Burcu Aydoğan, M.Sc.

Department of Mathematics

July 2014

*Supervisor*: Ümit Aksoy (Department of Mathematics, Atılım University, Ankara)*Co-supervisor*: Ömür Uğur (Institute of Applied Mathematics, Middle East Technical University, Ankara)

### Abstract

In financial mathematics, option pricing is a popular problem in theory of finance and mathematics. In option pricing theory, the valuation of American options is one of the most important problems. American options are the most traded option styles in all financial markets. In spite of the recent developments, the valuation of American options continues to exist as a challenging problem. There are no closed-form analytical solutions of American options, so that a usual way to deal with this problem is to develop numerical and approximation techniques.

In this thesis, we analyze binomial, finite difference and approximation methods, for pricing American options. We first consider the binomial approximation which is very easy to implement and assumes that the asset prices follow from geometric Brownian motion. Then, we present American options as a free boundary value problem based on Black-Scholes partial differential equation, which leads to a very famous model in finance theory, and formalize it as a linear complementarity problem. We refer to the projected successive over relaxation (PSOR) method to solve this problem. Although there are no closed-form solutions for American options, we deal with some analytical approximation methods to approach the option values. We demonstrate the applications of the each method and compare their solutions.

*Keywords*: American Options, Black-Scholes Equation, Binomial Method, Finite Difference Methods, Approximation Methods