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Credit default swaps with and without counterparty and collateral adjustments

We present an affine jump-diffusion framework for calculating the prices of credit default swaps (CDSs) with and without credit value adjustments (CVA) to account for counterparty risk. Computing CVA for credit derivatives is a particularly challenging problem because the likelihood of counterparty default is typically highly correlated with the size of the exposure. We show that in order to generate sufficient correlation to capture this so-called wrong-way risk, simultaneous jumps in the credit spreads of buyer, seller and reference entity are required. We illustrate this with relevant numerical examples. Closed-form expressions for the CVA in this framework are obtained in the limit of pure diffusion, and an expansion of the Green function in powers of the jump intensity is applicable in the general case. We also present a finite difference scheme for the general solution of the relevant partial integro-differential equation and an efficient Monte Carlo scheme applicable to the problem of evaluating CVA on an arbitrarily large portfolio of CDS.