Monte Carlo Methods in Financial Engineering

Transcription

1 Paul Glassennan Monte Carlo Methods in Financial Engineering With 99 Figures <Q Springer

2 1 Foundations Principles of Monte Carlo Introduction First Examples Efficiency of Simulation Estimators Principles of Derivatives Pricing Pricing and Replication Arbitrage and Risk-Neutral Pricing Change of Numeraire The Market Price of Risk 36 2 Generating Random Numbers and Random Variables Random Number Generation General Considerations Linear Congruential Generators Implementation of Linear Congruential Generators Lattice Structure Combined Generators and Other Methods General Sampling Methods Inverse Transform Method Acceptance-Rejection Method Normal Random Variables and Vectors Basic Properties Generating Univariate Normals Generating Multivariate Normals 71 3 Generating Sample Paths Brownian Motion One Dimension Multiple Dimensions Geometric Brownian Motion 93

3 x Basic Properties Path-Dependent Options Multiple Dimensions Gaussian Short Rate Models Basic Models and Simulation Bond Prices Ill Multifactor Models Square-Root Diffusions Transition Density Sampling Gamma and Poisson Bond Prices Extensions Processes with Jumps A Jump-Diffusion Model Pure-Jump Processes Forward Rate Models: Continuous Rates The HJM Framework The Discrete Drift Implementation Forward Rate Models: Simple Rates LIBOR Market Model Dynamics Pricing Derivatives Simulation ' Volatility Structure and Calibration Variance Reduction Techniques Control Variates Method and Examples Multiple Controls Small-Sample Issues Nonlinear Controls Antithetic Variates Stratified Sampling Method and Examples Applications Poststratification Latin Hypercube Sampling Matching Underlying Assets Moment Matching Through Path Adjustments Weighted Monte Carlo Importance Sampling Principles and First Examples Path-Dependent Options Concluding Remarks 276

4 xi Quasi-Monte Carlo General Principles Discrepancy Van der Corput Sequences The Koksma-Hlawka Bound Nets and Sequences Low-Discrepancy Sequences Halton and Hammersley Faure Sobol' Further Constructions Lattice Rules Randomized QMC The Finance Setting Numerical Examples Strategic Implementation Concluding Remarks : 335 Discretization Methods Introduction The Euler Scheme and a First Refinement Convergence Order Second-Order Methods The Scalar Case The Vector Case Incorporating Path-Dependence Extrapolation Extensions General Expansions Jump-Diffusion Processes Convergence of Mean Square Error Extremes and Barrier Crossings: Brownian Interpolation Changing Variables ' Concluding Remarks 375 Estimating Sensitivities Finite-Difference Approximations Bias and Variance Optimal Mean Square Error Pathwise Derivative Estimates Method and Examples Conditions for Unbiasedness Approximations and Related Methods The Likelihood Ratio Method Method and Examples 401

5 7.3.2 Bias and Variance Properties Gamma Approximations and Related Methods Concluding Remarks 418 Pricing American Options Problem Formulation Parametric Approximations Random Tree Methods High Estimator Low Estimator Implementation State-Space Partitioning Stochastic Mesh Methods General Framework Likelihood Ratio Weights Regression-Based Methods and Weights Approximate Continuation Values Regression and Mesh Weights Duality Concluding Remarks 478 Applications in Risk Management Loss Probabilities and Value-at-Risk Background Calculating VAR Variance Reduction Using the Delta-Gamma Approximation Control Variate Importance Sampling Stratified Sampling A Heavy-Tailed Setting Modeling Heavy Tails Delta-Gamma Approximation Variance Reduction Credit Risk Default Times and Valuation Dependent Defaults Portfolio Credit Risk Concluding Remarks 535 Appendix: Convergence and Confidence Intervals 539 A.I Convergence Concepts 539 A.2 Central Limit Theorem and Confidence Intervals 541

6 xiii B Appendix: Results from Stochastic Calculus 545 B.I 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.I Term Structure Terminology 559 C.2 Interest Rate Derivatives 564 References 569 Index 587