Analytic Representation of a General Multi-Factor Pricing Kernel

Abstract

We present a Green's function solution (aka pricing kernel) for a generic multi-factor short rate model based on correlated state variables of Ornstein-Uhlenbeck type, whose drift is assumed to be an affine function of the state variables (and of time). The solution is obtained in general as a perturbation expansion valid in the limit of low rates (an assumption almost invariably satisfied). We exhibit explicit solutions in the case where the interest rate model is of Hull-White and of Black-Karasinski type. We observe that the theory is equally applicable to the modelling of stochastic credit default intensity governed by a Black-Karasinski model as it is to interest rate modelling. It is not difficult either to extend it to multi-asset pricing problems.

Keywords: multi-factor, pricing kernel, Hull-White, Black-Karasinski, perturbation methods