Efficient Simulation and Calibration of General HJM Models by Splitting Schemes

Abstract

We introduce efficient numerical methods for generic HJM equations of interest rate theory by means of high-order weak approximation schemes. These schemes allow for QMC implementations due to the relatively low dimensional integration space. The complexity of the resulting algorithm is considerably lower than the complexity of multi-level MC algorithms as long as the optimal order of QMC-convergence is guaranteed. In order to make the methods applicable to real world problems, we introduce and use the setting of weighted function spaces, such that unbounded payoffs and unbounded characteristics of the equations in question are still allowed. We also provide an implementation, where we efficiently calibrate an HJM equation to caplet data.

Cubature methods for stochastic (partial) differential equations in weighted spaces

“…The following result follows from the theory of Röckner and Sobol [29] (see also Dörsek and Teichmann [11] and [12]). …”

“…We extended the functional analytic framework of Röck-ner and Sobol [29], used for the numerical analysis of stochastic evolution equations in Dörsek [10] and Dörsek and Teichmann [11,12], to more general characteristics through a flexible formulation of directional derivatives in weighted spaces. This setting was then used to prove optimal rates of convergence of cubature schemes for more general equations.…”

“…We recall the following definition of spaces of functions with controlled growth, see also Röckner and Sobol [29] for their use in the construction of the solution of martingale problems in infinite dimension, and Dörsek [9,10] and Dörsek and Teichmann [11,12] for their application to the analysis of splitting schemes for stochastic partial differential equations. …”

“…In this work, we shall relax the regularity assumptions of the cubature method, similarly as was done in Dörsek [9] and Dörsek and Teichmann [11,12] for the splitting approach of Ninomiya and Victoir [26]. Consider a stochastic differential equation on R n in its Stratonovich form,…”

“…The paper is organised as follows. Sections 2 and 3 contain an exposition of the theory of B ψ spaces, originally introduced in Röckner and Sobol [29], and explain how directional derivatives can be analysed in this setting, extending results from Dörsek and Teichmann [11,12]. In Sect.…”