Credit derivatives with recovery of market value for multiple firms

It is well known that a defaultable bond subject to recovery of market value (RMV) is priced by discounting the payoff with an adjusted short rate by the loss rate and the default intensity rate of the issuer. We show that the formula can be generalized for a defaultable contract subject to RMV with heterogeneous multiple reference firms. The discounting short rate is adjusted by sum of the loss rate times the default intensity of each firm. Associated with the multiple defaultable firms with the RMV rule, a survival contingent forward measure is constructed. As applications tractable pricing formulae are derived for a vulnerable option on a defaultable bond and a vulnerable option contract to enter into a CDS.