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

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

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

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

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

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 790
References 793
About the CD-ROM 803
GNU General Public License 807
Index 813