Modeling the credit risk process: analysis of models based on stochastic processes to predict default rates

Abstract

This thesis analyses mathematical models based on stochastic processes to quantify credit risk, focusing on the probability of default of a firm taking into account the financial moment it is encountering. Two main models are studied: structural models (including the Merton, barrier and jump-diffusion models) and reduced-form or hazard function models.

Structural models link default to the firm’s asset dynamics and debt levels, interpreting equity as an option on the firm’s assets. Hazard function models, by contrast, consider default as a random and unpredictable event, modeling the time of default as a random variable following a certain distribution. This type of models use survival probabilities and hazard rates to estimate default timing.

The theoretical basis of each model are developed in detail, with emphasis on their assumptions, mathematical formulation and limitations, particularly in the context of credit risk, defaultable bonds, and credit default swaps. To evaluate their practical performance, a real-world case study is performed using historical data from Lehman Brothers prior to its 2008 bankruptcy. The models are based on this data to assess their accuracy and ability to anticipate default.

The results show the strengths and limitations of each model. While structural models provide a good approximation of the probability of default, they often misestimate recovery rates or expected losses. Hazard models better fit market data but lack a direct link to the firm’s actual asset value, since these are based on statistical approximations rather than in economic data.