[usemoney=30] [/usemoney] Preface xiii Acknowledgments xix CHAPTER 1 Black-Scholes and Pricing Fundamentals 1 1.1 Forward Contracts 1 1.2 Black-Scholes Partial Differential Equation 4 1.3 Risk-Neutral Pricing 10 1.4 Black-Scholes and Diffusion Process Implementation 17 1.5 American Options 30 1.6 Fundamental Pricing Formulas 33 1.7 Change of Numeraire 35 1.8 Girsanov’s Theorem 38 1.9 The Forward Measure 41 1.10 The Choice of Numeraire 42 CHAPTER 2 Monte Carlo Simulation 45 2.1 Monte Carlo 45 2.2 Generating Sample Paths and Normal Deviates 47 2.3 Generating Correlated Normal Random Variables 50 2.4 Quasi-Random Sequences 56 2.5 Variance Reduction and Control Variate Techniques 67 2.6 Monte Carlo Implementation 69 2.7 Hedge Control Variates 76 2.8 Path-Dependent Valuation 84 2.9 Brownian Bridge Technique 92 2.10 Jump-Diffusion Process and Constant Elasticity of Variance Diffusion Model 98 2.11 Object-Oriented Monte Carlo Approach 102 CHAPTER 3 Binomial Trees 123 3.1 Use of Binomial Trees 123 3.2 Cox-Ross-Rubinstein Binomial Tree 131 vii 3.3 Jarrow-Rudd Binomial Tree 132 3.4 General Tree 133 3.5 Dividend Payments 135 3.6 American Exercise 137 3.7 Binomial Tree Implementation 138 3.8 Computing Hedge Statistics 140 3.9 Binomial Model with Time-Varying Volatility 144 3.10 Two-Variable Binomial Process 145 3.11 Valuation of Convertible Bonds 150 CHAPTER 4 Trinomial Trees 165 4.1 Use of Trinomial Trees 165 4.2 Jarrow-Rudd Trinomial Tree 166 4.3 Cox-Ross-Rubinstein Trinomial Tree 168 4.4 Optimal Choice of λ 169 4.5 Trinomial Tree Implementations 170 4.6 Approximating Diffusion Processes with Trinomial Trees 174 4.7 Implied Trees 178 CHAPTER 5 Finite-Difference Methods 183 5.1 Explicit Difference Methods 183 5.2 Explicit Finite-Difference Implementation 186 5.3 Implicit Difference Method 191 5.4 LU Decomposition Method 194 5.5 Implicit Difference Method Implementation 196 5.6 Object-Oriented Finite-Difference Implementation 202 5.7 Iterative Methods 232 5.8 Crank-Nicolson Scheme 235 5.9 Alternating Direction Implicit Method 241 CHAPTER 6 Exotic Options 246 6.1 Barrier Options 246 6.2 Barrier Option Implementation 255 6.3 Asian Options 258 6.4 Geometric Averaging 258 6.5 Arithmetic Averaging 260 6.6 Seasoned Asian Options 261 6.7 Lookback Options 262 6.8 Implementation of Floating Lookback Option 265 6.9 Implementation of Fixed Lookback Option 268 viii CONTENTS CHAPTER 7 Stochastic Volatility 274 7.1 Implied Volatility 274 7.2 Volatility Skews and Smiles 276 7.3 Empirical Explanations 283 7.4 Implied Volatility Surfaces 284 7.5 One-Factor Models 303 7.6 Constant Elasticity of Variance Models 305 7.7 Recovering Implied Volatility Surfaces 307 7.8 Local Volatility Surfaces 309 7.9 Jump-Diffusion Models 313 7.10 Two-Factor Models 315 7.11 Hedging with Stochastic Volatility 321 CHAPTER 8 Statistical Models 324 8.1 Overview 324 8.2 Moving Average Models 329 8.3 Exponential Moving Average Models 331 8.4 GARCH Models 334 8.5 Asymmetric GARCH 337 8.6 GARCH Models for High-Frequency Data 340 8.7 Estimation Problems 353 8.8 GARCH Option Pricing Model 354 8.9 GARCH Forecasting 362 CHAPTER 9 Stochastic Multifactor Models 367 9.1 Change of Measure for Independent Random Variables 368 9.2 Change of Measure for Correlated Random Variables 370 9.3 N-Dimensional Random Walks and Brownian Motion 371 9.4 N-Dimensional Generalized Wiener Process 373 9.5 Multivariate Diffusion Processes 374 9.6 Monte Carlo Simulation of Multivariate Diffusion Processes 375 9.7 N-Dimensional Lognormal Process 376 9.8 Ito’s Lemma for Functions of Vector-Valued Diffusion Processes 388 9.9 Principal Component Analysis 389 CHAPTER 10 Single-Factor Interest Rate Models 395 10.1 Short Rate Process 398 10.2 Deriving the Bond Pricing Partial Differential Equation 399 10.3 Risk-Neutral Drift of the Short Rate 401 10.4 Single-Factor Models 402 Contents ix 10.5 Vasicek Model 404 10.6 Pricing Zero-Coupon Bonds in the Vasicek Model 411 10.7 Pricing European Options on Zero-Coupon Bonds with Vasicek 420 10.8 Hull-White Extended Vasicek Model 425 10.9 European Options on Coupon-Bearing Bonds 429 10.10 Cox-Ingersoll-Ross Model 431 10.11 Extended (Time-Homogenous) CIR Model 436 10.12 Black-Derman-Toy Short Rate Model 438 10.13 Black’s Model to Price Caps 439 10.14 Black’s Model to Price Swaptions 443 10.15 Pricing Caps, Caplets, and Swaptions with Short Rate Models 448 10.16 Valuation of Swaps 455 10.17 Calibration in Practice 457 CHAPTER 11 Tree-Building Procedures 467 11.1 Building Binomial Short Rate Trees for Black, Derman, and Toy 468 11.2 Building the BDT Tree Calibrated to the Yield Curve 471 11.3 Building the BDT Tree Calibrated to the Yield and Volatility Curve 476 11.4 Building a Hull-White Tree Consistent with the Yield Curve 485 11.5 Building a Lognormal Hull-White (Black-Karasinski) Tree 495 11.6 Building Trees Fitted to Yield and Volatility Curves 501 11.7 Vasicek and Black-Karasinski Models 509 11.8 Cox-Ingersoll-Ross Implementation 515 11.9 A General Deterministic-Shift Extension 520 11.10 Shift-Extended Vasicek Model 524 11.11 Shift-Extended Cox-Ingersoll-Ross Model 541 11.12 Pricing Fixed Income Derivatives with the Models 549 CHAPTER 12 Two-Factor Models and the Heath-Jarrow-Morton Model 554 12.1 The Two-Factor Gaussian G2++ Model 556 12.2 Building a G2++ Tree 563 12.3 Two-Factor Hull-White Model 575 12.4 Heath-Jarrow-Morton Model 579 12.5 Pricing Discount Bond Options with Gaussian HJM 584 12.6 Pricing Discount Bond Options in General HJM 585 12.7 Single-Factor HJM Discrete-State Model 586 12.8 Arbitrage-Free Restrictions in a Single-Factor Model 591 12.9 Computation of Arbitrage-Free Term Structure Evolutions 595 12.10 Single-Factor HJM Implementation 598 12.11 Synthetic Swap Valuation 606 x CONTENTS 12.12 Two-Factor HJM Model 612 12.13 Two-Factor HJM Model Implementation 616 12.14 The Ritchken and Sankarasubramanian Model 620 12.15 RS Spot Rate Process 623 12.16 Li-Ritchken-Sankarasubramanian Model 624 12.17 Implementing an LRS Trinomial Tree 626 CHAPTER 13 LIBOR Market Models 630 13.1 LIBOR Market Models 632 13.2 Specifications of the Instantaneous Volatility of Forward Rates 636 13.3 Implementation of Hull-White LIBOR Market Model 640 13.4 Calibration of LIBOR Market Model to Caps 641 13.5 Pricing Swaptions with Lognormal Forward-Swap Model 642 13.6 Approximate Swaption Pricing with Hull-White Approach 646 13.7 LFM Formula for Swaption Volatilities 648 13.8 Monte Carlo Pricing of Swaptions Using LFM 650 13.9 Improved Monte Carlo Pricing of Swaptions with a Predictor-Corrector 655 13.10 Incompatibility between LSM and LSF 663 13.11 Instantaneous and Terminal Correlation Structures 665 13.12 Calibration to Swaption Prices 669 13.13 Connecting Caplet and S × 1-Swaption Volatilities 670 13.14 Including Caplet Smile in LFM 673 13.15 Stochastic Extension of LIBOR Market Model 677 13.16 Computing Greeks in Forward LIBOR Models 688 CHAPTER 14 Bermudan and Exotic Interest Rate Derivatives 710 14.1 Bermudan Swaptions 710 14.2 Implementation for Bermudan Swaptions 713 14.3 Andersen’s Method 718 14.4 Longstaff and Schwartz Method 721 14.5 Stochastic Mesh Method 730 14.6 Valuation of Range Notes 733 14.7 Valuation of Index-Amortizing Swaps 742 14.8 Valuation of Trigger Swaps 752 14.9 Quanto Derivatives 754 14.10 Gaussian Quadrature 760 APPENDIX A Probability Review 771 A.1 Probability Spaces 771 A.2 Continuous Probability Spaces 773 Contents xi A.3 Single Random Variables 773 A.4 Binomial Random Variables 774 A.5 Normal Random Variables 775 A.6 Conditional Expectations 776 A.7 Probability Limit Theorems 778 A.8 Multidimensional Case 779 A.9 Dirac’s Delta Function 780 APPENDIX B Stochastic Calculus Review 783 B.1 Brownian Motion 783 B.2 Brownian Motion with Drift and Volatility 784 B.3 Stochastic Integrals 785 B.4 Ito’s Formula 788 B.5 Geometric Brownian Motion 789 B.6 Stochastic Leibnitz Rule 789 B.7 Quadratic Variation and Covariation