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.