Pricing credit default swaps with bilateral value adjustments

The paper studies the problem of computing adjustments for bilateral counterparty risk for a standard CDS in a three-factor first-passage time default risk model. Extending the existing literature that gives analytical expression for the transition probability density function (or Green's function) for two-dimensional Brownian motions absorbed at the boundaries in the positive quadrant, this paper gives a semi-analytical expression for Green's function for three-dimensional Brownian motions absorbed at first exit time from the positive octant. This is done by separating the problem into a radial and an angular part, of which the latter is universal and depends only on the correlation matrix. These mathematical results are then used to provide semi-analytical expressions for bilateral CVA/DVA of a credit default swap. An example of market data is analysed in detail and it is shown that these value adjustments can be surprisingly large.