Numerical integration of Heath-Jarrow-Morton model of interest rates

TL;DR

This paper develops efficient numerical methods for the Heath-Jarrow-Morton interest rate model by discretizing the infinite-dimensional SDEs and applying high-order quadrature rules, with proven convergence and numerical validation.

Contribution

It introduces a novel discretization approach for the HJM model using quadrature rules, enabling high-order, efficient numerical algorithms with proven convergence.

Findings

High-order quadrature rules improve computational efficiency.

Convergence theorems validate the numerical methods.

Numerical experiments demonstrate accuracy with interest rate derivatives.

Abstract

We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic differential equations (SDEs) which we approximate in the weak and mean-square sense using the general theory of numerical integration of SDEs. The proposed numerical algorithms are computationally highly efficient due to the use of high-order quadrature rules which allow us to take relatively large discretization steps in the maturity time without affecting overall accuracy of the algorithms. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.