The pricing of perpetual convertible bond with credit risk

Convertible bond gives holder the right to choose a conversion strategy to maximize the bond value, and issuer also has the right to minimize the bond value in order to maximize equity value. When there is default occurring, conversion and calling strategies are invalid. In the framework of reduced form model, we reduce the price of convertible bond to variational inequalities, and the coefficients of variational inequalities are unbounded at the original point. Then the existence and uniqueness of variational inequality are proven. Finally, we prove that the conversion area, the calling area and the holding area are connected subsets of the state space.