Arbitrage-Free Bilateral Counterparty Risk Valuation under Collateralization and Application to Credit Default Swaps∗

We develop an arbitrage-free valuation framework for bilateral counterparty risk, where collateral is included with possible re-hypothecation. We show that the adjustment is given by the sum of two option payoff terms, where each term depends on the netted exposure, i.e. the difference between the on-default exposure and the pre-default collateral account. We then specialize our analysis to Credit Default Swaps (CDS) as underlying portfolios, and construct a numerical scheme to evaluate the adjustment under a doubly stochastic default framework. In particular, we show that for CDS contracts a perfect collateralization cannot be achieved, even under continuous collateralization, if the reference entity’s and counterparty’s default times are dependent. The impact of re-hypothecation, collateral margining frequency, and default correlation induced contagion is illustrated with numerical examples. keywords: Counterparty Risk, CVA, Bilateral CVA, Arbitrage-Free Credit Valuation Adjustment, Credit Default Swaps, Credit Spread Volatility, Default Correlation, Contagion, Stochastic Intensity, Collateral Margining, Netting, Re-hypotecation, Wrong Way Risk. ∗This paper is an updated and improved version of the report in Brigo and Capponi (2008), see also Brigo and Capponi (2010). †Dept. of Mathematics, King’s College, London, E-mail: [email protected] ‡Dept. of Industrial Engineering, Purdue University, E-mail:[email protected] §Banca IMI, Milano, Italy, E-mail:[email protected]