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

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 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. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.

Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations

“…HJMM-type equations are used to model the stochastic evolution of interest rates. Weak error rates of numerical discretizations of HJMM-type equations were studied in [9,10,20]; see also the references therein. The following proposition provides an upper bound on the weak error of noise discretizations of HJMM equations with additive noise, i.e., of infinite-dimensional Ornstein-Uhlenbeck forward rate models.…”

“…However, in the case of non-regularizing semigroups there remain many open questions. While temporal and spatial discretizations have been studied in [14,17,18,19] for additive noise and in [5,25,24,8,9,10,15,20,29] for multiplicative noise, this is the first result on the discretization of multiplicative noise in this setting. Moreover, our framework is general and encompasses a variety of equations from mathematical finance and physics.…”

Stochastic Transport with Lévy Noise -- Fully Discrete Numerical Approximation

“…Naturally, the numerical treatment then becomes more involved than in the parabolic case, as we face lower regularity of the solution and the transport semigroup is not analytic. Consequently, there is very little literature on the numerical analysis of stochastic transport problems as for example [4,30].…”