Discrete-time bond and option pricing for jump-diffusion processes

This paper provides a methodology for numerically pricing generalized interest rate contingent claims for jump-diffusion processes. The method enhances the standard finite-differencing approach to deal with partial differential-difference equations derived in a jump-diffusion setting. Numerical illustrations compare jump-diffusion and pure-diffusion models.I am especially grateful to Darrell Duffie, who provided me immensely valuable input on the paper. I would also like to thank Dilip Madan and Rangarajan Sundaram for alleviating my confusion with helpful comments.     

A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework

This paper considers a class of term structure models that is a parameterisation of the Shirakawa (1991) extension of the Heath et al. (1992) model to the case of jump-diffusions. We consider specific forward rate volatility structures that incorporate state dependent Wiener volatility functions and time dependent Poisson volatility functions. Within this framework, we discuss the Markovianisation issue, and obtain the corresponding affine term structure of interest rates. As a result we are able to obtain a broad tractable class of jump-diffusion term structure models. We relate our approach to the existing class of jump-diffusion term structure models whose starting point is a jump-diffusion process for the spot rate. In particular we obtain natural jump-diffusion versions of the Hull and White (1990, 1994) one-factor and two-factor models and the Ritchken and Sankarasubramanian (1995) model within the HJM framework. We also give some numerical simulations to gauge the effect of the jump-component on yield curves and the implications of various volatility specifications for the spot rate distribution.     

LINEAR-QUADRATIC JUMP-DIFFUSION MODELING

We aim at accommodating the existing affine jump-diffusion and quadratic models under the same roof, namely the linear-quadratic jump-diffusion (LQJD) class. We give a complete characterization of the dynamics of this class by stating explicitly the structural constraints, as well as the admissibility conditions. This allows us to carry out a specification analysis for the three-factor LQJD models. We compute the standard transform of the state vector relevant to asset pricing up to a system of ordinary differential equations. We show that the LQJD class can be embedded into the affine class using an augmented state vector. This establishes a one-to-one equivalence relationship between both classes in terms of transform analysis.