Approximate value adjustments for European claims

Abstract

We consider the problem of computing the Value Adjustment of European contingent claims when the possibility of default of either party is considered, possibly including also funding and collateralization requirements.As shown in Brigo et al. (Brigo, Liu, Pallavicini, and Sloth (2016b), Brigo, Francischello, and Pallavicini (2016a)), 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 quite expensive, In this paper, we suggest an alternative method based on an appropriate change of numéraire and a Taylor polynomial expansion when intensities are represented by affine processes correlated with the asset price. The numerical discussion at the end of this work shows that, at least in the case of the Cox-Ingersoll-Ross (CIR) intensity model, even the simple first-order approximation has a remarkable computational efficiency.