The Dynamic Spread of the Forward CDS with General Random Loss

Abstract

We assume that the filtration $\mathbb{F}$ is generated by a $d$-dimensional Brownian motion $W={(W_{\mathrm{1}},\dots ,W_d)}^{\prime }$ as well as an integer-valued random measure $\mu (du,dy)$. The random variable $\tilde{\tau }$ is the default time and $L$ is the default loss. Let $\mathbb{G}=\{{\mathcal{G}}_t;t\ge \mathrm{0}\}$ be the progressive enlargement of $\mathbb{F}$ by $(\tilde{\tau },L)$; that is, $\mathbb{G}$ is the smallest filtration including $\mathbb{F}$ such that $\tilde{\tau }$ is a $\mathbb{G}$-stopping time and $L$ is ${\mathcal{G}}_{\tilde{\tau }}$-measurable. We mainly consider the forward CDS with loss in the framework of stochastic interest rates whose term structures are modeled by the Heath-Jarrow-Morton approach with jumps under the general conditional density hypothesis. We describe the dynamics of the defaultable bond in $\mathbb{G}$ and the forward CDS with random loss explicitly by the BSDEs method.