An arbitrage-free conic martingale model with application to credit risk

TL;DR

This paper introduces an arbitrage-free conic martingale model, called the Φ-martingale, for credit risk, demonstrating its practical application in CVA pricing and comparison with existing models.

Contribution

It develops the Φ-martingale, an arbitrage-free conic martingale model, and shows how it fits within the Gaussian copula framework for credit risk applications.

Findings

The Φ-martingale is arbitrage-free and explicitly constructible.

It effectively models default times within the Gaussian copula framework.

Application to CVA pricing demonstrates practical advantages.

Abstract

Conic martingales refer to Brownian martingales evolving between bounds. Among other potential applications, they have been suggested for the sake of modeling conditional survival probabilities under partial information, as usual in reduced-form models. Yet, conic martingale default models have a special feature; in contrast to the class of Cox models, they fail to satisfy the so-called \emph{immersion property}. Hence, it is not clear whether this setup is arbitrage-free or not. In this paper, we study the relevance of conic martingales-driven default models for practical applications in credit risk modeling. We first introduce an arbitrage-free conic martingale, namely the $\Phi$-martingale, by showing that it fits in the class of Dynamized Gaussian copula model of Cr\'epey et al., thereby providing an explicit construction scheme for the default time. In particular, the $\Phi$-martingale features interesting properties inherent on its construction easing the practical implementation. Eventually, we apply this model to CVA pricing under wrong-way risk and CDS options, and compare our results with the JCIR++ (a.k.a. SSRJD) and TC-JCIR recently introduced as an alternative.