Exploring the use of Monte Carlo simulation in finance, this text reviews the essential mathematics and presents simple financial models. Beginning with the basics of Monte Carlo, the author gradually introduces advanced variance reduction techniques, covering such topics as importance sampling and stratified sampling. He also discusses numerical approximation, option pricing, and sensitivity analysis. The text presents diffusion techniques for diffusion models, American options, and sensitivity analysis. It also contains various types of exercises, progressive MATLAB(R)-based coding assignments, and computational projects.
Review of Basic Probability Theory Probability Space Random Variables Random Vectors Conditional Distributions Conditional Expectation Limit Theorems Basic Statistical Estimation Brownian Motion and Multivariate Normal Introduction Brownian Motion Multivariate Normal Distributions Multidimensional Brownian Motions Arbitrage Free Pricing Introduction Asset Pricing with Binomial Trees Black-Scholes Formula Monte Carlo Simulation Introduction Efficiency of an Estimate A Trivial Example Bias of an Estimate Pros and Cons of Monte Carlo Collection of Simple Examples Generating Random Variable Introduction Inverse Transform Method Generating Samples from Mixture Generating Multivariate Normal Distributions Variance Reduction in Monte Carlo Simulation Introduction Antithetic Sampling Stratified Sampling Importance Sampling Introduction Basic Idea of Importance Sampling Guideline for Selecting Alternative Distribution Importance Sampling for Normal Distributions Advanced Examples and General Discussions Preliminary Stochastic Calculus and SDE Introduction Stochastic Integral An Example of Stochastic Integral Ito Formula Stochastic Differential Equations Examples Numerical Approximation of SDE Introduction Euler Scheme Special Cases with No Discretization Error Refinement of Euler Scheme Examples American Option Pricing and Dynamic Programming Introduction The Price of an American Option American Option with Binomial Trees Diffusion Models: Binomial Approximation Call Options Difficulty of Monte Carlo for American Option Pricing Sensitivity Analysis Introduction Calibration vs. Estimation Sensitivity Analysis Monte Carlo Method for Greeks
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