Abstract
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.