A note on a PDE approach to option pricing under xVA

TL;DR

This paper develops analytical solutions for PDEs modeling xVA adjustments in option pricing within the Black-Scholes framework, providing semi-closed formulas and comparing them to Monte Carlo and numerical PDE solutions.

Contribution

It introduces an analytical PDE approach for xVA in option pricing, offering new semi-closed formulas and insights into collateral valuation.

Findings

Semi-closed formulas for xVA PDEs derived

Comparison shows accuracy of analytical solutions

Application to collateral valuation examples

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.