Simulation and Monte Carlo with applications in finance and MCMC

Simulation and Monte Carlo......Page 3
Contents......Page 9
Preface......Page 13
Glossary......Page 15
1 Introduction to simulation and Monte Carlo......Page 17
1.1 Evaluating a definite integral......Page 18
1.2 Monte Carlo is integral estimation......Page 20
1.3 An example......Page 21
1.4 A simulation using Maple......Page 23
1.5 Problems......Page 29
2 Uniform random numbers......Page 33
2.1.1 Mixed linear congruential generators......Page 34
2.1.2 Multiplicative linear congruential generators......Page 38
2.2 Theoretical tests for random numbers......Page 41
2.2.1 Problems of increasing dimension......Page 42
2.3 Shuffled generator......Page 44
2.4.1 Frequency test......Page 45
2.4.3 Other empirical tests......Page 46
2.5 Combinations of generators......Page 47
2.7 Problems......Page 48
3.1 Inversion of the cumulative distribution function......Page 53
3.2 Envelope rejection......Page 56
3.3 Ratio of uniforms method......Page 60
3.4 Adaptive rejection sampling......Page 64
3.5 Problems......Page 68
4.1.1 Box–Müller method......Page 75
4.1.2 An improved envelope rejection method......Page 77
4.2 Lognormal distribution......Page 78
4.3 Bivariate normal density......Page 79
4.4 Gamma distribution......Page 80
4.4.1 Cheng’s log-logistic method......Page 81
4.5.1 Beta log-logistic method......Page 83
4.7 Student’s t distribution......Page 85
4.8 Generalized inverse Gaussian distribution......Page 87
4.9 Poisson distribution......Page 89
4.11 Negative binomial distribution......Page 90
4.12 Problems......Page 91
5.1 Antithetic variates......Page 95
5.2 Importance sampling......Page 98
5.2.1 Exceedance probabilities for sums of i.i.d. random variables......Page 102
5.3 Stratified sampling......Page 105
5.3.1 A stratification example......Page 108
5.3.2 Post stratification......Page 112
5.4 Control variates......Page 114
5.5 Conditional Monte Carlo......Page 117
5.6 Problems......Page 119
6 Simulation and finance......Page 123
6.1 Brownian motion......Page 124
6.2 Asset price movements......Page 125
6.3 Pricing simple derivatives and options......Page 127
6.3.1 European call......Page 129
6.3.2 European put......Page 130
6.3.4 Delta hedging......Page 131
6.3.5 Discrete hedging......Page 132
6.4.1 Naive simulation......Page 134
6.4.2 Importance and stratified version......Page 135
6.5 Basket options......Page 139
6.6 Stochastic volatility......Page 142
6.7 Problems......Page 146
7 Discrete event simulation......Page 151
7.1 Poisson process......Page 152
7.2 Time-dependent Poisson process......Page 156
7.3 Poisson processes in the plane......Page 157
7.4.1 Discrete-time Markov chains......Page 158
7.4.2 Continuous-time Markov chains......Page 159
7.5 Regenerative analysis......Page 160
7.6 Simulating a G/G/1 queueing system using the three-phase method......Page 162
7.7 Simulating a hospital ward......Page 165
7.8 Problems......Page 167
8.1 Bayesian statistics......Page 173
8.2 Markov chains and the Metropolis–Hastings (MH) algorithm......Page 175
8.3 Reliability inference using an independence sampler......Page 179
8.4 Single component Metropolis–Hastings and Gibbs sampling......Page 181
8.4.1 Estimating multiple failure rates......Page 183
8.4.2 Capture–recapture......Page 187
8.4.3 Minimal repair......Page 188
8.5.1 Slice sampling......Page 192
8.5.2 Completions......Page 194
8.6 Problems......Page 195
9.2 Solutions 2......Page 203
9.3 Solutions 3......Page 206
9.4 Solutions 4......Page 207
9.5 Solutions 5......Page 211
9.6 Solutions 6......Page 212
9.7 Solutions 7......Page 218
9.8 Solutions 8......Page 221
Appendix 1: Solutions to problems in Chapter 1......Page 225
Appendix 2: Random number generators......Page 243
Appendix 3: Computations of acceptance probabilities......Page 245
Appendix 4: Random variate generators (standard distributions)......Page 249
Appendix 5: Variance reduction......Page 255
Appendix 6: Simulation and finance......Page 265
Appendix 7: Discrete event simulation......Page 299
Appendix 8: Markov chain Monte Carlo......Page 315
References......Page 341
Index......Page 345