Wrong-Way Risk CVA Models with Analytical EPE Profiles under Gaussian Exposure Dynamics

Vrins, Frédéric [UCL]

We consider two classes of wrong-way risk models in the context of CVA: static (resampling) and dynamic (reduced form). Although both potentially suffer from arbitrage problems, their tractability makes them appealing to the industry and therefore deserve additional study. For example, Gaussian copula-based resampling and reduced-form with ``Hull-White intensities'' yield analytical expected positive exposure (EPE) profiles when the portfolio price process (i.e. exposure process) is Gaussian. However, the first approach disregards credit volatility whilst the second can provide default probabilities larger than 1. We therefore enlarge the study by introducing a new dynamic approach for credit risk, consisting in the straight modeling of the survival (Az\'ema supermartingale) process using the $\Phi$-martingale. This method is appealing in that it helps fixing some drawbacks of the above models. Indeed, it is a dynamic method (it disentangles correlation and credit volatility) that preserves probabilities in [0,1] without affecting the analytical tractability of the model. In particular, calibration to any valid default probability curve is automatic and the closed-form expression for the EPE profiles remains available under Gaussian exposures. For each approach, we derive analytically the EPE profiles (conditional upon default) associated to prototypical exposure processes of FRA and IRS in all cases, provide a comparison and discuss the implied CVA figures.