An explicit scheme for pathwise cross valuation adjustment computations

Motivated by the equations of cross valuation adjustments (XVAs) accounting for the fungibility of capital at risk with variation margin, we introduce a simulation/regression scheme for a class of anticipated backward stochastic differential equations, where the coefficient entails a conditional expected shortfall of the martingale part of the solution. The scheme is explicit in time and uses neural network least-squares and quantile regressions for the embedded conditional expectations and expected shortfall computations. An a posteriori Monte Carlo validation procedure allows assessment of the scheme’s regression error at each time step. The superiority of this scheme with respect to Picard iterations is illustrated in the context of a high-dimensional market and a default risk XVA use case.