Monte Carlo methods in finance

Contents......Page 3
Preface......Page 9
Mathematical Notation......Page 13
Introduction......Page 17
A few basic terms in probability and statistics......Page 21
Monte Carlo simulations......Page 23
Monte Carlo supremacy......Page 24
Some common distributions......Page 25
Kolmogorov\'s strong law......Page 34
Error estimation for Monte Carlo methods......Page 35
The Feynman-Kac theorem......Page 37
The Moore-Penrose pseudo-inverse......Page 38
Brownian motion......Page 39
Itô\'s lemma......Page 40
Normal processes......Page 41
Lognormal processes......Page 42
The Markovian Wiener process embedding dimension......Page 43
Constant elasticity of variance processes......Page 44
Displaced diffusion......Page 45
Strong versus weak convergence......Page 47
The Euler scheme......Page 48
Transformations......Page 49
Spurious paths......Page 52
Strong convergence for Euler and Milstein......Page 53
Correlation and co-movement......Page 57
Measures for co-dependence......Page 58
Copulæ......Page 61
The Gaussian copula......Page 62
The t-copula......Page 65
Archimedean copulæ......Page 66
Salvaging a linear correlation matrix......Page 73
Hypersphere decomposition......Page 74
Spectral decomposition......Page 75
Angular decomposition of lower triangular form......Page 77
Examples......Page 78
Angular coordinates on a hypersphere of unit radius......Page 79
Pseudo-random numbers......Page 81
Chaos......Page 82
The mid-square method......Page 86
Congruential generation......Page 87
Ran0 to Ran3......Page 88
The Mersenne twister......Page 89
Which one to use?......Page 90
Low-discrepancy numbers......Page 91
Discrepancy......Page 92
Halton numbers......Page 93
Sobol\' numbers......Page 94
Primitive polynomials modulo two......Page 95
The construction of Sobol\' numbers......Page 96
The Gray code......Page 98
The initialisation of Sobol\' numbers......Page 99
Pairwise projections......Page 102
Empirical discrepancies......Page 104
Explicit formula for the L2-norm discrepancy on the unit hypercube......Page 109
Expected L2-norm discrepancy of truly random numbers......Page 110
Inversion of the cumulative probability function......Page 111
Using a sampler density......Page 112
Importance sampling......Page 115
Rejection sampling......Page 116
The Box-Muller method......Page 117
The Neave effect......Page 119
Simulating multi-variate copula draws......Page 120
Antithetic sampling......Page 123
Variate recycling......Page 124
Control variates......Page 125
Stratified sampling......Page 127
Importance sampling......Page 128
Moment matching......Page 129
Latin hypercube sampling......Page 132
Path construction......Page 133
Spectral......Page 134
The Brownian bridge......Page 137
A comparison of path construction methods......Page 140
Multivariate path construction......Page 142
Eigenvalues and eigenvectors of a discrete-time covariance matrix......Page 145
The conditional distribution of the Brownian bridge......Page 148
Importance of Greeks......Page 151
An Up-Out-Call option......Page 152
Finite differencing with path recycling......Page 153
Finite differencing with importance sampling......Page 155
Pathwise differentiation......Page 156
The likelihood ratio method......Page 157
Comparative figures......Page 159
Summary......Page 161
The likelihood ratio formula for Vega......Page 163
The likelihood ratio formula for Rho......Page 165
The Brace-Gatarek-Musiela/Jamshidian market model......Page 167
Factorisation......Page 170
Bermudan swaptions......Page 171
Calibration to European swaptions......Page 172
The Predictor-Corrector scheme......Page 178
Heuristics of the exercise boundary......Page 181
The algorithm......Page 184
Numerical results......Page 185
Summary......Page 189
Introduction......Page 191
Evolving the forward rates......Page 193
Optimal simplex alignment......Page 195
Implementation......Page 198
Convergence performance......Page 199
Variance matching......Page 200
Exact martingale conditioning......Page 204
Clustering......Page 205
Summary......Page 207
Interpolation of the term structure of implied volatility......Page 209
Watch your cpu usage......Page 212
Numerical overflow and underflow......Page 213
A single number or a convergence diagram?......Page 214
How slow is exp()?......Page 215
Parallel computing and multi-threading......Page 217
Bibliography......Page 221
Index......Page 231