Derivatives: Principles and Practice

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1 Derivatives: Principles and Practice Rangarajan K. Sundaram Stern School of Business New York University New York, NY Sanjiv R. Das Leavey School of Business Santa Clara University Santa Clara, CA I McGraw-Hill I Irwin

2 Contents Author Biographies xv Preface xvi Acknowledgments xxi Chapter 1 Introduction Forward and Futures Contracts Options Swaps Using Derivatives: Some Comments 1.5 The Structure of this Book Exercises 15 PART ONE Futures and Forwards 17 Chapter 2 Futures Markets Introduction The Changing Face of Futures Markets The Functioning of Futures Exchanges The Standardization of Futures Contracts Closing Out Positions Margin Requirements and Default Risk Case Studies in Futures Markets Exercises 53 Appendix 2A Futures Trading and US Regulation: A Brief History 57 Chapter 3 Pricing Forwards and Futures I: The Basic Theory Introduction Pricing Forwards by Replication Examples Forward Pricing on Currencies and Related Assets Forward-Rate Agreements Concept Check The Marked-to-Market Value of a Forward Contract Futures Prices 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 Introduction From Theory to Reality The Implied Repo Rate Transactions Costs Forward Prices and Future Spot Prices Index Arbitrage Exercises 97 Appendix 4A Forward Prices with Convenience Yields 100 Chapter 5 Hedging with Futures and Forwards Introduction A Guide to the Main Results The Cash Flow from a Hedged Position The Case of No Basis Risk The Minimum-Variance Hedge Ratio Examples Implementation Further Issues in Implementation Index Futures and Changing Equity Risk Fixed-Income Futures and Duration-Based Hedging Exercises 115 Appendix 5A Derivation of the Optimal Tailed Hedge Ratio h** 120 Chapter 6 Interest-Rate Forwards and Futures 6.1 Introduction Eurodollars and Libor Rates Forward-Rate Agreements Eurodollar Futures viii

3 Contents ix 6.5 Treasury Bond Futures Treasury Note Futures Treasury Bill Futures Duration-Based Hedging 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 Introduction Definitions and Terminology Options as Financial Insurance Naked Option Positions Options as Views on Market Direction and Volatility Exercises 165 Appendix 7A Options Markets 167 Chapter 8 Options: Payoffs and Trading Strategies Introduction Trading Strategies I: Covered Calls and Protective Puts Trading Strategies II: Spreads Trading Strategies III: Combinations Trading Strategies IV: Other Strategies Which Strategies Are the Most Widely Used? The Barings Case Exercises 192 Appendix 8A Asymmetric Butterfly Spreads 195 Chapter 9 No-Arbitrage Restrictions on Option Prices Introduction Motivating Examples Notation and Other Preliminaries Maximum and Minimum Prices for Options The Insurance Value of an Option Option Prices and Contract Parameters Numerical Examples Exercises 210 Chapter 10 Early Exercise and Put-Call Parity Introduction A Decomposition of Option Prices The Optimality of Early Exercise Put-Call Parity Exercises 226 Chapter 11 Option Pricing: An Introduction Overview The Binomial Model Pricing by Replication in a One-Period Binomial Model Comments Riskless Hedge Portfolios Pricing Using Risk-Neutral Probabilities The One-Period Model in General Notation The Delta of an Option An Application: Portfolio Insurance 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

4 x Contents Chapter 12 Binomial Option Pricing Introduction The Two-Period Binomial Tree Pricing Two-Period European Options European Option Pricing in General w-period Trees Pricing American Options: Preliminary Comments American Puts on Non-Dividend-Paying Stocks Cash Dividends in the Binomial Tree An Alternative Approach to Cash Dividends Dividend Yields in Binomial Trees Exercises 282 Appendix 12A A General Representation of European Option Prices 286 Chapter 13 Implementing the Binomial Model Introduction The Lognormal Distribution Binomial Approximations of the Lognormal Computer Implementation of the Binomial Model Exercises 303 Appendix 13A Estimating Historical Volatility 306 Chapter 14 The Black-Scholes Model Introduction Option Pricing in the Black-Scholes Setting Remarks on the Formula Working with the Formulae I: Plotting Option Prices Working with the Formulae II: Algebraic Manipulation Dividends in the Black-Scholes Model Options on Indices, Currencies, and Futures Testing the Black-Scholes Model: Implied Volatility The VIX and Its Derivatives 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 Introduction Geometric Brownian Motion Defined The Black-Scholes Formula via Replication The Black-Scholes Formula via Risk-Neutral Pricing The Black-Scholes Formula via CAPM Exercises 354 Chapter 16 Options Modeling: Beyond Black-Scholes Introduction Jump-Diffusion Models Stochastic Volatility GARCH Models Other Approaches Implied Binomial Trees/Local Volatility Models Summary 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" Introduction Interpreting the Greeks: A Snapshot View 404

5 Contents xi 17.3 The Option Delta The Option Gamma The Option Theta The Option Vega The Option Rho Portfolio Greeks Exercises 432 Appendix 17A Deriving the Black-Scholes Option Greeks 436 Chapter 18 Exotic Options I: Path-Independent Options Introduction Forward Start Options Binary Options Chooser Options Compound Options Exchange Options Quanta Options Variants on the Exchange Option Theme Exercises 465 Chapter 19 Exotic Options II: Path-Dependent Options Path-Dependent Exotic Options 470 _ 19.2 Barrier Options Asian Options Lookback Options Cliquets Shout Options Exercises 492 Appendix 19A Barrier Option Pricing Formulae 496 Chapter 20 Value-at-Risk Introduction Value-at-Risk Risk Decomposition Coherent Risk Measures Exercises 515 Chapter 21 Convertible Bonds Introduction Convertible Bond Terminology Main Features of Convertible Bonds Breakeven Analysis Pricing Convertibles: A First Pass Incorporating Credit Risk Convertible Greeks Convertible Arbitrage Summary 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 Introduction Preliminary Analysis and Examples A Real Options "Case Study" Creating the State Space Applications of Real Options Summary 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 Introduction Floating-Rate Notes Interest Rate Swaps Uses of Swaps Swap Payoffs Valuing and Pricing Swaps Extending the Pricing Arguments Case Study: The Procter & Gamble-Bankers Trust "5/30" Swap 589

6 xii Contents 23.9 Case Study: A Long-Term Capital Management "Convergence Trade" Credit Risk and Credit Exposure Hedging Swaps Caps, Floors, and Swaptions The Black Model for Pricing Caps, Floors, and Swaptions Summary Exercises 609 Chapter 24 Equity Swaps Introduction Uses of Equity Swaps Payoffs from Equity Swaps Valuation and Pricing of Equity Swaps 24.5 Summary Exercises 628 Chapter 25 Currency and Commodity Swaps 25.1 Introduction Currency Swaps Commodity Swaps Summary Exercises 644 PART FOUR Interest Rate Modeling Chapter 26 The Term Structure of Interest Rates: Concepts Introduction The Yield-to-Maturity The Term Structure of Interest Rates Discount Functions Zero-Coupon Rates Forward Rates Yield-to-Maturity, Zero-Coupon Rates, and Forward Rates Constructing the Yield-to-Maturity Curve: An Empirical Illustration Summary Exercises 662 Appendix 26A The Raw YTM Data Chapter 27 Estimating the Yield Curve Introduction Bootstrapping Splines Polynomial Splines Exponential Splines Implementation Issues with Splines The Nelson-Siegel-Svensson Approach Summary Exercises 676 Appendix 27A Bootstrapping by Matrix Inversion 680 Appendix 27B Implementation with Exponential Splines 681 Chapter 28 Modeling Term-Structure Movements Introduction Interest-Rate Modeling versus Equity Modeling Arbitrage Violations: A Simple Example A Gentle Introduction to No-Arbitrage Modeling "No-Arbitrage" and "Equilibrium" Models Summary Exercises 697 Chapter 29 Factor Models of the Term Structure Overview The Black-Derman-Toy Model The Ho-Lee Model One-Factor Models in Continuous Time Multifactor Models Affine Factor Models Summary Exercises 726 Appendix 29A Deriving the Fundamental PDE in Factor Models 729 Chapter 30 The Heath-Jarrow-Morton and Libor Market Models Overview 731

7 Contents xiii 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 Mathematical Excursion: Marting; ales Libor Rates: Notation Risk-Neutral Pricing in the LMM Simulation of the Market Model Calibration Swap Market Models Swaptions Summary Exercises Appendix 30A Risk-Neutral Drifts PART FIVE Credit Risk 769 and Volatilities in HJM Chapter 31 Credit Derivative Products 771 Algebraic Introduction Total Return Swaps Credit Spread Options/Forwards Credit Default Swaps / Credit-Linked Notes ' Correlation Products Summary Exercises 797 Appendix 31A The CDS Big Bang 800 Chapter 32 Structural Models of Default Risk Introduction The Merton (1974) Model Issues in Implementation A Practitioner Model Extensions of the Merton Model Evaluation of the Structural Model Approach Summary Exercises 824 Appendix 32A The Delianedis-Geske Model 826 Chapter 33 Reduced-Form Models of Default Risk Introduction 829 Modeling Default I: Intensity Processes \ 830 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 Appendix 33A Duffle-Singleton in Discrete Time 859 Appendix 33B Derivation of the Drift-Volatility Relationship 860 Chapter 34 Modeling Correlated Default 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 Index 1-1 B-l 829

8 xiv Contents The following Web chapters are available at PART SIX Computation 901 Chapter 35 Derivative Pricing with Finite Differencing Introduction Solving Differential Equations A First Approach to Pricing Equity Options Implicit Finite Differencing The Crank-Nicholson Scheme Finite Differencing for Term-Structure Models Summary Exercises 922 Chapter 36 Derivative Pricing with Monte Carlo Simulation Introduction Simulating Normal Random Variables Bivariate Random Variables Cholesky Decomposition Stochastic Processes for Equity Prices ARCH Models Interest-Rate Processes Estimating Historical Volatility for Equities Estimating Historical Volatility for Interest Rates Path-Dependent Options Variance Reduction Monte Carlo for American Options Summary Exercises 943 Chapter 37 Using Octave Some Simple Commands Regression and Integration Reading in Data, Sorting, and Finding Equation Solving Screenshots 955

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