§ 1. Modeling Tools for Financial Options

1.1 Options

1.2 Model of the Financial Market

1.3 The Binomial Method

1.4 Risk-Neutral Valuation

1.5 Stochastic Processes

1.6 Diffusion Models and Itô's Lemma

§ 2. Generating Random Numbers with Specified Distributions

2.1 Uniform Deviates

2.2 Extension to Random Variables from Other Distributions

2.3 Normally Distributed Random Variables

§ 3. Monte Carlo Simulations with Stochastic Differential Equations

3.1 Approximation Error

3.2 Stochastic Taylor Expansion

3.3 Intermediate Values

3.4 Monte Carlo Simulation

3.5 Monte Carlo Methods for American Options

§ 4. Standard Methods for Standard Options

4.1 Foundations of Finite Difference Methods

4.2 Boundary Conditions

4.3 American Options as Free Boundary Problem

4.4 Numerical Computation of American Options

4.5 Accuracy and Error Control

4.6 Analytic Methods

§ 5. Finite Element Methods

5.1. Weighted Residuals

5.2 Galerkin Approach with Hat Function

5.3 Application to Standard Options

5.4 Application to Exotic Call Option

5.5 Error Estimates

§ 6. Pricing of Exotic Options

6.1 Exotic Options

6.2 Options depending on Several Assets

6.3 Asian Options

6.4 Stability Analysis of Convection-Diffusion Problems

6.5 Upwind Schemes and other Methods

6.6 High-Resolution Methods