• volatility risk
• structural models
• endogenous bankruptcy
• credit risk

The dissertation deals with modeling credit risk through a structural model approach. The thesis consists of three papers in which we build on the capital structure of a firm proposed by Leland and we study different extensions of his seminal paper with the purpose of obtaining
results more in line with historical norms and empirical evidence, studying in details all mathematical aspects. The thesis analyzes credit risk modeling following a structural model approach with endogenous default. We extend the classical Leland framework in three main
directions with the aim at obtaining results more in line with empirical evidence. We introduce payouts and then also consider corporate tax rate asymmetry: numerical results show that these lead to predicted leverage ratios closer to historical norms, through their joint influence on optimal capital structure. Finally, we introduce volatility risk. Following Leland suggestions we consider a framework in which the assumption of constant volatility in the underlying firm’s assets value stochastic evolution is removed. Analyzing defaultable claims involved in the capital structure of the firm we derive their corrected prices under a fairly large class of stochastic volatility models by applying singular perturbation theory. Exploiting optimal capital
structure, the stochastic volatility framework seems to be a robust way to improve results in the direction of both higher spreads and lower leverage ratios in a quantitatively significant way.

This PhD Thesis has been done under a cotutelle program between University of Pisa
and Institut de Mathématiques de Toulouse (IMT), Université Paul Sabatier, Toulouse III (Toulouse, France).