A Class of Markovian Models for the Term Structure of Interest Rates Under Jump-Diffusions

Jump-Diffusion processes capture the standardized empirical statistical features of interest rate dynamis, thus providing an improved setting to overcome some of the mispricing of derivative securities that arises with the extensively develped pure diffusion models. A combination of jump-diffusion models with state dependent volatility specifications generates a class of models that accommodates the empirical statistical evidence of jump components and the more general and realistic setting of stochastic volatiliy. For modelling the term structure of interest rates, the Heath, Jarrow and Morton (1992) (hereafter HJM) framework constitutes the most general and adaptable platform for the study of interest rate dynamics that accommodates, by construction, consistency with the currently observed yield curve within an arbitrage free environment. The HJM model requires two main inputs, the market information of the initial forward curve and the specification of the forward rate volatility. This second requirement of the volaility specification enables the model builder to generate a wide class of models and in particular to derive within HJM framework a number of the popular interest rate models. However, the general HJM model is Markovian only in the entire yield curve, thus requiring an infinite number of state variables to determine the future evolution of the yield curve. By imposing appropriate conditions on the forward rate volatility, the HJM model can admit finite dimensional Markovian structures, where the generality of the HJM models coexists with the computational tractability of Markovian structures. The main contributions of this thesis include: Markovianisation of jump-diffusion versions of the HJM model - Chapters 2 and 3. Under a specific formulation of state and time dependent forward rate volatility specifications, Markovian representations of a generalised Shirakawa (1991) model are developed. Further, finite dimensional affine realisations of the term structure in terms of forward rates are obtained. Within this framework, some specific classes of jump-diffusion term structure models are examined such as extensions of the Hull and White (1990), (194) class of models and the Ritchken and Sankarasubramanian (1995) class of models to the jump-diffusion case. Markovianisation of defaultable HJM models - Chapters 4. Suitable state dependent volatility specifications, under deterministic default intensity, lead to Markovian defaultable term structures under the Schonbucher (2000), (2003) general HJM framework. The state variables of this model can be expressed in terms of a finite number of benchmark defaultable forward rates. Moving to the more general setting of stochastic intensity of defaultable term structures, we discuss model limitations and an approximate Markovianisation of the system is proposed. Bond option pricing under jump-diffusions - Chapter 5. Within the jump-diffusion framework, the pricing of interest rate derivative securities is discussed. A tractable Black-Scholes type pricing formula is derived under the assumption of constant jump volatility specifications and a viable control variate method is propsed for pricing by Monte Carlo simulation under more general volatity specifications.