The structural credit risk model, or asset value model. is a model for assessing credit risk, typically of a corporation's debt. It was proposed in Black and Scholes' (1973) seminal paper on option pricing and a more detailed paper by Merton (1974). Merton anticipated the model in Merton (1970), and he actively supported the work of Black and Scholes, which is why the model is often called the Merton model. A popular implementation of the model is the commercial KMV model. KMV was a boutique software firm that is now owned by Moodys.
The model considers a corporation financed through a single debt and a single equity issue. The debt comprises a zero-coupon bond that matures at time t = t*, at which time it is to pay investors b dollars. The equity pays no dividends.
An unobservable process V describes the firm's value 0 at any time t. We ascribe the outstanding debt and equity values and , respectively. Accordingly, at any time t
At time t*, the firm's debt matures. At that time, either will exceed the bond's maturity value b, or it won't. In the former case, the firm will pay off the bondholders. The remaining value of the firm will belong to the equity holders, so
In the latter case, the firm defaults on its debt. The bondholders take ownership of the firm, and the stockholders are left with nothing:
Combining the above two results, we obtain a general expression for the value of the firm's stock at the maturity of its debt:
Look closely at this formula. It is precisely the payoff of a call option on the firm's value with strike price b. Based upon this realization, the asset value model treats the firm's equity as a call option on the value of the firm struck at the maturity value b of its debt. By put-call parity, the firm's debt comprises a risk-free bond that guarantees payment of b plus a short put option on the value of the firm struck at b. Accordingly
The asset value model treats just like any underlier. It assumes follows a geometric Brownian motion with volatility . Further, it makes all the other simplifying assumptions of the Black-Scholes (1973) option pricing formula. Accordingly, the firm's equity can be valued at any time t as
where p is the Black-Scholes formula for the value of a put. Note that we discount the payment b at the risk free rate because that payment is risk-free in formula —we have stripped out the credit risk as a put option.
At any time t, the distance to default for a the firm's debt is defined as
This is a metric indicating how many standard deviations the equity holders' call option is in-the-money. The smaller the distance to default, the more likely a default is to occur. The probability of default is precisely the probability of the call option expiring out-of-the-money. This is approximately equal to one minus the option's normalized delta (if investors were risk neutral, equality would be exact). See this glossary's article Black-Scholes (1973) option pricing formula for the Black-Scholes formula for delta. To normalize that value, divide the delta by the underlier's value.
Three shortcomings of the asset value model are:
1. Its assumption that the firm's debt financing consists of a one-year zero-coupon bond is, for most firms, an oversimplification..
2. The Black-Scholes (1973) simplifying assumptions are questionable in the context of corporate debt, and
3. The firm's value is not observable, which makes assigning values to it and its volatility problematic.
Still, the model provides a useful context for considering and modeling credit risk. Practical implementations of the model are used by financial institutions and institutional investors. These extend the model in some manner to facilitate the assigning of values to and . Such techniques generally relate to the observable market capitalization of the firm.