Use of numerical PDE software for the solution of a classic problem in mathematical finance

Abstract

Mathematical modelling is an important part of the finance industry. These models can be very complex and one often needs to use numerical methods and numerical software packages to get approximate solutions to these models. In this thesis we use a high quality numerical software package called EPDCOL which is designed to solve systems of linear and non-linear partial differential equations (PDEs) and has temporal error control. Specifically, we will use it to numerically solve the Black-Scholes equation, a linear PDE, that can be used to value a financial instrument known as an option. An option is a contract which gives the holder the option to buy or sell a stock at a future time for an agreed upon price. A stock represents ownership of a corporation's assets and gives the opportunity to share in the corporation's earnings. EPDCOL implements a combination of high quality numerical methods which allow us to solve the Black-Scholes equation for two types of options (puts and calls) involving various parameters. Additionally, we are able to get an estimate of the spatial error associated with the numerical solutions.