Option Pricing through Numerical Methods for PDEs: An Object-Oriented Python Implementation

This thesis explores the numerical methods used for solving partial differential equations (PDEs) that arise in financial mathematics, specifically in the context of option pricing. We begin by introducing the Black-Scholes framework, which serves as the foundation for pricing European options, as well as more complex derivatives such as Barrier and Asian options. The work then delves into the finite difference and finite element methods (FEM), focusing on their application to the Black-Scholes PDE and its variants. Our approach includes the development of an object-oriented Python implementation that leverages the NumPy and SciPy libraries for efficient numerical computation. We also integrate the PyFreeFem API to utilize the capabilities of the FreeFem++ language for finite element analysis. Through this implementation, we solve a variety of option pricing problems, examining the convergence and accuracy of different numerical methods. The results demonstrate the effectiveness of these methods in approximating option prices under various market conditions. We provide detailed comparisons between numerical and analytical solutions where available, as well as an analysis of the computational efficiency of different techniques. The thesis concludes with a discussion on the limitations of the current implementation and suggestions for future enhancements, particularly in extending the framework to more complex financial instruments and higher-dimensional problems.