The modeling of the probability of joint default or total number of defaults among the firms is one of the crucial problems to mitigate the credit risk since the default correlations significantly affect the portfolio loss distribution and hence play a significant role in allocating capital for solvency purposes. In this article, we derive a closed-form expression for the default probability of a single firm and probability of the total number of defaults by time $t$ in a homogeneous portfolio. We use a contagion process to model the arrival of credit events causing the default and develop a framework that allows firms to have resistance against default unlike the standard intensity-based models. We assume the point process driving the credit events is composed of a systematic and an idiosyncratic component, whose intensities are independently specified by a mean-reverting affine jump-diffusion process with self-exciting jumps. The proposed framework is competent of capturing the feedback effect. We further demonstrate how the proposed framework can be used to price synthetic collateralized debt obligation (CDO). Finally, we present the sensitivity analysis to demonstrate the effect of different parameters governing the contagion effect on the spread of tranches and the expected loss of the CDO.

公司间联合违约概率或违约总数的建模是减轻信用风险的关键问题之一，因为违约相关性显着影响投资组合损失分布，因此在为偿付能力目的分配资本方面发挥重要作用。在本文中，我们推导出单个公司的违约概率和同质投资组合中按时间 $t$ 的违约总数概率的封闭式表达式。我们使用传染过程来模拟导致违约的信用事件的到来，并开发一个框架，使公司能够抵抗违约，这与基于标准强度的模型不同。我们假设驱动信用事件的点过程由一个系统的和一个特殊的部分组成，其强度由具有自激跳跃的均值回复仿射跳跃扩散过程独立指定。所提出的框架能够捕获反馈效果。我们进一步证明了所提出的框架如何用于为合成债务抵押债券 (CDO) 定价。最后，我们提出了敏感性分析，以证明控制传染效应的不同参数对分档的扩散和 CDO 的预期损失的影响。