A Modified Structural Model for Credit Risk—Utility Indifference Valuation

This paper modifies the classical structural models for credit risk by embedding them into the framework of optimal portfolio problems in an incomplete market. The price of corporate bonds is derived based on the indifference between the investor’s two utility maximization problems. Besides the uncertainty of the firm’s asset value, we introduce another diffusion process, which comes from the firm’s stock value, to drive credit risk. This results in the different behaviors of the credit spread from classical structural models. Two new parameters are introduced into the model, namely the investor’s risk aversion parameter and the correlation coefficient between the firm’s assets and the stocks it issues, which results in the nonlinearity of the pricing rule. Under the Markovian framework, default occurring at the maturity and default occurring at the first-passage-time are considered, and the corresponding closed formulae are derived by solving the HJB equation Cauchy problem and the HJB equation Dirichlet problem respectively. Keyword : Credit risk, Structural models, Credit spread, Utility indifference valuation, HJB equations.