The Dynamic Spread of the Forward CDS with General Random Loss

We assume that the filtration F is generated by a d-dimensional Brownian motion W=(W1,…,Wd)′ as well as an integer-valued random measure μ(du,dy). The random variable τ~ is the default time and L is the default loss. Let G={Gt;t≥0} be the progressive enlargement of F by (τ~,L); that is, G is the smallest filtration including F such that τ~ is a G-stopping time and L is Gτ~-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 G and the forward CDS with random loss explicitly by the BSDEs method.