Dynamic Hedging of Counterparty Exposure∗

We study mathematical aspects of dynamic hedging of Credit Valuation Adjustment (CVA) in a portfolio of OTC financial derivatives. Since the sub-prime crisis, counterparty risk and wrong-way risk are a crucial issue in connection with valuation and risk management of credit derivatives. In this work we first derive a general, model free equation for the dynamics of the CVA of a portfolio of OTC derivatives. We then particularize these dynamics to the counterparty risk of a portfolio of credit derivatives including, for instance, CDSs and/or CDOs, possibly netted and collateralized, considered in the so called Markovian copula model. Wrongway risk is represented in the model by the possibility of simultaneous defaults. We establish a rigorous connection between the CVA, which represents the price of the counterparty risk, and a suitable notion of Expected Positive Exposure (EPE). Specifically, the EPE emerges as the key ingredient of the min-variance hedging ratio of the CVA by a CDS on the counterparty. Related notions of EPE have actually long been used in an ad-hoc way by practitioners for hedging their CVA. Our analysis thus justifies rigorously this market practice, making also precise the proper Tomasz R. Bielecki Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA, e-mail: [email protected] Stéphane Crépey Equipe Analyse et Probabilité, Université d’Évry Val d’Essonne, 91025 Évry Cedex, France, and CRIS Consortium, e-mail: [email protected] ∗ The authors warmly thank Giovanni Cesari for stimulating discussions throughout the preparation of this work. We also thank an anonymous referee for a very careful reading of the manuscript and for valuable comments and remarks. † The research of T.R. Bielecki was supported by NSF Grant DMS–0603789 and NSF Grant DMS–0908099. ‡ The research of S. Crépey benefited from the support of the ‘Chaire Risque de crédit’, Fédération Bancaire Française, and of the DGE.