Numerical Integration of 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. Keywords. Infinite dimensional stochastic equations, HJM model, weak approximation, Monte Carlo technique, interest rate derivatives, method of lines, meansquare convergence. AMS 2000 subject classification. 65C30, 60H35, 60H30, 91G80.T t 0 |σ(s, T )| 2 ds < ∞; and t * ∧ T := min(t * , T ).Assumption 2.1 The functions σ i (t, T, z), i = 1, . . . , d, are uniformly bounded, i.e., there is a constant C > 0 such that(2.6) Assumption 2.2 For sufficiently large p 1 , p 2 ≥ 1, the partial derivatives 7)are continuous and uniformly bounded in, is deterministic and sufficiently smooth.The imposed conditions are sufficient to ensure that the SDE (2.4)-(2.5) has a unique strong solution f (t, T ), which is sufficiently smooth in the last argument (see [13,24] and also [16,8] for differentiating SDE solutions with respect to a parameter). Further, it is not difficult to show that they imply boundedness of exponential moments of f (t, T ), i.e., for a c ∈ R there is a constant C > 0 such thatThe constant C in (2.8) depends on the initial forward curve f 0 (T ), volatility σ(t, T, z), and on c.Remark 2.1 As it was shown in [24], for the SDE (2.4)-(2.5) to have the unique strong solution it suffices to require a weaker assumption than Assumption 2.1:However, in the paper we restrict ourselves to the stronger set of conditions which allow us to consider methods of higher order. Assumptions 2.1-2.3 are sufficient for all the statements in this paper. The choice of p 1 and p 2 depends on a particular algorithm (as usual, the more accurate an algorithm the more derivatives are needed). At the same time, the imposed conditions are not necessary and the proposed numerical methods themselves can be used under broader assumptions.

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.…”