Incorporating boundary conditions in a stochastic volatility model for the numerical approximation of bond prices

Abstract

In this paper, we consider a two-factor interest rate model with stochastic volatil-ity, and we assume that the instantaneous interest rate follows a jump-diffusionprocess. In this kind of problems, a two-dimensional partial integro-differentialequation is derived for the values of zero-coupon bonds. To apply standardnumerical methods to this equation, it is customary to consider a boundeddomain and incorporate suitable boundary conditions. However, for thesetwo-dimensional interest rate models, there are not well-known boundary con-ditions, in general. Here, in order to approximate bond prices, we propose newboundary conditions, which maintain the discount function property of thezero-coupon bond price. Then, we illustrate the numerical approximation ofthe corresponding boundary value problem by means of an alternative directionimplicit method, which has been already applied for pricing options. We testthese boundary conditions with several interest rate pricing models.