Numerical Methods for Heath-Jarrow-Morton Model of Interest Rates
2012-06-14T08:36:59Z (GMT) by
The celebrated HJM framework models the evolution of the term structure of interest rates through the dynamics of the forward rate curve. These dynamics are described by a multifactor infinite-dimensional stochastic equation with the entire forward rate curve as state variable. Under no-arbitrage conditions, the HJM model is fully characterized by specifying forward rate volatility functions and the initial forward curve. In short, it can be described as a unifying framework with one of its most striking features being the generality: any arbitrage-free interest rate model driven by Brownian motion can be described as a special case of the HJM model. The HJM model has closed-form solutions only for some special cases of volatility, and valuations under the HJM framework usually require a numerical approximation. We propose and analyze numerical methods for the 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. 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 overall accuracy of the algorithms. They also have a high degree of flexibility and allow 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.