Local Risk Minimization of Contingent Claims Simultaneously Exposed to Endogenous and Exogenous Default Times

We study the local risk minimization approach for contingent claims that might be simultaneously prone to both endogenous (or structural) and exogenous (or reduced form) default events. The exogenous default time is defined through a hazard rate process that can depend on both the underlying risky asset values and its running infimum process. On the other hand, the endogenous default time could be modeled by a first-passage-time approach. In particular, our framework provides a unification of structural and reduced form credit risk modeling. In our work, the evolution of the underlying risky asset values is modeled by an exponential Lévy process, for example exponential jump-diffusion models. Our aim is to determine locally risk minimizing hedging strategies of the contingent claims that are affected by both structural and reduced form default events, through solutions of either partial differential equations or partial-integro differential equations. Finally, we show that these solutions are numerically implementable, and we provide some numerical examples.