Efficient Computation of Various Valuation Adjustments Under Local Levy Models

Various valuation adjustments (XVAs) can be written in terms of nonlinear partial integro-differential equations equivalent to forward-backward SDEs (FBSDEs). In this paper we develop a Fourier based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local Levy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.