  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.
