Default Probability In Structural Credit Models And Implementation

This paper mainly discuss two structural credit risk models under the essential as-sumption that the firm's asset value follows a geometric Brownian motion: Merton credit risk model and credit risk model in a barrier option framework (DOC credit risk model). The fundamental difference between these two models is the diverse understanding of corporate securities. Corporate securities have been viewed in terms of standard options written on the underlying assets of the firm in Merton model. But in the barrier option framework, it argues that corporate equity is a down-and-out call option on corporate assets and corporate bond is a down-and-in call option on firm assets. they are all barrier options. We provide the valuation formula for corporate securities and formula for implied risk-neutral default proba-bilities respectively, the more important is that embeds implied risk-neutral default probabilities derived from these two models into backward stochastic differential equations framework, regarding its as g-expectation with specially terminal fune-tion. In particular, we discuss how to implement generalized default probability in market on uncomplete information, which the investors often face. This requires estimates for model unknown parameters and unobserved variables. The main al-gorithm using in this paper is maximum likelihood estimates (MLE) by employing observed corporate equity data as transformed-data. it is more better because it has accompanying distributional information for parameters. After that, we get the estimates for generalized default probabilities through the algorithm for BSDE. These new default probabilities can improve the ability for default forecasting by calibrating generators using historical data, this can overcome the drawback of structural models. In addition, we test the assumption that asset values follow a geometric Brownian motion. On the other hand. we also discuss definition of de- fault probability under the assumption that firm's asset value follows the dynamic equation with uncertain volatility, which also means that the investors confront volatility risk. Moreover, we study the Black-Scholes-Barenblatt equation related to this new default probability, and give finite difference method for numerical solution of BSB equations and numerical examples.