Matthias Ehrhardt: 14.-22.01.08
Finite Difference and Finite Element Methods in Option Pricing

MO 14.01 Lecture 1: Foundations of finite difference methods
         Motivation
           space and time discretizations
           Construction of finite difference approximations
         The Matrix Approach for Stability Analysis
            space and time discretizations
            shift operator, amplification factor

Di 15.01 Lecture2: (Exercise) Stability Analysis of specific Methods using the Matrix Approach
         Example 1: explicit Euler scheme
         Example 2: implicit Euler scheme
         Example 3: Crank-Nicolson method
         Example 4: Predictor Corrector method
         Example 5: Multiple-level methods

Mi 16.01 Lecture 3: (Exercise) The Fourier Approach for Stability Analysis
         Example 1: theta scheme
         Example 2: Multiple-level methods
         The modified equation approach 

         Lecture 4: Implementation of the time advancement
         Solving Spares Systems of Linear Equations
           Direct Solvers
              tridiagonal solver (Thomas Algorithm)
              (sparse) LU decomposition
           Iterative Solvers
              stationary methods
                 Jacobi-, Gauss-Seidel- SOR methods
              nonstationary methods
                 conjugate gradient method
           Preconditioner

DO 17.01 Lecture 5: Boundary Conditions for Option Pricing
         Boundary Conditions for European Options
         American Options as free boundary value problems
         continuation and stopping regions
         Boundary Conditions for American Put/Call Options

FR 18.01 Lecture 6: American Options as obstacle problems
         Black-Scholes inequality
         Obstacle Problems
            linear complementarity problem (LCP)
         Formulation as Variational Inequality
         Discretization of the Obstacle Problem
         Linear Complementarity for American Put Options

         Lecture 7 (Exercise): MATLAB-Implementation of finite difference
                               codes for European Options

MO 21.01 Lecture 8: The method of finite elements in option pricing
         principle of weighted residuals
           basis functions, residual, weighting functions
         Examples of weighting functions
           Galerkin, Collocation, least squares, subdomain
         Examples of basis functions
           hat functions
         Galerkin approach with Hat functions
            Properties of Hat functions
            element-mass matrix, element-stiffness matrix
            simple application
            assembling procedure
            mass matrix, stiffness matrix
         Application to Standard Options


DI 22.01 Lecture 9: Compact Finite difference methods for nonlinear BS equations
         

         Lecture 10: Pricing of Basket Options with the Mellin Transform