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, the counterparty risk and the wrong way risk are crucial issues 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. Wrong way 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 definition of the EPE which should be used in this regard, and the way in which the EPE should be used in the hedging strategy.