Option pricing in random field models with stochastic volatility for the term structure of interest rates

In this dissertation, we introduce a general interest rate modeling framework by looking at yield curves in a Hilbert space, and bridge the popular HJM factor models with more recent random field models. Then we study the problem of vanilla interest rate option (cap) pricing under the random field model. This will be a generalization of Kennedy paper in the sense that the volatility will also follow a random field process instead of being deterministic. In particular, we consider both cases in which the two random fields for forward rates and volatilities are independent or correlated. In the computation of option prices, we have proposed a log-normal approximation of the summary statistics - integrated volatility, for the independent case and have proposed a trivariate Gaussian approximation for the correlated case. The approximations will enable us to compute option prices much faster than the usual brute force Monte Carlo method which introduces certain discretization error. Finally, we perform simulation studies of a MCMC estimation procedure for a special random field model with one factor stochastic volatility