Thesis (Ph. D.)--University of Toronto, 2004. / Adviser: Ruth Childs. Includes bibliographical references.

M.Sc. (Mathematical Statistics) / Stochastic Differential Equations (SDE’s) are commonly found in most of the modern finance used today. In this dissertation we use SDE’s to model a random phenomenon known as the short-term interest rate where the explanatory power of a particular short-term interest rate model is largely dependent on the description of the SDE to the real data. The challenge we face is that in most cases the transition density functions of these models are unknown and therefore, we need to find reliable and accurate alternative estimation techniques. In this dissertation, we discuss estimating techniques for discretely sampled continuous diffusion processes that do not require the true transition density function to be known. Moreover, the reader is introduced to the following techniques: (i) continuous time maximum likelihood estimation; (ii) discrete time maximum likelihood estimation; and (iii) estimating functions. We show through a Monte Carlo simulation study that the parameter estimates obtained from these techniques provide a good approximation to the estimates obtained from the true transition density. We also show that the bias in the mean reversion parameter can be reduced by implementing the jackknife bias reduction technique. Furthermore, the data analysis carried out on South-African interest rate data indicate strongly that single factor models do not explain the variability in the short-term interest rate. This may indicate the possibility of distinct jumps in the South-African interest rate market. Therefore, we leave the reader with the notion of incorporating jumps into a SDE framework.