Since the formulation by Black, Scholes, and Merton in 1973 of the first rational option pricing formula which depended only on observable values, the volume of options traded daily on the Chicago Board of Exchange has grown rapidly. In fact, over the past three decades, options have undergone a transformation from specialized and obscure securities to ubiquitous components of the portfolios of not only large fund managers, but of ordinary individual investors. Essential ingredients of any successful modern investment strategy include the ability to generate income streams and reduce risk, as well as some level of speculation, all of which can be accomplished by effective use of options.Naturally practitioners require an accurate method of pricing options. Furthermore, because today's market conditions evolve very rapidly, they also need to be able to obtain the price estimates quickly. This dissertation is devoted primarily to improving the efficiency of popular valuation procedures for stock options. In particular, we develop a method of simulating values of European stock options under the Heston stochastic volatility model in a fraction of the time required by the existing method. We also develop an efficient method of simulating the values of American stock option values under the same dynamic in conjunction with the Least-Squares Monte Carlo (LSM) algorithm. We attempt to improve the efficiency of the LSM algorithm by utilizing quasi-Monte Carlo techniques and spline methodology. We also consider optimal investor behavior and consider the notion of option trading as opposed to the much more commonly studied valuation problems.
