Numerical integration of the Heath–Jarrow–Morton model of interest rates

We propose and analyse numerical methods for the Heath–Jarrow–Morton (HJM) model. To construct the methods, we first discretize the infinite-dimensional HJM equation in the maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite-dimensional system of stochastic differential equations which we approximate in the weak and mean-square sense. The proposed numerical algorithms are highly computationally 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 the overall accuracy of the algorithms. They also have a high degree of flexibility and allow us to choose appropriate approximations in maturity and calendar times separately. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.