A Free Boundary Problem for Corporate Bond Pricing and Credit Rating Under Different Upgrade and Downgrade Thresholds

Abstract

In this paper, a new model for corporate bond pricing and credit rating is proposed. In this model, credit rating migrations are assumed to depend on the ratio of debt and asset value of the underlying company, where debt is assumed to be a zero coupon corporate bond and asset value follows a geometric Brownian motion with volatility and dividend depending on credit rating. There is a buffer zone (called deadband in engineering) in credit rating migration, so upgrade and downgrade thresholds are different. Mathematically, this model is a system of partial differential equations with two free boundaries that correspond to the hitting boundaries in state space to upgrade and downgrade credit rating, respectively. The existence, uniqueness, regularity, and asymptotic behavior of the solution and free boundaries with zero dividend are obtained. Some nonzero dividend cases are discussed with numerical results.