1 Introduction to simulation and Monte Carlo.
1.1 Introduction.

1.2 Evaluating a de.nite integral.

1.3 Monte Carlo is integral estimation.

1.4 An example.

1.5 A simulation using Maple.

1.6 Problems.

2 Uniform random numbers.

2.1 Linear congruential generators.

2.1.1 Mixed linear congruential generators.

2.1.2 Multiplicative linear congruential generators.

2.2 Theoretical tests for random numbers.

2.2.1 Problems of increasing dimension.

2.3 Shu ed generator.

4 CONTENTS.

2.4 Empirical tests.

2.4.1 Frequency test.

2.4.2 Serial test.

2.4.3 Other 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.

4.1.1 Box-Müller method.

4.1.2 An improved envelope rejection method.

4.2 Lognormal distribution.

4.3 Bivariate normal density.

4.4 Gamma distribution.

4.4.1 Cheng.s log-logistic method.

4.5 Beta distribution.

4.5.1 Beta log-logistic method.

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.

5.2.1 Exceedance probabilities for sums of i.i.d. randomvari-ables.

5.3 Strati.ed sampling.

5.3.1 A Strati.cation example.

5.3.2 Post strati.cation.

5.4 Control variates.

5.5 Conditional Monte Carlo.

5.6 Problems.

6 Simulation and.nance.

6.1 Brownian motion.

6.2 Asset price movements.

6.3 Pricing simple derivatives and options.

6.3.1 European call.

6.3.2 European put.

6.3.3 Continuous income.

6.3.4 Delta hedging.

6.3.5 Discrete hedging.

6.4 Asian options.

6.4.1 Naive simulation.

6.4.2 Importance and strati.ed version.

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.

7.4.1 Discrete time Markov chains.

7.4.2 Continuous time 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 algorithm.

8.3 Reliability inference using an independence sampler.

8.4 Single component Metropolis-Hastings and Gibbs sampling.

8.4.1 Estimating multiple failure rates.

8.4.2 Capture-recapture.

8.4.3 Minimal repair.

8.5 Other aspects of Gibbs sampling.

8.5.1 Slice sampling.

8.5.2 Completions.

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.

10 References.