structure of this Book 14
1.6 Exercises 15
11
PART ONE
Futures and Forwards
17
Chapter 2
Futures Markets
19
2.1 Introduction 192.2 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 Introduction 60
3.2 Pricing Forwards by Replication 61 3.3 Examples 63
3.4 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
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
Chapter 4
Pricing Forwards and Futures II: Building on the Foundations 85
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 Introduction 101
5.2 A Guide to the Main Results 103
5.3 The Cash Flow from a Hedged Position 104 5.4 The Case of No Basis Risk 105
5.5 The Minimum-Variance Hedge Ratio 106
5.6 Examples 109 5.7 Implementation 111
5.8 Further Issues in Implementation 112 5.9 Index Futures and Changing Equity Risk 114
5.10 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 Introduction 1226.2 Eurodollars and Libor Rates 122 6.3 Forward-Rate Agreements 123 6.4 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
PART TWO
Options 153
Chapter 7
Options Markets
155
7.1 Introduction 1557.2 Definitions and Terminology 155
7.3 Options as Financial Insurance 156 7.4 Naked Option Positions 158 7.5 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 Introduction 171
8.2 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 9
No-Arbitrage Restrictions on Option
Prices 196
9.1 Introduction 196
9.2 Motivating Examples 196
9.3 Notation and Other Preliminaries 198 9.4 Maximum and Minimum Prices for
Options 199
9.5 The Insurance Value of an Option 204
9.6 Option Prices and Contract Parameters 205
9.7 Numerical Examples 208
9.8 Exercises 210
Chapter 10
Early Exercise and Put-Call Parity
213
10.1 Introduction 213
10.2 A Decomposition of Option Prices 213
10.3 The Optimality of Early Exercise 216 10.4 Put-Call Parity. 220
10.5 Exercises 226
Chapter 11
Option Pricing: An Introduction
228
11.1 Overview 228
11.2 The Binomial Model 229
11.3 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
Chapter 12
Binomial Option Pricing
259
12.1 Introduction 259
12.2 The Two-Period Binomial Tree 261 12.3 Pricing Two-Period European Options 262 12.4 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 Introduction 289
13.2 The Lognormal Distribution 289 13.3 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 Introduction 308
14.2 Option Pricing in the Black-Scholes Setting 310
14.3 Remarks on the Formula 313
14.4 Working with the Formulae I: Plotting Option
Prices 314
14.5 Working with the Formulae II: Algebraic Manipulation 315
14.6 Dividends in the Black-Scholes Model 319
14.7 Options on Indices, Currencies,
and Futures 324
14.8 Testing the Black-Scholes Model: Implied
Volatility 327
14.9 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
344344
15.1 Introduction 344
15.2 Geometric Brownian Motion Defined 15.3 The Black-Scholes Formula via
Replication 348
15.4 The Black-Scholes Formula via Risk-Neutral Pricing 351
15.5 The Black-Scholes Formula via CAPM 353 15.6 Exercises 354
Chapter 16
Options Modeling:
Beyond Black-Scholes
357 16.1 Introduction 357 16.2 Jump-Diffusion Models 358 16.3 Stochastic Volatility 368 16.4 GARCH Models 374 16.5 Other Approaches 37816.6 Implied Binomial Trees/Local Volatility
Models 379
16.7 Summary 389 16.8 Exercises 389
Appendix 16A Program Code for
Jump-Diffusions 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 Introduction 404
17.2 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 Introduction 440
18.2 Forward Start Options 442 18.3 Binary Options 445 18.4 Chooser Options 450 18.5 Compound Options 453 18.6 Exchange Options 458 18.7 Quanta Options 460
18.8 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 492Appendix 19A Barrier Option Pricing
Formulae 496
Chapter 20
Value-at-Risk
498
20.1 Introduction 498 20.2 Value-at-Risk 498 20.3 Risk Decomposition 505 20.4 Coherent Risk Measures 511 20.5 Exercises 515Chapter 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 Introduction 548
22.2 Preliminary Analysis and Examples 22.3 A Real Options "Case Study" 554 22.4 Creating the State Space 560 22.5 Applications of Real Options 563 22.6 Summary 564
22.7 Exercises 564
550
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 Introduction 571 23.2 Floating-Rate Notes 571 23.3 Interest Rate Swaps 575 23.4 Uses of Swaps 576 23.5 Swap Payoffs 579
23.6 Valuing and Pricing Swaps 582 23.7 Extending the Pricing Arguments 586 23.8 Case Study: The Procter & Gamble-Bankers
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 Introduction 61324.2 Uses of Equity Swaps 614 24.3 Payoffs from Equity Swaps 616 24.4 Valuation and Pricing of Equity Swaps 24.5 Summary 628
24.6 Exercises 628
622
Chapter 25
Currency and Commodity Swaps
25.1 Introduction 631 25.2 Currency Swaps 631 25.3 Commodity Swaps 639 25.4 Summary 643 25.5 Exercises 644
631
PART FOUR
Interest Rate Modeling
647
Chapter 26
The Term Structure of Interest Rates:
Concepts 649
26.1 Introduction 649
26.2 The Yield-to-Maturity 649
26.3 The Term Structure of Interest Rates 651 26.4 Discount Functions 652
26.5 Zero-Coupon Rates 653
26.6 Forward Rates 654
26.7 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
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
28.1 Introduction 684
28.2 Interest-Rate Modeling versus Equity
Modeling 684
28.3 Arbitrage Violations: A Simple
Example 685
28.4 A Gentle Introduction to No-Arbitrage
Modeling 687
28.5 "No-Arbitrage" and "Equilibrium"
Models 693
28.6 Summary 697 28.7 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
Contents xiii 30.2 30.3 30.4 30.5 30.6 30.7 30.8 30.9 30.10 30.11 30.12 30.13 30.14 30.15
The HJM Framework: Preliminary Comments 731
A One-Factor HJM Model 733 A Two-Factor HJM Setting 742 The HJM Risk-Neutral Drifts: An Derivation 746
Libor Market Models 749 Mathematical Excursion: Marting; Libor Rates: Notation 751 Risk-Neutral Pricing in the LMM Simulation of the Market Model Calibration 757
Swap Market Models 758 Swaptions 760
Summary 761 Exercises 761
Appendix 30A Risk-Neutral Drifts
PART
Credit
and Volatilities in HJMFIVE
Risk 769
Algebraic ales 750 753 757 765Chapter 33
Reduced-Form Models of Default Risk
Chapter 31
Credit Derivative Products
771
779
31.1 Introduction 771 31.2 Total Return Swaps 775 31.3 Credit Spread Options/Forwards 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 32.7 32.8 Introduction 802 The Merton (1974) Model Issues in Implementation A Practitioner Model 817
803 812 Extensions of the Merton Model 819 Evaluation of the Structural
Approach 820 Summary 823 Exercises 824 Model 33.1 33.2 33.3 33.4 33.5 33.6 33.7 33.8 33.9 33.10 Introduction 829
Modeling Default I: Intensity Processes \ Modeling Default II: Recovery Rate Conventions 834
The Litterman-Iben Model 836 The Duffie-Singleton Result 841 Defaultable HJM Models 843 Ratings-Based Modeling: The JLT Model 845
An Application of Reduced-Form Models: Pricing CDS 853
Summary 855 Exercises 855
Appendix 33A Duffle-Singleton
in Discrete Time 859
Appendix 33B Derivation of the Drift-Volatility
Relationship 860
Chapter 34
Modeling Correlated Default 863
34.1 34.2 34.3 34.4 34.5 34.6 34.7 34.8 34.9 34.10 34.11 Introduction 863
Examples of Correlated Default Products 863
Simple Correlated Default Math 865 Structural Models Based on
Asset Values 868
Reduced-Form Models 874 Multiperiod Correlated Default 875 Fast Computation of Credit Portfolio Loss Distributions without Simulation 878 Copula Functions 881
Top-Down Modeling of Credit Portfolio Loss 893 Summary 897 Exercises 898
Bibliography B-l
Index
1-1
829
830Appendix 32A The Delianedis-Geske
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 Introduction 903
35.2 Solving Differential Equations 904 35.3 A First Approach to Pricing Equity
Options 907
35.4 Implicit Finite Differencing 913 35.5 The Crank-Nicholson Scheme 917 35.6 Finite Differencing for Term-Structure
Models 919
35.7 Summary 921 35.8 Exercises 922
Chapter 36
Derivative Pricing with Monte Carlo
Simulation 923
36.1 Introduction 923
36.2 Simulating Normal Random Variables 924 36.3 Bivariate Random Variables 925
36.4 Cholesky Decomposition 925
36.5 Stochastic Processes for Equity Prices 927 36.6 ARCH Models 929
36.7 Interest-Rate Processes 930 36.8 Estimating Historical Volatility for
Equities 932
36.9 Estimating Historical Volatility for Interest
Rates 932
36.10 Path-Dependent Options 933 36.11 Variance Reduction 935
36.12 Monte Carlo for American Options 938 36.13 Summary 942
36.14 Exercises 943
Chapter 37
Using Octave 945
37.1 Some Simple Commands 945 37.2 Regression and Integration 948
37.3 Reading in Data, Sorting, and Finding 950 37.4 Equation Solving 955
