This dissertation considers an American put option written on a single underlying which does not pay dividends, for which no closed form solution exists. As a conse- quence, numerical techniques have been developed to estimate the value of the Amer- ican put option. These include analytical approximations, tree or lattice methods, ¯nite di®erence methods, Monte Carlo simulation and integral representations. We ¯rst present the mathematical descriptions underlying these numerical techniques. We then provide an examination of a selection of algorithms from each technique, including implementation details, possible enhancements and a description of the convergence behaviour. Finally, we compare the estimates and the execution times of each of the algorithms considered.