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 𝔽 (see [2]). Under the assumption that the density process of the default is a bounded 𝔽-predictable process, we obtain a quadratic-exponential type system of BSDEs similar to [2], which always has a unique solution (X, θ, ϑ). 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 [6]) explicitly. Then we introduce another quadratic-exponential type system of BSDEs (called 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 𝕋 = {a = T 0 < T 1 < … <T n = b} explicitly.
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