A Study About The Dynamics Of The CDS Pricing With Stochastic Rate

We consider that the default-free instantaneous forward rate is modeled in Heath-Jarrow-Merton model with no jumps. And then we introduce a bounded F-predictableprocess as a intensity process of the default and give some relevant conclusion on theenlargement of the filtration. Based on these assumptions, we discuss the generalexpression for the pre-default value of the defaultable bond. We obtain a new BSDEsystem by simplifying the one in Xiong and Kohlmann（2012）[23]. We get theexplicit solution of the system and describe the pre-default value of the defaultablebond explicitly with this solution. Particularly, the expression for the pre-default valueof the defaultable bond is more clear when the intensity process of the default andthe short-term rate are determined functions. Next, we study the dynamics of theCDS pricing problem. We introduce a reduced form of BSDE system compared withthe one in Xiong and Kohlmann（2012）[23]. Using the solutions of the above twosystems of BSDEs, we describe the premium of the CDS explicitly. We find that thepremium is a constant, independent of time, when the intensity process of the defaultand the short-term rate are both determined functions.