A finite element approach to pricing Barrier options

Abstract

In this dissertation we consider the valuation of discretely monitored barrier options under the in nite element method. The in nite element method is an extension to the standard nite element method that accepts problems with unbounded spacial domains (such as the Black-Scholes PDE), without resorting to domain truncation. The degeneracy of the Black-Scholes PDE when the underlying asset reaches zero, requires that the method be formulated within the context of weighted Sobolev spaces. We will demonstrate the convergence of the proposed method and provide a rigorous investigation into the underlying weighted Sobolev spaces in which the convergence is to be demonstrated.