There are two types of models for modelling default probabilities and defaultable bonds.
- Structural Models also known as Asset Based Models. The time to default is an endogenous random variable. These models are used mainly by ratings agencies to determine default probabilities and functions in corporate finance when designing the capital structure for example. A modelling practitioner would require the firm’s balance sheet, bankruptcy and historic credit data in order to obtain information on assets and liabilities.
- Reduced-form Models also known as Intensity Based Models. The time to default is an exogenous random variable. These models are typically used for pricing and hedging. A modelling practitioner would only require market prices and publicly available information.
Duffie and Lando (1997) show that there is a close connection between both model types.
Asset Based Models (Firm Value)
The time of default is related in a deterministic way to the asset value $V_t$ of the company. In general, the asset value is modelled as an Ito diffusion.
- MERTON MODEL
Default can only occur at bond maturity $T$ at which principal $L$ is repaid. For example, the holder of a zero coupon bond can only default at maturity (as long as there are no other covenants as part of the overall debt contract). This model assumes Absolute Priority Rules (APR).- Deviation from absolute priority, we call this strategic default, is the subject of the models which make use of a default barrier. Simply put this is an asset level $V_T$ which is selected by equity holders (junior claimants) as the default triggering debt level prior to which they cover operating shortfalls by issuing new equities or other means of injecting cash. One such model, Leland (1994) models the asset level $V_T$ in this manner. Another, introduced by Mella-Barral and Perraudin in their Journal of Finance paper Strategic Debt Service available from Wiley, examines the cashflow process driving $V_T$.
- BLACK & COX MODEL
A First-Passage Default Model which generalises the asset based model such that default can occur at any point $\tau$ in time and not just at maturity $T$ of the debt contract.
More generally, $V_t$ is known as the state variable and could represent assets or a cashflow process. The payoff structure
means any claim could be determined.