Monte Carlo Methods in Finance

Peter Jackel currently works at Commerzbank Securities in London as a quant in the front office product development and derivatives modelling group. Prior to that he worked within the NatWest Group/Royal Bank of Scotland Quantitative Research Centre. He started his career in finance with his employment at Nikko Securities' London operation.

• Preface xiAcknowledgements xiiiMathematical Notation xv1 Introduction 12 The Mathematics Behind Monte Carlo Methods 52.1 A Few Basic Terms in Probability and Statistics 52.2 Monte Carlo Simulations 72.2.1 Monte Carlo Supremacy 82.2.2 Multi-dimensional Integration 82.3 Some Common Distributions 92.4 Kolmogorov’s Strong Law 182.5 The Central Limit Theorem 182.6 The Continuous Mapping Theorem 192.7 Error Estimation for Monte Carlo Methods 202.8 The Feynman–Kac Theorem 212.9 The Moore–Penrose Pseudo-inverse 213 Stochastic Dynamics 233.1 Brownian Motion 233.2 Itô’s Lemma 243.3 Normal Processes 253.4 Lognormal Processes 263.5 The Markovian Wiener Process Embedding Dimension 263.6 Bessel Processes 273.7 Constant Elasticity Of Variance Processes 283.8 Displaced Diffusion 294 Process-driven Sampling 314.1 Strong versus Weak Convergence 314.2 Numerical Solutions 324.2.1 The Euler Scheme 324.2.2 The Milstein Scheme 334.2.3 Transformations 334.2.4 Predictor–Corrector 354.3 Spurious Paths 364.4 Strong Convergence for Euler and Milstein 375 Correlation and Co-movement 415.1 Measures for Co-dependence 425.2 Copulæ 455.2.1 The Gaussian Copula 465.2.2 The t-Copula 495.2.3 Archimedean Copulae 516 Salvaging a Linear Correlation Matrix 596.1 Hypersphere Decomposition 606.2 Spectral Decomposition 616.3 Angular Decomposition of Lower Triangular Form 626.4 Examples 636.5 Angular Coordinates on a Hypersphere of Unit Radius 657 Pseudo-random Numbers 677.1 Chaos 687.2 The Mid-square Method 727.3 Congruential Generation 727.4 Ran0 To Ran3 747.5 The Mersenne Twister 747.6 Which One to Use? 758 Low-discrepancy Numbers 778.1 Discrepancy 788.2 Halton Numbers 798.3 Sobol’ Numbers 808.3.1 Primitive Polynomials Modulo Two 818.3.2 The Construction of Sobol’ Numbers 828.3.3 The Gray Code 838.3.4 The Initialisation of Sobol’ Numbers 858.4 Niederreiter (1988) Numbers 888.5 Pairwise Projections 888.6 Empirical Discrepancies 918.7 The Number of Iterations 968.8 Appendix 968.8.1 Explicit Formula for the L2-norm Discrepancy on the Unit Hypercube 968.8.2 Expected L2-norm Discrepancy of Truly Random Numbers 979 Non-uniform Variates 999.1 Inversion of the Cumulative Probability Function 999.2 Using a Sampler Density 1019.2.1 Importance Sampling 1039.2.2 Rejection Sampling 1049.3 Normal Variates 1059.3.1 The Box–Muller Method 1059.3.2 The Neave Effect 1069.4 Simulating Multivariate Copula Draws 10910 Variance Reduction Techniques 11110.1 Antithetic Sampling 11110.2 Variate Recycling 11210.3 Control Variates 11310.4 Stratified Sampling 11410.5 Importance Sampling 11510.6 Moment Matching 11610.7 Latin Hypercube Sampling 11910.8 Path Construction 12010.8.1 Incremental 12010.8.2 Spectral 12210.8.3 The Brownian Bridge 12410.8.4 A Comparison of Path Construction Methods 12810.8.5 Multivariate Path Construction 13110.9 Appendix 13410.9.1 Eigenvalues and Eigenvectors of a Discrete-time Covariance Matrix 13410.9.2 The Conditional Distribution of the Brownian Bridge 13711 Greeks 13911.1 Importance Of Greeks 13911.2 An Up-Out-Call Option 13911.3 Finite Differencing with Path Recycling 14011.4 Finite Differencing with Importance Sampling 14311.5 Pathwise Differentiation 14411.6 The Likelihood Ratio Method 14511.7 Comparative Figures 14711.8 Summary 15311.9 Appendix 15311.9.1 The Likelihood Ratio Formula for Vega 15311.9.2 The Likelihood Ratio Formula for Rho 15612 Monte Carlo in the BGM/J Framework 15912.1 The Brace–Gatarek–Musiela/Jamshidian Market Model 15912.2 Factorisation 16112.3 Bermudan Swaptions 16312.4 Calibration to European Swaptions 16312.5 The Predictor–Corrector Scheme 16912.6 Heuristics of the Exercise Boundary 17112.7 Exercise Boundary Parametrisation 17412.8 The Algorithm 17612.9 Numerical Results 17712.10 Summary 18213 Non-recombining Trees 18313.1 Introduction 18313.2 Evolving the Forward Rates 18413.3 Optimal Simplex Alignment 18713.4 Implementation 19013.5 Convergence Performance 19113.6 Variance Matching 19213.7 Exact Martingale Conditioning 19513.8 Clustering 19613.9 A Simple Example 19913.10 Summary 20014 Miscellanea 20114.1 Interpolation of the Term Structure of Implied Volatility 20114.2 Watch Your CPU Usage 20214.3 Numerical Overflow and Underflow 20514.4 A Single Number or a Convergence Diagram? 20514.5 Embedded Path Creation 20614.6 How Slow is Exp()? 20714.7 Parallel Computing And Multi-threading 209Bibliography 213Index 219