Valuation of derivative securities using stochastic analytic and numerical methods

Abstract

This thesis details methods and procedures to compute prices and hedging strategies for derivative securities in financial mathematics using stochastic analytic, numerical and variance reduction techniques. Results are obtained on explicit hedge ratio representations for non-smooth payoff functionals and mult idimensional diffusion processes with stopping boundaries. These methods are used to determine hedge ratios for the maximum of several assets and lookback options. A number of powerful variance reduction techniques are described. These include the use of measure transformations, discrete versions of importance sampling estimators, control variates based on Ito integral representations, stratified sampling and quasi Monte Carlo. For many of these techniques explicit formulas for the variance of the resulting estimators are obtained. Pricing and hedging procedures are developed for a class of foreign exchange barrier options under stochastic volatility. These procedures are applied to the calculation of down-and-out call options for the Heston model. A general methodology for pricing discount bonds and options on discount bonds for multifactor term structure models is established. This approach is used for both European and American style securities for a version of the two-factor Fong and Vasicek model, extended to include time dependent parameters. For American pricing an exact representation of the early exercise premium is derived for a class of one-factor models. This enables both American option prices and the corresponding two-dimensional critical exercise boundary to be computed for the extended Fong and Vasicek model.