Free boundary problem pricing defaultable corporate bonds with multiple credit rating migration risk and stochastic interest rate

In this paper, valuation for a defaultable corporate bond subject to multiple credit rating migration risk and stochastic volatility of interest rate is addressed in the structure framework through a free boundary problem, which is derived by a series of transformations. The existence, uniqueness and regularity of solution to the free boundary problem are obtained to verify the rationality of the bond pricing model. Furthermore, we show that the solution of the free boundary problem is convergent to a close form steady status, which may provide some information on the developing characteristics of the bond price. As the coexistence of stochastic interest rate and defaulting boundary, this convergence is achieved through an auxiliary free boundary problem and a Lyapunov argument. Interestingly, the converged steady status can be explicitly solved, which is not the case in the existing literatures on multiple credit rating migration. Finally, we present an explicit formula for valuating this defaultable bond with multiple credit rating migration risk and stochastic interest rate.