Arbitrage-Free Pricing of XVA - Part II: PDE Representation and Numerical Analysis

TL;DR

This paper develops a PDE framework for arbitrage-free XVA pricing, incorporating funding, repo, collateral asymmetries, and early termination, with proofs of solution uniqueness and numerical analysis of key risk factors.

Contribution

It introduces a PDE approach for XVA pricing that accounts for complex market asymmetries and counterparty risk, with rigorous proof of solution existence and uniqueness.

Findings

Funding costs significantly impact XVA values.

Repo rate asymmetries influence valuation adjustments.

Counterparty credit risk affects early termination valuation.

Abstract

We study the semilinear partial differential equation (PDE) associated with the non-linear BSDE characterizing buyer's and seller's XVA in a framework that allows for asymmetries in funding, repo and collateral rates, as well as for early contract termination due to counterparty credit risk. We show the existence of a unique classical solution to the PDE by first proving the existence and uniqueness of a viscosity solution and then its regularity. We use the uniqueness result to conduct a thorough numerical study illustrating how funding costs, repo rates, and counterparty credit risk contribute to determine the total valuation adjustment.