Monte Carlo Simulation with Applications to Finance

Developed from the author’s course on Monte Carlo simulation at Brown University, Monte Carlo Simulation with Applications to Finance provides a self-contained introduction to Monte Carlo methods in financial engineering. It is suitable for advanced undergraduate and graduate students taking a one-semester course or for practitioners in the financial industry.

The author first presents the necessary mathematical tools for simulation, arbitrary free option pricing, and the basic implementation of Monte Carlo schemes. He then describes variance reduction techniques, including control variates, stratification, conditioning, importance sampling, and cross-entropy. The text concludes with stochastic calculus and the simulation of diffusion processes.

Only requiring some familiarity with probability and statistics, the book keeps much of the mathematics at an informal level and avoids technical measure-theoretic jargon to provide a practical understanding of the basics. It includes a large number of examples as well as MATLAB® coding exercises that are designed in a progressive manner so that no prior experience with MATLAB is needed.

Hui Wang is an associate professor in the Division of Applied Mathematics at Brown University. He earned a Ph.D. in statistics from Columbia University. His research and teaching cover Monte Carlo simulation, mathematical finance, probability and statistics, and stochastic optimization.

Review of Probability Probability SpaceIndependence and Conditional ProbabilityRandom VariablesRandom VectorsConditional DistributionsConditional ExpectationClassical Limit Theorems

Brownian Motion Brownian MotionRunning Maximum of Brownian Motion Derivatives and Black–Scholes PricesMultidimensional Brownian Motions

Arbitrage Free PricingArbitrage Free PrincipleAsset Pricing with Binomial TreesThe Black–Scholes Model

Monte Carlo SimulationBasics of Monte Carlo Simulation Standard Error and Confidence Interval Examples of Monte Carlo Simulation Summary

Generating Random VariablesInverse Transform MethodAcceptance-Rejection MethodSampling from Multivariate Normal Distributions

Variance Reduction TechniquesAntithetic SamplingControl VariatesStratified Sampling

Importance SamplingBasic Ideas of Importance SamplingThe Cross-Entropy Method Applications to Risk Analysis

Stochastic Calculus Stochastic IntegralsItô FormulaStochastic Differential EquationsRisk-Neutral Pricing Black–Scholes Equation

Simulation of DiffusionsEuler SchemeEliminating Discretization ErrorRefinements of Euler SchemeThe Lamperti TransformNumerical Examples

Sensitivity AnalysisCommonly Used GreeksMonte Carlo Simulation of Greeks

Appendix A: Multivariate Normal DistributionsAppendix B: American Option PricingAppendix C: Option Pricing Formulas

Bibliography
Index

Exercises appear at the end of each chapter.