NO FULL TEXT AVAILABLE. This thesis contains 3rd party copyright material. ----- Interest rate models have been long of interest to researchers as interest rates play
a vital role in all aspects of the economy. The no arbitrage framework for interest
rate modelling has gained huge popularity due to, first of all, its focus on eliminating
arbitrage opportunities in the market, and no less importantly, its flexibility in
capturing the current shape of the yield curve as well as its ability not to make any
particular assumptions on investor preferences. Among arbitrage models, the Heath-
Jarrow-Morton (1992) (hereafter HJM) framework is a well-developed, quite general
and parsimonious one, which in addition to the currently observed forward curve only
requires a specification for the volatility functions of the stochastic differential equations
describing the evolution of the instantaneous forward interest rates. The evolution
of related economic quantities, such as spot rates, bond prices, option prices are then
derived as a consequence.
The volatility structure of interest rate markets therefore becomes the most crucial
element in the specification of no-arbitrage interest rate models, the pricing of interest
rate derivative securities and the management of interest rate risk. Despite its crucial
role, there is still a paucity of empirical analyses of this volatility structure, mainly due
to the difficulties in the estimation of the models involved These difficulties stem from
the fact that the stochastic differential system describing the interest rate models in the
most general form of the HJM framework is nonlinear, non-Markovian and involves
latent variables. The aim of this thesis is to propose estimation techniques for a broad
class of HJM models and apply them to analyse the volatility structure of different
interest rate markets.
The challenge of the estimation of HJM models is apparent even when the stochastic
differential equation for the instantaneous forward rates is time deterministic.
This evolution for the unobservable forward rates can only be used directly in the estimation
with some form of proxy, and this will lead to non-negligible estimation bias,
as demonstrated in the thesis. The evolution equation for the observed bond prices is
nonlinear and non-Markovian. However, it is shown in the thesis that the evolution
of futures contract prices can be derived analytically, and this evolution, though still
nonlinear, is Markovian. Consequently, a maximum likelihood estimator can be used
to estimate the system efficiently. The framework is also expanded to allow a jump
volatility component to be present in the market. This work, reported in Chapter 4 and
Chapter 5, constitutes the first two original contribution chapters of the thesis.
The framework outlined above uses data from the futures markets, which are very
liquid, but contain mostly short-maturity contracts. It is usually argued that different
volatility components cannot be distinguished using short maturity futures contracts.
However, the difficulty in separating different factors may just arise from the numerical
technique involved in the estimation. The likelihood function that needs to be maximized
may not be smooth, a situation required by traditional hill-climbing optimization
methods to locate the global optimum. Here a genetic algorithm is proposed and then
used successfully to estimate a three-factor model for the futures market. Even though
the computational task involved is quite demanding, it turns out to be worthwhile, as
the inclusion of a multi-factor Wiener volatility is found to obviate the need to include
a jump volatility component, making the model more attractive for practical uses, such
as pricing options or making forecasts. This study, reported in Chapter 6, constitutes
the third original chapter of the thesis.
If the stochastic system describing the interest rate models has a volatility function
(the diffusion term) dependent on the underlying state variables, the estimation
methods of the earlier chapters cannot be applied. In Chapter 7 the system is then
converted into a Markovian one, so that filtering techniques became applicable. As the
system is nonlinear, the local linearization filter of Jimenez and Ozaki (2003), which
is known to have some desirable statistical and numerical features, is adopted to estimate
the model via the maximum likelihood method The likelihood function again
is non-smooth (multi-modal and discontinuous), which calls for the use of the genetic
algorithm developed in the previous chapter. The estimator is then applied to the U.S,
U.K and Australian interest rate markets. This constitutes the final original contribution
chapter of the thesis.