Default Barrier Models
The modeling of default from an economic point of view is a great challenge due to the binary and low probability nature of such an event. Default barrier models provide an elegant solution to this challenge since they link the default event to the point at which some continuously evolving quantity hits a known barrier. In structural models of credit risk (see Structural Default Risk Models) the process and the barrier are interpreted in terms of capital structure of the firm as the value of the firm and its liabilities. More generally, one can view the process and the barrier as state variables that need not necessarily observable.
In the classic Merton framework , the value of a firm (asset value) is considered to be stochastic and default is modeled as the point where the firm is unable to pay its outstanding liabilities when they mature. The asset value is modeled as a geometric Brownian motion:
where µ and σ represent the drift and volatility of the asset, respectively, and dW is a standard Brownian motion. The original Merton model assumes that a firm has issued only a zero-coupon bond and will not therefore default prior to the maturity of this debt as illustrated in Figure 1. Denoting the maturity and face value of the debt by ...