Handbook in Monte Carlo Simulation

PAOLO BRANDIMARTE is Full Professor of Quantitative Methods for Finance and Logistics in the Department of Mathematical Sciences at Politecnico di Torino in Italy. He has extensive teaching experience in engineering and economics faculties, including master’s- and PhD-level courses. Dr. Brandimarte is the author or coauthor of Introduction to Distribution Logistics, Quantitative Methods: An Introduction for Business Management, and Numerical Methods in Finance and Economics: A MATLAB-Based Introduction, Second Edition, all published by Wiley.

• Preface xiiiPart I Overview and Motivation1 Introduction to Monte Carlo Methods 31.1 Historical origin of Monte Carlo simulation 41.2 Monte Carlo Simulation vs. Monte Carlo Sampling 71.3 System dynamics and the mechanics of Monte Carlo simulation 101.4 Simulation and optimization 211.5 Pitfalls in Monte Carlo simulation 301.6 Software tools for Monte Carlo simulation 351.7 Prerequisites 37For further reading 38Chapter References 382 Numerical Integration Methods 412.1 Classical quadrature formulae 432.2 Gaussian quadrature 482.3 Extension to higher dimensions: Product rules 532.4 Alternative approaches for high-dimensional integration 552.5 Relationship with moment matching 672.6 Numerical integration in R 69For further reading 71Chapter References 71Part II Input Analysis: Modeling and Estimation3 Stochastic Modeling in Finance and Economics 753.1 Introductory examples 773.2 Some common probability distributions 863.3 Multivariate distributions: Covariance and correlation 1113.4 Modeling dependence with copulae 1273.5 Linear regression models: a probabilistic view 1363.6 Time series models 1373.7 Stochastic differential equations 1583.8 Dimensionality reduction 177S3.1 Risk-neutral derivative pricing 190S3.1.1 Option pricing in the binomial model 192S3.1.2 A continuous-time model for option pricing: The Black–Scholes–Merton formula 194S3.1.3 Option pricing in incomplete markets 199For further reading 202Chapter References 2034 Estimation and Fitting 2054.1 Basic inferential statistics in R 2074.2 Parameter estimation 2154.3 Checking the fit of hypothetical distributions 2244.4 Estimation of linear regression models by ordinary least squares 2294.5 Fitting time series models 2324.6 Subjective probability: the Bayesian view 235For further reading 244Chapter References 245Part III Sampling and Path Generation5 Random Variate Generation 2495.1 The structure of a Monte Carlo simulation 2505.2 Generating pseudo-random numbers 2525.3 The inverse transform method 2635.4 The acceptance–rejection method 2655.5 Generating normal variates 2695.6 Other ad hoc methods 2745.7 Sampling from copulae 276For further reading 277Chapter References 2796 Sample Path Generation for Continuous-Time Models 2816.1 Issues in path generation 2826.2 Simulating geometric Brownian motion 2876.3 Sample paths of short-term interest rates 2986.4 Dealing with stochastic volatility 3066.5 Dealing with jumps 308For further reading 310Chapter References 311Part IV Output Analysis and Efficiency Improvement7 Output Analysis 3157.1 Pitfalls in output analysis 3177.2 Setting the number of replications 3237.3 A world beyond averages 3257.4 Good and bad news 327For further reading 327Chapter References 3288 Variance Reduction Methods 3298.1 Antithetic sampling 3308.2 Common random numbers 3368.3 Control variates 3378.4 Conditional Monte Carlo 3418.5 Stratified sampling 3448.6 Importance sampling 350For further reading 363Chapter References 3639 Low-Discrepancy Sequences 3659.1 Low-discrepancy sequences 3669.2 Halton sequences 3679.3 Sobol low-discrepancy sequences 3749.4 Randomized and scrambled low-discrepancy sequences 3799.5 Sample path generation with low-discrepancy sequences 381For further reading 385Chapter References 385Part V Miscellaneous Applications10 Optimization 38910.1 Classification of optimization problems 39010.2 Optimization model building 40510.3 Monte Carlo methods for global optimization 41210.4 Direct search and simulation-based optimization methods 41610.5 Stochastic programming models 42010.6 Scenario generation and Monte Carlo methods for stochastic programming 42810.7 Stochastic dynamic programming 43310.8 Numerical dynamic programming 44010.9 Approximate dynamic programming 451For further reading 453Chapter References 45311 Option Pricing 45511.1 European-style multidimensional options in the BSM world 45611.2 European-style path-dependent options in the BSM world 46211.3 Pricing options with early exercise features 47511.4 A look outside the BSM world 48711.5 Pricing interest-rate derivatives 490For further reading 497Chapter References 49812 Sensitivity Estimation 50112.1 Estimating option greeks by finite differences 50312.2 Estimating option greeks by pathwise derivatives 50912.3 Estimating option greeks by the likelihood ratio method 513For further reading 517Chapter References 51813 Risk Measurement and Management 51913.1 What is a risk measure? 52013.2 Quantile-based risk measures: value at risk 52213.3 Monte Carlo methods for V@R 53313.4 Mean-risk models in stochastic programming 53713.5 Simulating delta-hedging strategies 54013.6 The interplay of financial and nonfinancial risks 546For further reading 548Chapter References 54814 Markov Chain Monte Carlo and Bayesian Statistics 55114.1 An introduction to Markov chains 55214.2 The Metropolis–Hastings algorithm 55514.3 A re-examination of simulated annealing 558For further reading 560Chapter References 561Index 563