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 Lévy 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.

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SIAM Journal on Financial Mathematics
Applied mathematics for risk measures in finance and insurance, in the wake of the crisis
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Borovykh, A., Pascucci, A., & Oosterlee, K. (2018). Efficient computation of various valuation adjustments under local Lévy models. SIAM Journal on Financial Mathematics, 9(1), 251–273. doi:10.1137/16M1099005