Modeling the forward CDS spreads with jumps

Abstract

We consider the forward CDS in the framework of stochastic interest rates whose term structures are modeled in the sense of the Heath-Jarrow-Morton model with jumps adapted to a filtration $\bb F$(see \cite{Xiong-Kohlmann2010b}). Under the assumption that the density process of the default is a bounded $\bb F$-predictable process, we obtain a quadratic-exponential type system of BSDEs similar to \cite{Xiong-Kohlmann2010b} which always has a unique solution $(X,\theta,\vartheta)$. By the solution of such a system of BSDEs, we will describe the dynamics of the the pre-default values of the defaultable bond, the defaultable forward Libor rates and the restricted defaultable forward measure (see in \cite{Eberlein-Kluge-Schoenbucher-2006}) explicitly. Then we introduce another quadratic-exponential type system of BSDEs (called \textbf{adjoint system of BSDEs}) which also always has a unique solution, and using this solution, we describe the dynamic of the fair spread of the forward CDS with the tenor structure $\bb T ={a=T_0 < T_1 < \cdots< T_n=b}$ explicitly.