Calibrating Drift of Short-Rate Models with Jump-Diffusion

Abstract

The drift or the mean-reversion level of short-rate models under jump-diffusion is derived to fit the initial term-structure of zero-coupon bond. In particular, the drift is obtained for Hull-White and Cox-Ingersoll-Ross short-rate models. The purpose of obtaining the drift is for the Monte-Carlo simulations for pricing exotic equity options under stochastic interest rates and exotic interest rate derivatives. The theoretical expressions for the drift are verified by Monte-Carlo simulations of the short-rate and bond paths, and show that the expectation of the discounted entities reproduce the initial zero-coupon bond term-structure. For the Cox-Ingersoll-Ross model the drift is obtained by a numerical computation of a second-order ordinary differential equation using a perturbation method. The correctness of this method is verified by the Monte-Carlo results. Further, the bond partial differential equation (PDE) for the Cox-Ingersoll-Ross jump-diffusion short-rate model is also solved numerically and the results reproduce the initial zero-coupon bond term-structure. This bond PDE computation verifies the self-consistency of the drift computation, and also validates the numerical boundary conditions used in the finite-difference computations of this final-value problem. Similar results are presented from the finite-difference computation of the Hull-White PDE also.