Monte Carlo Simulation with Applications to Finance

Review of Probability
Probability Space
Independence and Conditional Probability
Random Variables
Random Vectors
Conditional Distributions
Conditional Expectation
Classical Limit Theorems

Brownian Motion
Brownian Motion
Running Maximum of Brownian Motion
Derivatives and Black–Scholes Prices
Multidimensional Brownian Motions

Arbitrage Free Pricing
Arbitrage Free Principle
Asset Pricing with Binomial Trees
The Black–Scholes Model

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

Generating Random Variables
Inverse Transform Method
Acceptance-Rejection Method
Sampling from Multivariate Normal Distributions

Variance Reduction Techniques
Antithetic Sampling
Control Variates
Stratified Sampling

Importance Sampling
Basic Ideas of Importance Sampling
The Cross-Entropy Method
Applications to Risk Analysis

Stochastic Calculus
Stochastic Integrals
Itô Formula
Stochastic Differential Equations
Risk-Neutral Pricing
Black–Scholes Equation

Simulation of Diffusions
Euler Scheme
Eliminating Discretization Error
Refinements of Euler Scheme
The Lamperti Transform
Numerical Examples

Sensitivity Analysis
Commonly Used Greeks
Monte Carlo Simulation of Greeks

Appendix A: Multivariate Normal Distributions
Appendix B: American Option Pricing
Appendix C: Option Pricing Formulas