Monte Carlo Methods in Finance

Transcription

1 Monte Carlo Methods in Finance Peter Jackel JOHN WILEY & SONS, LTD

2 Preface Acknowledgements Mathematical Notation xi xiii xv 1 Introduction 1 2 The Mathematics Behind Monte Carlo Methods A Few Basic Terms in Probability and Statistics Monte Carlo Simulations Monte Carlo Supremacy Multi-dimensional Integration Some Common Distributions Kolmogorov's Strong Law The Central Limit Theorem The Continuous Mapping Theorem Error Estimation for Monte Carlo Methods The Feynman-Kac Theorem The Moore-Penrose Pseudo-inverse Stochastic Dynamics 3.1 Brownian Motion 3.2 Ito's Lemma 3.3 Normal Processes 3.4 Lognormal Processes 3.5 The Markovian Wiener Process Embedding Dimension 3.6 Bessel Processes 3.7 Constant Elasticity Of Variance Processes 3.8 Displaced Diffusion Process-driven Sampling Strong versus Weak Convergence Numerical Solutions 32

3 viii Contents The Euler Scheme The Milstein Scheme Transformations Predictor-Corrector Spurious Paths Strong Convergence for Euler and Milstein 37 5 Correlation and Co-movement Measures for Co-dependence Copulae The Gaussian Copula The f-copula Archimedean Copulae 51 6 Salvaging a Linear Correlation Matrix Hypersphere Decomposition Spectral Decomposition Angular Decomposition of Lower Triangular Form Examples Angular Coordinates on a Hypersphere of Unit Radius 65 7 Pseudo-random Numbers Chaos The Mid-square Method Congruential Generation RanO To Ran The Mersenne Twister Which One to Use? 75 8 Low-discrepancy Numbers Discrepancy Halton Numbers Sobol' Numbers Primitive Polynomials Modulo Two The Construction of Sobol' Numbers The Gray Code The Initialisation of Sobol' Numbers Niederreiter (1988) Numbers Pairwise Projections Empirical Discrepancies The Number of Iterations Appendix Explicit Formula for the L2-norm Discrepancy on the Unit Hypercube Expected L2-norm Discrepancy of Truly Random Numbers 97 9 Non-uniform Variates Inversion of the Cumulative Probability Function Using a Sampler Density 101

4 9.2.1 Importance Sampling Rejection Sampling Normal Variates The Box-Muller Method The Neave Effect Simulating Multivariate Copula Draws Variance Reduction Techniques Antithetic Sampling Variate Recycling Control Variates Stratified Sampling Importance Sampling Moment Matching Latin Hypercube Sampling Path Construction Incremental Spectral The Brownian Bridge A Comparison of Path Construction Methods Multivariate Path Construction Appendix Eigenvalues and Eigenvectors of a Discrete-time Covariance Matrix The Conditional Distribution of the Brownian Bridge Greeks Importance Of Greeks An Up-Out-Call Option Finite Differencing with Path Recycling Finite Differencing with Importance Sampling Pathwise Differentiation The Likelihood Ratio Method Comparative Figures Summary Appendix The Likelihood Ratio Formula for Vega The Likelihood Ratio Formula for Rho Monte Carlo in the BGM/J Framework The Brace-Gatarek-Musiela/Jamshidian Market Model Factorisation Bermudan Swaptions Calibration to European Swaptions The Predictor-Corrector Scheme Heuristics of the Exercise Boundary Exercise Boundary Parametrisation The Algorithm 176 ix

5 12.9 Numerical Results Summary Non-recombming Trees 13.1 Introduction 13.2 Evolving the Forward Rates 13.3 Optimal Simplex Alignment 13.4 Implementation 13.5 Convergence Performance 13.6 Variance Matching 13.7 Exact Martingale Conditioning 13.8 Clustering 13.9 A Simple Example Summary Miscellanea Interpolation of the Term Structure of Implied Volatility Watch Your CPU Usage Numerical Overflow and Underflow A Single Number or a Convergence Diagram? Embedded Path Creation How Slow is Exp ()? Parallel Computing And Multi-threading 209 Bibliography 213 Index 219