A note on a PDE approach to option pricing under xVA

Falko Baustian, Martin Fencl, Jan Posp\'i\v{s}il and Vladim\'ir \v{S}v\'igler

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Abstract: In this paper we study partial differential equations (PDEs) that can be used to model value adjustments. Different value adjustments denoted generally as xVA are nowadays added to the risk-free financial derivative values and the PDE approach allows their easy incorporation. The aim of this paper is to show how to solve the PDE analytically in the Black-Scholes setting to get new semi-closed formulas that we compare to the widely used Monte-Carlo simulations and to the numerical solutions of the PDE. Particular example of collateral taken as the values from the past will be of interest.

Date: 2021-04, Revised 2021-07
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