Spectral Discretizations of Option Pricing Models for European Put Options

Abstract

The aim of this thesis is to solve option pricing models efficiently by using spectral methods. The option pricing models that will be solved are the Black-Scholes model and Heston's stochastic volatility model. We will restrict us to pricing European put options. We derive the partial differential equations governing the two models and their corresponding weak formulations. The models are then solved using both the spectral Galerkin method and a polynomial collocation method. The numerical solutions are compared to the exact solution. The exact solution is also used to study the numerical convergence. We compare the results from the two numerical methods, and look at the time consumptions of the different methods. Analysis of the methods are also given. This includes coercivity, continuity, stability and convergence estimates.For Black-Scholes equation, we study both the original equation and the log transformed equation, and we also compare the results to a solution obtained by using a finite element method solver.