Default and Information

In a traditional structural model of default it is implicitly assumed that the information used to calibrate and run the model is publicly available. In reality, model inputs and parameters are unobservable. In this article we analyze the role of information in structural models, which we specify through a model definition of the default time and a model filtration. The model definition relates the default of a firm to its assets and liabilities. The model filtration describes the information of investors relative to the model definition. It parameterizes a family of default models for a given default time. An important situation is when the default is not observable with respect to the model filtration. Examples include models with incomplete information about firm assets and models with incomplete information about the liability-dependent barrier that triggers default. Here the default time is typically totally inaccessible, as in the intensity-based, reduced-form models of default. In this case the model admits generalized reduced-form security pricing formulae in terms of the trend, which is the cumulative intensity. The trend can be explicitly characterized through the conditional default probability given the model filtration. If the trend is absolutely continuous with respect to the Lebesgue measure, then its density is the intensity and our formulae simplify to the classical intensity-based formulae. Not every incomplete information model admits an intensity. A model in which investors cannot observe the default barrier is a first example.