Three essays in computational finance
This thesis consists of three essays. The first essay experimentally investigates the optimal use of Richardson extrapolation in order to compute option prices, balancing computational requirements and precision. We specifically investigate the number of points to use in Richardson extrapolation and the combination of points to be used for obtaining the best results. We consider the cases of the American, Barrier and options on 2 underlying assets. We develop models to explain the behavior of the option prices as a function of the step size used in the tree. Our models predict that for a fixed computing budget it is optimal to use 2-point Richardson extrapolation rather than 3-point Richardson extrapolation. From the results obtained from the models, we recommend that 2-point Richardson extrapolation be used for pricing of the above three classes of options with the ratio of the number of time steps in two option pricing trees chosen to lie between 2 and 4.
In the second essay, we develop two methods to replicate the High Yield Index, which is a carefully constructed portfolio of High Yield bonds, meant to reflect the U.S. High Yield market. The first methodology, which we refer to as the Sub-Index methodology, is based on choosing a subset of bonds in the High Yield Index with appropriate weights to replicate the High Yield Index as closely as possible. This methodology exploits the special structure of the High Yield Index to construct a replicating portfolio using a linear programming technique that relies only on the current composition of the High Yield Index. The second methodology is based on acquiring an understanding of the relationship between the High Yield market, equity market, derivatives in the equity market and the U.S. Government bond market, using historical data and then using optimization techniques to form the index-tracking portfolio, with the assumption that the historical behavior of the relationship is a reasonable predictor of future behavior.
In the third essay, we develop a Monte Carlo simulation based methodology to estimate the probability of default/defaults and the distribution of loss due to default/defaults in a basket of N bonds. We further develop a variance reduction technique that combines importance sampling based on a change of drift and stratified sampling along an appropriate direction to enhance the accuracy of our estimates. The optimal drift vector is selected by solving an appropriate optimization problem. The optimal drift vector also determines the stratification direction. We show the applicability of this method for computing the probability of one and two defaults and the distribution of losses due to one and two defaults in a basket of N bonds.