1 Foundations 1
1.1 Principles of Monte Carlo 1
1.1.1 Introduction 1
1.1.2 First Examples 3
1.1.3 Efficiency of Simulation Estimators 9
1.2 Principles of Derivatives Pricing 19
1.2.1 Pricing and Replication 21
1.2.2 Arbitrage and Risk-Neutral Pricing 25
1.2.3 Change of Numeraire 32
1.2.4 The Market Price of Risk 36
2 Generating Random Numbers and Random Variables 39
2.1 Random Number Generation 39
2.1.1 General Considerations 39
2.1.2 Linear Congruential Generators 43
2.1.3 Implementation of Linear Congruential Generators 44
2.1.4 Lattice Structure 47
2.1.5 Combined Generators and Other Methods 49
2.2 General Sampling Methods 53
2.2.1 Inverse Transform Method 54
2.2.2 Acceptance-Rejection Method 58
2.3 Normal Random Variables and Vectors 63
2.3.1 Basic Properties 63
2.3.2 Generating Univariate Normals 65
2.3.3 Generating Multivariate Normals 71
3 Generating Sample Paths 79
3.1 Brownian Motion 79
3.1.1 One Dimension 79
3.1.2 Multiple Dimensions 90
3.2 Geometric Brownian Motion 93
3.2.1 Basic Properties 93
3.2.2 Path-Dependent Options 96
3.2.3 Multiple Dimensions 104
3.3 Gaussian Short Rate Models 108
3.3.1 Basic Models and Simulation 108
3.3.2 Bond Prices 111
3.3.3 Multifactor Models 118
3.4 Square-Root Diffusions 120
3.4.1 Transition Density 121
3.4.2 Sampling Gamma and Poisson 125
3.4.3 Bond Prices 128
3.4.4 Extensions 131
3.5 Processes with Jumps 134
3.5.1 A Jump-Diffusion Model 134
3.5.2 Pure-Jump Processes 142
3.6 Forward Rate Models: Continuous Rates 149
3.6.1 The HJM Framework 150
3.6.2 The Discrete Drift 155
3.6.3 Implementation 160
3.7 Forward Rate Models: Simple Rates 165
3.7.1 LIBOR Market Model Dynamics 166
3.7.2 Pricing Derivatives 172
3.7.3 Simulation 174
3.7.4 Volatility Structure and Calibration 180
4 Variance Reduction Techniques 185
4.1 Control Variates 185
4.1.1 Method and Examples 185
4.1.2 Multiple Controls 196
4.1.3 Small-Sample Issues 200
4.1.4 Nonlinear Controls 202
4.2 Antithetic Variates 205
4.3 Stratified Sampling 209
4.3.1 Method and Examples 209
4.3.2 Applications 220
4.3.3 Poststratification 232
4.4 Latin Hypercube Sampling 236
4.5 Matching Underlying Assets 243
4.5.1 Moment Matching Through Path Adjustments 244
4.5.2 Weighted Monte Carlo 251
4.6 Importance Sampling 255
4.6.1 Principles and First Examples 255
4.6.2 Path-Dependent Options 267
4.7 Concluding Remarks 276
5 Quasi-Monte Carlo 281
5.1 General Principles 281
5.1.1 Discrepancy 283
5.1.2 Van der Corput Sequences 285
5.1.3 The Koksma-Hlawka Bound 287
5.1.4 Nets and Sequences 290
5.2 Low-Discrepancy Sequences 293
5.2.1 Halton and Hammersley 293
5.2.2 Faure 297
5.2.3 Sobol 303
5.2.4 Further Constructions 314
5.3 Lattice Rules 316
5.4 Randomized QMC 320
5.5 The Finance Setting 323
5.5.1 Numerical Examples 323
5.5.2 Strategic Implementation 331
5.6 Concluding Remarks 335
6 Discretization Methods 339
6.1 Introduction 339
6.1.1 The Euler Scheme and a First Refinement 339
6.1.2 Convergence Order 344
6.2 Second-Order Methods 348
6.2.1 The Scalar Case 348
6.2.2 The Vector Case 351
6.2.3 Incorporating Path-Dependence 357
6.2.4 Extrapolation 360
6.3 Extensions 362
6.3.1 General Expansions 362
6.3.2 Jump-Diffusion Processes 363
6.3.3 Convergence of Mean Square Error 365
6.4 Extremes and Barrier Crossings: Brownian Interpolation 366
6.5 Changing Variables 371
6.6 Concluding Remarks 375
7 Estimating Sensitivities 377
7.1 Finite-Difference Approximations 378
7.1.1 Bias and Variance 378
7.1.2 Optimal Mean Square Error 381
7.2 Pathwise Derivative Estimates 386
7.2.1 Method and Examples 386
7.2.2 Conditions for Unbiasedness 393
7.2.3 Approximations and Related Methods 396
7.3 The Likelihood Ratio Method 401
7.3.1 Method and Examples 401
7.3.2 Bias and Variance Properties 407
7.3.3 Gamma 411
7.3.4 Approximations and Related Methods 413
7.4 Concluding Remarks 418
8 Pricing American Options 421
8.1 Problem Formulation 421
8.2 Parametric Approximations 426
8.3 Random Tree Methods 430
8.3.1 High Estimator 432
8.3.2 Low Estimator 434
8.3.3 Implementation 437
8.4 State-Space Partitioning 441
8.5 Stochastic Mesh Methods 443
8.5.1 General Framework 443
8.5.2 Likelihood Ratio Weights 450
8.6 Regression-Based Methods and Weights 459
8.6.1 Approximate Continuation Values 459
8.6.2 Regression and Mesh Weights 465
8.7 Duality 470
8.8 Concluding Remarks 478
9 Applications in Risk Management 481
9.1 Loss Probabilities and Value-at-Risk 481
9.1.1 Background 481
9.1.2 Calculating VAR 484
9.2 Variance Reduction Using the Delta-Gamma Approximation 492
9.2.1 Control Variate 493
9.2.2 Importance Sampling 495
9.2.3 Stratified Sampling 500
9.3 A Heavy-Tailed Setting 506
9.3.1 Modeling Heavy Tails 506
9.3.2 Delta-Gamma Approximation 512
9.3.3 Variance Reduction 514
9.4 Credit Risk 520
9.4.1 Default Times and Valuation 520
9.4.2 Dependent Defaults 525
9.4.3 Portfolio Credit Risk 529
9.5 Concluding Remarks 535
A Appendix: Convergence and Confidence Intervals 539
A.1 Convergence Concepts 539
A.2 Central Limit Theorem and Confidence Intervals 541
B Appendix: Results from Stochastic Calculus 545
B.1 Ito's Formula 545
B.2 Stochastic Differential Equations 548
B.3 Martingales 550
B.4 Change of Measure 553
C Appendix: The Term Structure of Interest Rates 559
C.1 Term Structure Terminology 559
C.2 Interest Rate Derivatives 564
References 569
Index 587