  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.
