A BSDE Approach to Counterparty Risk under Funding Constraints

This paper deals with the valuation and hedging of counterparty risk on OTC derivatives. Our study is done in a multiple-curve setup reflecting the various funding constraints (or costs) involved, allowing one to investigate the question of interaction between counterparty risk and funding. The correction in value of a contract due to counterparty risk under funding constraints is represented as the value of an option on the value of the contract clean of counterparty risk and excess funding costs. We develop a reduced-form backward stochastic differential equations (BSDE) approach to the problem of pricing and hedging this correction, the so-called Credit Valuation Adjustment (CVA for short). In the Markov setup, explicit CVA pricing and hedging schemes are formulated in terms of semilinear CVA PDEs. The take-away message of the paper is twofold. Firstly, for properly valuing and hedging a counterparty risky contract under funding constraints, it is necessary to focus on a party of interest (rather than on the contract in itself) and to consider explicitly the three pillars of its position consisting of the contract, its hedging portfolio and its funding portfolio. Secondly, the counterparty risk two stages valuation and hedging methodology (counterparty risky price obtained as clean price minus CVA) which is currently emerging for practical reasons in banks, is also useful in the mathematical analysis of the problem.