Robust 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 bond, but is confident that it lies within a 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 superreplicating 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.

American options in nonlinear markets

“…In the aftermath of the global financial crisis of 2007-2009, there was a rapidly growing interest in financial models accounting for the counterparty credit risk, collateralization, differential funding costs and other trading adjustments; see, e.g., Bichuch et al [7,8], Brigo and Pallavicini [14], Burgard and Kjaer [15], Capponi [16], Crépey [18,19], Crépey et al [20], Pallavicini et al [64], and Piterbarg [67]. Due to abovementioned intricacies of trading, the problem of risk mitigation through hedging of a financial contract is no longer as straightforward as it was in the past.…”

“…The inequalities satisfied by unilateral prices and the range fair bilateral prices were studied in papers by Nie and Rutkowski [60,61] for models with either an exogenous or endogenous collateralization, respectively. More recently, Bichuch et al [7,8] (see also Lee et al [52] and Lee and Zhou [53] for related studies) explicitly addressed the issue of hedging the counterparty credit risk and analyze the CVA for European claims in the Black-Scholes model complemented by EJP 26 (2021), paper 90. defaultable bonds issued by the counterparties and they also examined bounds for fair bilateral prices. We stress that the above-mentioned papers are mainly concerned with prices of contingent claims of a European style and thus it is natural to ask analogous questions regarding American contingent claims in a nonlinear market model with idiosyncratic funding costs, counterparty credit risk and other market frictions affecting the trading mechanism.…”