Higher-order Discretization Methods of Forward-backward SDEs Using KLNV-scheme and Their Applications to XVA Pricing

Abstract

This study proposes new higher-order discretization methods of forward-backward stochastic differential equations. In the proposed methods, the forward component is discretized using the Kusuoka-Lyons-Ninomiya-Victoir scheme with discrete random variables and the backward component using a higher-order numerical integration method consistent with the discretization method of the forward component, by use of the tree based branching algorithm. The proposed methods are applied to the XVA pricing, in particular to the credit valuation adjustment. The numerical results show that the expected theoretical order and computational efficiency could be achieved.

Numerical methods for backward stochastic differential equations: A survey

“…For backward Euler type methods, we find that in general, few extra assumptions are required, if any [23,61,76,84]. Some backward Euler type methods however do require some regularity assumptions, in particular smoothness and boundedness conditions [22,45,93] to ensure the theoretical convergence results hold, which can, for example, rely on the continuity and differentiability of the terminal condition.…”

“…Also, the conditional expectations are expressed in terms of Fourier transforms [20,63], while the authors of [45] analyse in detail the error induced by the time discretization (simple backward Euler approximation) of a FBSDE. In [23,76], the authors combine a Euler style discretization for FBSDEs with the cubature method, which is an approximation method for the law of the solution of an SDE and consequently generates a tree on which expectations and conditional expectations are evaluated. Finally, in [17], we find a number of backward Runge-Kutta type schemes for FBSDEs where the order of each scheme matches the number p of intermediate stages for small p.…”

“…We will discuss such implementation matters in Section 5.3 to avoid overloading this section. Backward Euler type methods have favourable theoretical convergence properties, often under very mild assumptions [23,45,61,76,84,93]. For example, the method proposed in [61] has weak error of convergence 1, while the method in [45] is of polynomial order.…”

Monte Carlo construction of cubature on Wiener space

“…The iterated integrals appearing in (2) have a rich algebraic structure (see Sect. 4.1), so algebraic approaches have been adopted in the literature [8,19,21,22,24,29]. However, solving complicated equations of Lie algebra is required in those approaches, and constructions of the formula are limited to a small range (1)…”

“…Ref. [21] represents other concrete examples when m = 5 with general d and the case (d, m) = (2, 7).…”