Spectral Approximation of European and American Option Pricing Problems

Abstract

In this thesis, properties of spectral methods applied to option pricing problems are inves-tigated. The Legendre Galerkin method with numerical integration is applied to Europeanand American option pricing problems under the Black-Scholes model. The method iscoupled with an implicit time stepping technique for a full discretization. As a remedyfor non-smooth payoff functions in option pricing, the method is combined with domaindecomposition, where the domain is split at slope discontinuities. For the American pric-ing problem, an approach based on penalization is proposed. Numerical results show thatthe method provides spectral convergence for European pricing problems and fourth orderconvergence for American options. Stability and convergence is proved for the numericalscheme of the single-asset European pricing problem.