Essays on stochastic volatility and random-field models in finance
In this thesis we develop random-field models for the implied volatility of equity options and the term structure of interest rates. Following a brief introduction to the topics of this thesis in chapter 1, chapter 2 models the Black-Scholes implied volatility of plain-vanilla European stock options as a random field with three parameters: current time, the maturity date and the exercise price of the corresponding option. In this model all plain-vanilla European options are needed to complete the market. Illiquid and exotic derivatives can be priced as a function of the stock price and the implied volatility surface. In chapter 3 we develop a random-field model for forward interest rates with stochastic volatility. It is assumed that the forward rate volatility function can be decomposed into a deterministic function of the time to maturity and a maturity- independent stochastic process driven by a standard Brownian motion. The separability of the forward-rate volatility function allows closed-form solutions to be obtained for the prices of a number of interest rate derivatives: bond options, interest rate caplets, and interest rate spread options. Forward LIBOR and swap rates are modelled in a similar way, and closed-form solutions are derived for the prices of LIBOR caplets and swaptions. In chapter 4 we estimate three random-field models of the term structure of interest rates: one model with deterministic forward-rate volatility, and two with stochastic forward-rate volatility. The models axe estimated using seven years of daily UK and US forward rate data, spanning times to maturity between zero and 120 months. The parameters of each model are obtained by maximizing the likelihood function. We develop an importance sampling technique that substantially reduces the variance of the Monte Carlo estimator of the likelihood function in the case of stochastic volatility.