An analytic solution does not exist for evaluating the American put option. Usually, the value is obtained by applying numerical methods. For instance, the PSOR algorithm is a widely used one in financial industry. In the past few years, many other methods to solve American option problems have been introduced, two examples are Linear Programming and Penalty method. The aims of this dissertation are: first, to provide an introduction to four algorithms - Explicit, PSOR, Penalty and Linear Pro...