Fast resolution of a single factor Heath-Jarrow-Morton model with stochastic volatility

TL;DR

This paper introduces an efficient numerical method for solving the three-dimensional PDE form of a single factor Heath-Jarrow-Morton interest rate model with stochastic volatility, significantly reducing computation time while maintaining accuracy.

Contribution

It develops an optimized finite-difference and Crank-Nicholson scheme combined with ADI methods to efficiently solve the Markovian PDE form of the model, improving practical applicability.

Findings

Reduced computation time for PDE solutions

Maintained accuracy with optimized discretization

Enhanced practical usability of the model

Abstract

This paper considers the single factor Heath-Jarrow-Morton model for the interest rate curve with stochastic volatility. Its natural formulation, described in terms of stochastic differential equations, is solved through Monte Carlo simulations, that usually involve rather large computation time, inefficient from a practical (financial) perspective. This model turns to be Markovian in three dimensions and therefore it can be mapped into a 3D partial differential equations problem. We propose an optimized numerical method to solve the 3D PDE model in both low computation time and reasonable accuracy, a fundamental criterion for practical purposes. The spatial and temporal discretization are performed using finite-difference and Crank-Nicholson schemes respectively, and the computational efficiency is largely increased performing a scale analysis and using Alternating Direction Implicit schemes. Several numerical considerations such as convergence criteria or computation time are analyzed and discussed.