无套利锥鞅模型及其在信用风险中的应用
An arbitrage-free conic martingale model with application to credit risk
摘要：
锥鞅是指在界之间演化的布朗鞅。在其他潜在的应用中，他们已经被建议为部分信息下的条件生存概率建模，就像通常在简化形式模型中一样。然而，锥鞅违约模型有一个特殊的功能，在对比类Cox模型，他们不满足所谓的\emph{immersion property}。因此，不清楚这种设置是否是无套利的。本文研究了锥鞅驱动的违约模型在信用风险建模中的实际应用。首先引入了一个无套利锥鞅，即$Φ$ -鞅，证明了它属于Crépey等人的动态高斯copula模型，从而为默认时间提供明确的构造方案。特别是，$Φ$ -鞅具有有趣的性质，其建设缓解实际实施。最后，我们将该模型应用于错误方向风险和CDS期权下的CVA定价，并将我们的结果与JCIR++（a.k.a. SSRJD）和TC-JCIR最近作为替代方案推出。
英文摘要：
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 $Φ$-martingale, by showing that it fits in the class of Dynamized Gaussian copula model of Crépey et al., thereby providing an explicit construction scheme for the default time. In particular, the $Φ$-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.
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