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Abstract
This thesis is concerned with the modelling of the term structure of interest rates, with a particular focus on empirical aspects of the modelling.
In this thesis, we explore the θ-parameterised (θ being the length of time to maturity) term structure of interest rates, corresponding to the traditional T-parameterised (T being the time of maturity) term structure of interest rates. The constructions of Australian yield curves are illustrated using generic yield curves produced by the Reserve Bank of Australia based on bonds on issue and by constructed yield curves of the Commonwealth Bank of Australia derived from swap rates.
The data used to build the models is Australian Treasury yields from January 1996 to December 2001 for maturities of 1, 2, 3, 5 and 10 years, and the second data used to validate the model is Australian Treasury yields from July 2000 to April 2004 for maturities of all years from 1-10. Both data were supplied by the Reserve Bank of Australia. Initially, univariate Generalised Autoregressive Conditional Heteroskedasticity (GARCH), with models of individual yield increment time series are developed for a set of fixed maturities. Then, a multivariate Matrix-Diagonal GARCH model with multivariate asymmetric t-distribution of the term structure of yield increments is developed. This model captures many important properties of financial data such as volatility mean reversion, volatility persistency, stationarity and heavy tails.
There are two innovations of GARCH modelling in this thesis: (i) the development of the Matrix-Diagonal GARCH model with multivariate asymmetric t-distribution using meta-elliptical distribution in which the degrees of freedom of each series varies with maturity, and the estimation is given; (ii) the development of a GARCH model of term structure of interest rates (TS-GARCH). The TS-GARCH model describes the parameters specifying the GARCH model and the degrees of freedom using simple smooth functions of time to maturity of component series. TS-GARCH allows an empirical description of complete interest rate yield curve increments therefore allowing the model to be used for interpolation to additional maturity beyond those used to construct the model. Diagnostics of TS-GARCH model are provided using Australian Treasury bond yields.