# Finite Difference Methods in Finance

**Dr. Christoph Wagner**

**Abstract**: The Feynman-Kac theorem relates the probabilistic representation of a pricing problem to the solution of a partial differential equation (PDE). In finance, the relevant PDEs are of the convection-diffusion-reaction type in n space variables and one time variable. The space variables correspond to underlying financial quantities such as an asset price or interest rate while the non-negative time variable t is bounded above by the expiration T of an option. The space variables take values in their respective positive price or rate half-planes. Since there exists rarely a closed form solution for the respective PDEs we need to resort to approximate methods. A popular and well established method is the finite difference method (FDM), which is essentially a discretisation of the PDE. We introduce this method and then apply it to various pricing problems in finance.

**Literature**:

- Duffy, D.: Finite difference methods in financial engineering, Wiley Finance (2006)
- Tavella, D., Randall, C.: Pricing FinancialInstruments: The Finite Difference Method, Wiley (2000)
- Evans, L.: Partial Differential Equations, American Mathematical Society (2010)

**Termin**: Montag 8-10 Uhr, Start 16.10.2017

**Prerequisites**: Financial Mathematics I+II, Econometrics, Probability

Theory