Partial Differential Equation Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs

Christoph Burgard and Mats Kjaer

Given the market conditions since the global financial crisis, counterparty credit risk implicitly embedded in derivative contracts has become increasingly relevant. This kind of risk represents the possibility that a counterparty will default while owing money under the terms of a derivative contract, or, more precisely, if the mark-to-market value of the derivative is positive to the seller at the time of default of the counterparty. While for exchange-traded contracts the counterparty credit risk is mitigated by the exchange’s presence as intermediary, this is not the case for over-the-counter products. For these, a number of different techniques are used to mitigate counterparty risk, most commonly by means of netting agreements and collateral mechanisms. The details of these agreements are specified, for example, by the International Swaps and Derivatives Association (ISDA) 2002 Master Agreement. However, the counterparty faces the similar risk of the seller defaulting when the mark-to-market value is positive to the counterparty. Taking into account the credit risk of both parties is commonly referred to as considering bilateral counterparty risk. When doing so, the value of