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

• Published in: Mathematical methods in the applied sciences 2020-09, Vol.43 (14), p.7993-8005
• Main Authors: Gómez‐Valle, Lourdes, López‐Marcos, Miguel Ángel, Martínez‐Rodríguez, Julia
• Format: Article
• Language: English
• Subjects: alternative direction implicit methods; Approximation; Boundary conditions; Boundary value problems; Differential equations; Diffusion rate; finite difference methods; Interest rates; interest rates (stochastic models); jump‐diffusion stochastic processes; Mathematical analysis; Numerical methods; Pricing; Stochastic models; Volatility; zero‐coupon bonds
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.5815

In this paper, we consider a two‐factor interest rate model with stochastic volatility, and we assume that the instantaneous interest rate follows a jump‐diffusion process. In this kind of problems, a two‐dimensional partial integro‐differential equation is derived for the values of zero‐coupon bonds. To apply standard numerical methods to this equation, it is customary to consider a bounded domain and incorporate suitable boundary conditions. However, for these two‐dimensional interest rate models, there are not well‐known boundary conditions, in general. Here, in order to approximate bond prices, we propose new boundary conditions, which maintain the discount function property of the zero‐coupon bond price. Then, we illustrate the numerical approximation of the corresponding boundary value problem by means of an alternative direction implicit method, which has been already applied for pricing options. We test these boundary conditions with several interest rate pricing models.