Consistency Problems for Jump-Diffusion Models

In this paper consistency problems for multi-factor jump-diffusion models, where the jump parts follow multivariate point processes are examined. First the gap between jump-diffusion models and generalized HeathJarrow-Morton (HJM) models is bridged. By applying the drift condition for a generalized arbitrage-free HJM model, the consistency condition for jumpdiffusion models is derived. Then we consider a case in which the forward rate curve has a separable structure, and obtain a specific version of the general consistency condition. In particular, a necessary and sufficient condition for a jump-diffusion model to be affine is provided. Finally the Nelson-Siegel type of forward curve structures is discussed. It is demonstrated that under regularity condition, there exists no jump-diffusion model consistent with the Nelson-Siegel curves. We are indebted to Erhan Çinlar and Damir Filipović for helpful discussions. 1 2 ERHAN BAYRAKTAR, LI CHEN, AND H. VINCENT POOR 1. The Arbitrage-free Condition for Generalized HJM Models The purpose of this paper is to study consistency problems for multi-factor jumpdiffusion term structure models of interest rates. The concept of consistency in this context was first introduced and studied in [4]. Previous works ([13], [14], [15]) in this area have focused on diffusion models without considering jumps. Because jump-diffusion models usually provide a better characterization of the randomness in financial markets than do diffusion models (see [1], [19]), there has been an upsurge in the modeling of interest rate dynamics with jumps (e.g. [3], [12], [17], [20]). Therefore it is of interest to clarify the consistency conditions for jumpdiffusion models. Consider a Heath-Jarrow-Morton (HJM) model ([16]) incorporating a marked point process. The dynamics of the forward curve for such a model can be given by (1.1) dr(t, T ) = α(t, T )dt+ σ(t, T )dBt + ∫ Θ ρ(t, T, y)μ(dt, dy), where B is a standard Brownian motion and μ(dt, dy) is a random measure on R+×Θ with the compensator ν(t, dy)dt. Thus the price of a zero-coupon bond can be written as (1.2) P (t, T ) = e ∫ T t . A measure Q is said to be a local martingale measure if the discounted bond price D(t, T ) = P (t, T ) e ∫ t 0 r(s,s)ds is a Q-local martingale, for each T ∈ R+. It is well known that the existence of an equivalent local martingale measure implies the absence of arbitrage (e.g. see [9]). Under regularity conditions, Björk et al. [5] give the following lemma for the arbitrage-free condition of a generalized HJM model defined by (1.1). Lemma 1.1. An equivalent local martingale measure exists if and only if the forward rate dynamics under this measure specified by (1.1) satisfy the following relation for ∀ 0 ≤ t < T . (1.3) α(t, T ) = σ(t, T ) ∫ T