Structural Models of the Firm Under State-Dependent Volatility and Jump Process Asset Dynamics

Abstract

We generalize the asset dynamics assumptions of Leland (1994b) and Leland and Toft (1996) to a much richer class of models. By assuming a stationary corporate debt structure with constant principal, coupon payment and average maturity through continuous retirement and refinancing as long as the firm remains solvent, we obtain analytical solutions with the state dependent diffusion volatility following the constant elasticity of variance (CEV) process for the variables of interest, including corporate debt value, total levered firm value and equity value. We also develop an efficient numerical algorithm with mixed jump diffusion asset dynamics, by adopting a restricted structure of default times, and derive numerical solutions for several variables of interest. We study the impact of state dependent volatility and jumps on the optimal capital structure, the debt capacity, the term structure of credit spread, the duration and convexity of risky debt, the equity volatility, the asset substitution impacts and the cumulative default probabilities (CDP). We also find that the term structure of CDP generated by the CEV diffusion process with a negative relationship between asset value and asset volatility could explain the CDP puzzle of structural models for short term debt by examining Moody’s historical CDP data.