Simulation and Monte Carlo – With Applications in Finance and MCMC

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Preface. Glossary. 1 Introduction to simulation and Monte Carlo. 1.1 Evaluating a definite integral. 1.2 Monte Carlo is integral estimation. 1.3 An example. 1.4 A simulation using Maple. 1.5 Problems. 2 Uniform random numbers. 2.1 Linear congruential generators. 2.2 Theoretical tests for random numbers. 2.3 Shuffled generator. 2.4 Empirical tests. 2.5 Combinations of generators. 2.6 The seed(s) in a random number generator. 2.7 Problems. 3 General methods for generating random variates. 3.1 Inversion of the cumulative distribution function. 3.2 Envelope rejection. 3.3 Ratio of uniforms method. 3.4 Adaptive rejection sampling. 3.5 Problems. 4 Generation of variates from standard distributions. 4.1 Standard normal distribution. 4.2 Lognormal distribution. 4.3 Bivariate normal density. 4.4 Gamma distribution. 4.5 Beta distribution. 4.6 Chi-squared distribution. 4.7 Student's t distribution. 4.8 Generalized inverse Gaussian distribution. 4.9 Poisson distribution. 4.10 Binomial distribution. 4.11 Negative binomial distribution. 4.12 Problems. 5 Variance reduction. 5.1 Antithetic variates. 5.2 Importance sampling. 5.3 Stratified sampling. 5.4 Control variates. 5.5 Conditional Monte Carlo. 5.6 Problems. 6 Simulation and finance. 6.1 Brownian motion. 6.2 Asset price movements. 6.3 Pricing simple derivatives and options. 6.4 Asian options. 6.5 Basket options. 6.6 Stochastic volatility. 6.7 Problems. 7 Discrete event simulation. 7.1 Poisson process. 7.2 Time-dependent Poisson process. 7.3 Poisson processes in the plane. 7.4 Markov chains. 7.5 Regenerative analysis. 7.6 Simulating a G/G/1 queueing system using the three-phase method. 7.7 Simulating a hospital ward. 7.8 Problems. 8 Markov chain Monte Carlo. 8.1 Bayesian statistics. 8.2 Markov chains and the Metropolis-Hastings (MH) algorithm. 8.3 Reliability inference using an independence sampler. 8.4 Single component Metropolis-Hastings and Gibbs sampling. 8.5 Other aspects of Gibbs sampling. 8.6 Problems. 9 Solutions. 9.1 Solutions 1. 9.2 Solutions 2. 9.3 Solutions 3. 9.4 Solutions 4. 9.5 Solutions 5. 9.6 Solutions 6. 9.7 Solutions 7. 9.8 Solutions 8. Appendix 1: Solutions to problems in Chapter 1. Appendix 2: Random Number Generators. Appendix 3: Computations of acceptance probabilities. Appendix 4: Random variate generators (standard distributions). Appendix 5: Variance Reduction. Appendix 6: Simulation and Finance. Appendix 7: Discrete event simulation. Appendix 8: Markov chain Monte Carlo. References. Index.