Approximate XVA for European claims

TL;DR

This paper introduces an efficient approximation method for computing XVA adjustments of European claims considering default risk, funding, and collateralization, using a change of numeraire and Taylor expansion in affine intensity models.

Contribution

It proposes a novel approximation technique leveraging change of numeraire and Taylor expansion for XVA computation, reducing computational costs in affine intensity models.

Findings

First-order approximation shows remarkable efficiency in CIR intensity model.

Method reduces computational complexity compared to PDE discretization and Monte Carlo.

Approach is effective for affine process-based default intensity modeling.

Abstract

We consider the problem of computing the Value Adjustment of European contingent claims when default of either party is considered, possibly including also funding and collateralization requirements. As shown in Brigo et al. (\cite{BLPS}, \cite{BFP}), this leads to a more articulate variety of Value Adjustments ({XVA}) that introduce some nonlinear features. When exploiting a reduced-form approach for the default times, the adjusted price can be characterized as the solution to a possibly nonlinear Backward Stochastic Differential Equation (BSDE). The expectation representing the solution of the BSDE is usually quite hard to compute even in a Markovian setting, and one might resort either to the discretization of the Partial Differential Equation characterizing it or to Monte Carlo Simulations. Both choices are computationally very expensive and in this paper we suggest an approximation method based on an appropriate change of numeraire and on a Taylor's polynomial expansion when intensities are represented by means of affine processes correlated with the asset's price. The numerical discussion at the end of this work shows that, at least in the case of the CIR intensity model, even the simple first-order approximation has a remarkable computational efficiency.