Preface xiii
 
Part I Overview and Motivation
 
1 Introduction to Monte Carlo Methods 3
 
1.1 Historical origin of Monte Carlo simulation 4
 
1.2 Monte Carlo Simulation vs. Monte Carlo Sampling 7
 
1.3 System dynamics and the mechanics of Monte Carlo simulation 10
 
1.4 Simulation and optimization 21
 
1.5 Pitfalls in Monte Carlo simulation 30
 
1.6 Software tools for Monte Carlo simulation 35
 
1.7 Prerequisites 37
 
For further reading 38
 
Chapter References 38
 
2 Numerical Integration Methods 41
 
2.1 Classical quadrature formulae 43
 
2.2 Gaussian quadrature 48
 
2.3 Extension to higher dimensions: Product rules 53
 
2.4 Alternative approaches for high-dimensional integration 55
 
2.5 Relationship with moment matching 67
 
2.6 Numerical integration in R 69
 
For further reading 71
 
Chapter References 71
 
Part II Input Analysis: Modeling and Estimation
 
3 Stochastic Modeling in Finance and Economics 75
 
3.1 Introductory examples 77
 
3.2 Some common probability distributions 86
 
3.3 Multivariate distributions: Covariance and correlation 111
 
3.4 Modeling dependence with copulae 127
 
3.5 Linear regression models: a probabilistic view 136
 
3.6 Time series models 137
 
3.7 Stochastic differential equations 158
 
3.8 Dimensionality reduction 177
 
S3.1 Risk-neutral derivative pricing 190
 
S3.1.1 Option pricing in the binomial model 192
 
S3.1.2 A continuous-time model for option pricing: The Black-Scholes-Merton formula 194
 
S3.1.3 Option pricing in incomplete markets 199
 
For further reading 202
 
Chapter References 203
 
4 Estimation and Fitting 205
 
4.1 Basic inferential statistics in R 207
 
4.2 Parameter estimation 215
 
4.3 Checking the fit of hypothetical distributions 224
 
4.4 Estimation of linear regression models by ordinary least squares 229
 
4.5 Fitting time series models 232
 
4.6 Subjective probability: the Bayesian view 235
 
For further reading 244
 
Chapter References 245
 
Part III Sampling and Path Generation
 
5 Random Variate Generation 249
 
5.1 The structure of a Monte Carlo simulation 250
 
5.2 Generating pseudo-random numbers 252
 
5.3 The inverse transform method 263
 
5.4 The acceptance-rejection method 265
 
5.5 Generating normal variates 269
 
5.6 Other ad hoc methods 274
 
5.7 Sampling from copulae 276
 
For further reading 277
 
Chapter References 279
 
6 Sample Path Generation for Continuous-Time Models 281
 
6.1 Issues in path generation 282
 
6.2 Simulating geometric Brownian motion 287
 
6.3 Sample paths of short-term interest rates 298
 
6.4 Dealing with stochastic volatility 306
 
6.5 Dealing with jumps 308
 
For further reading 310
 
Chapter References 311
 
Part IV Output Analysis and Efficiency Improvement
 
7 Output Analysis 315
 
7.1 Pitfalls in output analysis 317
 
7.2 Setting the number of replications 323
 
7.3 A world beyond averages 325
 
7.4 Good and bad news 327
 
For further reading 327
 
Chapter References 328
 
8 Variance Reduction Methods 329
 
8.1 Antithetic sampling 330
 
8.2 Common random numbers 336
 
8.3 Control variates 337
 
8.4 Conditional Monte Carlo 341
 
8.5 Stratified sampling 344
 
8.6 Importan