Volatility structure of interest rate markets under a arbitrage-free framework

Publication Type:
Thesis
Issue Date:
2005
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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.
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