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 standard approximations by Monte-Carlo simulations and by numerical finite-differences solutions of the PDE. Particular example of collateral taken as the values from the past will be of interest.
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