Counterparty Risk Modeling: Beyond Immersion

Counterparty risk reduced-form models typically hinge on an immersion property of a reference filtration into the full model filtration enlarged by the default times of the counterparties, as well as on a continuity assumption on some of the data at default time. This is too restrictive for cases of strong wrong-way risk, i.e. adverse dependence between the exposure and the credit riskiness of the counterparty. In this paper we generalize the approach by switching from the class of pseudo-stopping times, which is classically used to model the defaults of the counterparties, to the much more flexible class of invariant times. For instance, these can be marked default times, where the role of the mark is to convey some additional information about the defaults, in order to account for various possible wrong-way and gap risk scenarios and features. Additional tools are introduced to analyze the cure period (time interval between the default and the liquidation) and the ensuing gap risk of diverging evolutions of the portfolio and of its collateral. In particular, the liquidation time is predictable (as announced by the default), which modifies the nature of the pricing problem. We illustrate our approach in two dynamic copula models of portfolio credit risk.