Forward Measures in a Ho and Lee Jump Diffusion Model

Abstract

Empirical studies have shown that macroeconomic announcements result in large jumps in the short-term interest rate process. For this reason, finding a tractable option pricing model for jump diffusion models is of considerable interest. In this paper, we consider a model that could be described as a (continuous-time) Ho and Lee model with jumps. We first derive forward measures for this process using the Heath-Jarrow-Morton methodology. It is well known that in a jump diffusion model there are many equivalent martingale measures. We thus make two different assumptions regarding the pricing measure: (i) first we assume that the distribution of jumps is time-independent under the equilibrium spot measure; (ii) then we assume that the distribution of jumps is time-independent under the equilibrium forward measure. We show that only the second assumption leads to option prices that can be explicitly calculated. Using this assumption we derive the price of European options on bonds. The resulting formulas are extensions of the Ho and Lee model and have analogies to the Bates (1991) equity option pricing model in which the market price of jump risk results in a sort of dividend yield that depends on the jump parameters.