Pricing multi-windowed barrier options using finite element method

Abstract

In this thesis we study pricing multi-windowed barrier options under three different models: Black-Scholes' model, Heston model, and the multi-dimensional Heston model proposed by De Col, Gnoatto and Grasselli. The PDE approach is employed where the option price is deemed as the solution of a partial differential equation. The PDEs arising in the area of option pricing are most parabolic equations. The interesting questions are a) how to deal with the semi-infinite boundary; b) how to determine the boundary conditions when the domain changes with time. Especially we also consider the situation where the Feller condition is violated in the foreign exchange markets, which gives degenerate parabolic equations. We use the finite element method to obtain the numerical results of the PDEs. It is implemented by C++. The main result of this thesis provides a practical scheme in pricing options in a real market scenario. All the coefficients used in the multi-dimensional Heston model can be calibrated once for all according to the real markets. Then the model dimension can be reduced when different types of options are priced.