Quantification of Counterparty Risk Via Bessel Bridges

We construct a dynamical credit model that can be calibrated exactly to CDS quotes. Modelling the default time as the first-passage time of a credit index process to the level zero, we show that the parameters of this credit index process can be chosen such that the risk-neutral (implied) distribution of the time of default is matched. Employing this default model we develop a model for asset prices conditional on the occurrence of default at a given time. We illustrate the use of the model in estimating the expected positive exposure of an oil swap traded with an airline as counterparty.