Description
Book SynopsisA guide which uses a problem solving approach and shows how to implement Monte Carlo methods, starting from first principles to advanced techniques.
Table of ContentsPreface xi
Acknowledgements xiii
Mathematical Notation xv
1 Introduction 1
2 The Mathematics Behind Monte Carlo Methods 5
2.1 A Few Basic Terms in Probability and Statistics 5
2.2 Monte Carlo Simulations 7
2.2.1 Monte Carlo Supremacy 8
2.2.2 Multi-dimensional Integration 8
2.3 Some Common Distributions 9
2.4 Kolmogorov’s Strong Law 18
2.5 The Central Limit Theorem 18
2.6 The Continuous Mapping Theorem 19
2.7 Error Estimation for Monte Carlo Methods 20
2.8 The Feynman–Kac Theorem 21
2.9 The Moore–Penrose Pseudo-inverse 21
3 Stochastic Dynamics 23
3.1 Brownian Motion 23
3.2 Itô’s Lemma 24
3.3 Normal Processes 25
3.4 Lognormal Processes 26
3.5 The Markovian Wiener Process Embedding Dimension 26
3.6 Bessel Processes 27
3.7 Constant Elasticity Of Variance Processes 28
3.8 Displaced Diffusion 29
4 Process-driven Sampling 31
4.1 Strong versus Weak Convergence 31
4.2 Numerical Solutions 32
4.2.1 The Euler Scheme 32
4.2.2 The Milstein Scheme 33
4.2.3 Transformations 33
4.2.4 Predictor–Corrector 35
4.3 Spurious Paths 36
4.4 Strong Convergence for Euler and Milstein 37
5 Correlation and Co-movement 41
5.1 Measures for Co-dependence 42
5.2 Copulæ 45
5.2.1 The Gaussian Copula 46
5.2.2 The t-Copula 49
5.2.3 Archimedean Copulae 51
6 Salvaging a Linear Correlation Matrix 59
6.1 Hypersphere Decomposition 60
6.2 Spectral Decomposition 61
6.3 Angular Decomposition of Lower Triangular Form 62
6.4 Examples 63
6.5 Angular Coordinates on a Hypersphere of Unit Radius 65
7 Pseudo-random Numbers 67
7.1 Chaos 68
7.2 The Mid-square Method 72
7.3 Congruential Generation 72
7.4 Ran0 To Ran3 74
7.5 The Mersenne Twister 74
7.6 Which One to Use? 75
8 Low-discrepancy Numbers 77
8.1 Discrepancy 78
8.2 Halton Numbers 79
8.3 Sobol’ Numbers 80
8.3.1 Primitive Polynomials Modulo Two 81
8.3.2 The Construction of Sobol’ Numbers 82
8.3.3 The Gray Code 83
8.3.4 The Initialisation of Sobol’ Numbers 85
8.4 Niederreiter (1988) Numbers 88
8.5 Pairwise Projections 88
8.6 Empirical Discrepancies 91
8.7 The Number of Iterations 96
8.8 Appendix 96
8.8.1 Explicit Formula for the L2-norm Discrepancy on the Unit Hypercube 96
8.8.2 Expected L2-norm Discrepancy of Truly Random Numbers 97
9 Non-uniform Variates 99
9.1 Inversion of the Cumulative Probability Function 99
9.2 Using a Sampler Density 101
9.2.1 Importance Sampling 103
9.2.2 Rejection Sampling 104
9.3 Normal Variates 105
9.3.1 The Box–Muller Method 105
9.3.2 The Neave Effect 106
9.4 Simulating Multivariate Copula Draws 109
10 Variance Reduction Techniques 111
10.1 Antithetic Sampling 111
10.2 Variate Recycling 112
10.3 Control Variates 113
10.4 Stratified Sampling 114
10.5 Importance Sampling 115
10.6 Moment Matching 116
10.7 Latin Hypercube Sampling 119
10.8 Path Construction 120
10.8.1 Incremental 120
10.8.2 Spectral 122
10.8.3 The Brownian Bridge 124
10.8.4 A Comparison of Path Construction Methods 128
10.8.5 Multivariate Path Construction 131
10.9 Appendix 134
10.9.1 Eigenvalues and Eigenvectors of a Discrete-time Covariance Matrix 134
10.9.2 The Conditional Distribution of the Brownian Bridge 137
11 Greeks 139
11.1 Importance Of Greeks 139
11.2 An Up-Out-Call Option 139
11.3 Finite Differencing with Path Recycling 140
11.4 Finite Differencing with Importance Sampling 143
11.5 Pathwise Differentiation 144
11.6 The Likelihood Ratio Method 145
11.7 Comparative Figures 147
11.8 Summary 153
11.9 Appendix 153
11.9.1 The Likelihood Ratio Formula for Vega 153
11.9.2 The Likelihood Ratio Formula for Rho 156
12 Monte Carlo in the BGM/J Framework 159
12.1 The Brace–Gatarek–Musiela/Jamshidian Market Model 159
12.2 Factorisation 161
12.3 Bermudan Swaptions 163
12.4 Calibration to European Swaptions 163
12.5 The Predictor–Corrector Scheme 169
12.6 Heuristics of the Exercise Boundary 171
12.7 Exercise Boundary Parametrisation 174
12.8 The Algorithm 176
12.9 Numerical Results 177
12.10 Summary 182
13 Non-recombining Trees 183
13.1 Introduction 183
13.2 Evolving the Forward Rates 184
13.3 Optimal Simplex Alignment 187
13.4 Implementation 190
13.5 Convergence Performance 191
13.6 Variance Matching 192
13.7 Exact Martingale Conditioning 195
13.8 Clustering 196
13.9 A Simple Example 199
13.10 Summary 200
14 Miscellanea 201
14.1 Interpolation of the Term Structure of Implied Volatility 201
14.2 Watch Your CPU Usage 202
14.3 Numerical Overflow and Underflow 205
14.4 A Single Number or a Convergence Diagram? 205
14.5 Embedded Path Creation 206
14.6 How Slow is Exp()? 207
14.7 Parallel Computing And Multi-threading 209
Bibliography 213
Index 219
