Sparse Grid Method for Highly Efficient Computation of Exposures for xVA

TL;DR

This paper introduces a sparse grid numerical technique combining Stochastic Collocation and Smolyak's extension to drastically reduce the computational effort in xVA exposure simulations for large, multi-factor portfolios.

Contribution

The paper presents a novel application of sparse grid methods to efficiently compute exposures in xVA, significantly decreasing the number of portfolio evaluations needed.

Findings

Potential to reduce portfolio evaluations by over 6000 times

Effective for portfolios with linear and non-linear derivatives

Applicable to multi-currency interest rate portfolios

Abstract

Every "x"-adjustment in the so-called xVA financial risk management framework relies on the computation of exposures. Considering thousands of Monte Carlo paths and tens of simulation steps, a financial portfolio needs to be evaluated numerous times during the lifetime of the underlying assets. This is the bottleneck of every simulation of xVA. In this article, we explore numerical techniques for improving the simulation of exposures. We aim to decimate the number of portfolio evaluations, particularly for large portfolios involving multiple, correlated risk factors. The usage of the Stochastic Collocation (SC) method, together with Smolyak's sparse grid extension, allows for a significant reduction in the number of portfolio evaluations, even when dealing with many risk factors. The proposed model can be easily applied to any portfolio and size. We report that for a realistic portfolio comprising linear and non-linear derivatives, the expected reduction in the portfolio evaluations may exceed 6000 times, depending on the dimensionality and the required accuracy. We give illustrative examples and examine the method with realistic multi-currency portfolios consisting of interest rate swaps and swaptions.