Wrong-Way Risk Models: A Comparison of Analytical Exposures

TL;DR

This paper compares static and dynamic models for wrong-way risk in CVA, introducing a new approach using the $ ext{Azé}$-martingale, and derives analytical positive exposure profiles for different market processes.

Contribution

It introduces a new dynamic modeling approach for wrong-way risk using the $ ext{Azé}$-martingale and provides analytical expressions for positive exposures.

Findings

All models allow calibration to default probability curves.

Explicit formulas for positive exposures are derived for prototypical market processes.

The process $ ext{zeta}$ is positive with unit expectation but not a martingale.

Abstract

In this paper, we compare static and dynamic (reduced form) approaches for modeling wrong-way risk in the context of CVA. Although all these approaches potentially suffer from arbitrage problems, they are popular (respectively) in industry and academia, mainly due to analytical tractability reasons. We complete the stochastic intensity models with another dynamic approach, consisting in the straight modeling of the survival (Az\'ema supermartingale) process using the $\Phi$-martingale. Just like the other approaches, this method allows for automatic calibration to a given default probability curve. We derive analytically the positive exposures $V^+_t$ "conditional upon default" associated to prototypical market price processes of FRA and IRS in all cases. We further discuss the link between the "default" condition and change-of-measure techniques. The expectation of $V^+_t$ conditional upon $\tau=t$ is equal to the unconditional expectation of $V^+_t\zeta_t$. The process $\zeta$ is explicitly derived in the dynamic approaches: it is proven to be positive and to have unit expectation. Unfortunately however, it fails to be a martingale, so that Girsanov machinery cannot be used. Nevertheless, the expectation of $V^+_t\zeta_t$ can be computed explicitly, leading to analytical expected positive exposure profiles in the considered examples.