Numerical methods and software for the pricing of American financial derivatives

In recent years leading-edge financial institutions routinely use advanced analytical and numerical techniques from science and engineering to create, deploy, and manage new financial instruments. The proliferation and complexity of the available financial instruments in conjunction with the ever-increasing globalization and interplay of the world capital and equity markets have largely mandated this trend. This dissertation addresses the numerical solution of mathematical models used for the pricing of financial products called financial derivatives or options. The thrust of our research focuses on the development, analysis, and performance evaluation of numerical methods for solving American option pricing models, and the design of a problem solving framework to support the pricing process and its variations. We assume a free boundary partial differential equation model of the American option pricing problem and develop and analyze a class of numerical methods referred to as front-tracking methods utilizing finite element and finite difference discretization schemes. The central thesis of this dissertation is that for certain types of option pricing problems front-tracking methods merit closer attention as candidates for solving the option pricing model, both in terms of efficiency and robustness. Furthermore, the natural properties of the problem and its inherent characteristics can be exploited in a meaningful manner in order to develop a unifying software framework for delivering the computational solutions.