Robust XVA

TL;DR

This paper develops an arbitrage-free, robust framework for valuation adjustments in credit risk, accounting for uncertainty in counterparty bond returns and the effects of collateralization and credit contagion.

Contribution

It introduces bounds for XVA processes under uncertainty, derived via nonlinear differential equations, and analyzes the impact of collateral and credit contagion on valuation and hedging strategies.

Findings

Bounds for XVA depend on the relation between the replicating portfolio and close-out value.

Collateralization and closeout payoffs significantly affect valuation.

Credit contagion exhibits nonlinear effects on strategies and XVA.

Abstract

We introduce an arbitrage-free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a risky counterparty, and hedges credit risk by taking a position in defaultable bonds. The investor does not know the return rate of her counterparty's bond, but is confident that it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs leads to important differences with respect to classical credit risk valuation. The value of the super-replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the close-out value throughout the life of the transaction. Our comparative statics analysis indicates that credit contagion has a nonlinear effect on the replication strategies and on the XVA.