Merton's (1974) model is the foundational insight that makes credit risk a derivative pricing problem. If the value of a firm's assets follows a geometric Brownian motion and the firm has a single class of zero-coupon debt with face value due at time , then equity at maturity is exactly the payoff of a European call on asset value:
Default occurs when — the firm cannot repay its bondholders. This means the probability of default, the value of risky debt, and the credit spread are all computable from Black-Scholes. The model explains the intuition that highly leveraged firms (small ) and highly volatile firms (large ) have higher credit spreads — and provides quantitative formulas for both.
The practical limitation — firm asset value is unobservable — is resolved by the KMV/EDF methodology, which backs out and from equity prices using Itô's formula, then computes the distance-to-default (DD) and maps it to an empirically estimated default frequency (EDF).
Learning objectives. By the end of this note you should be able to:
- derive the equity and debt valuation formulas in Merton's model from the call/put option analogy;
- compute the physical and risk-neutral default probability in terms of ;
- derive the credit spread and describe its dependence on leverage and volatility ;
- explain the distance-to-default and describe how KMV/EDF uses it in practice;
- state two key empirical limitations of the Merton model (short-term spreads, lognormal tails).
In this lesson
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- The Merton model
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- Pricing equity and risky debt
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- Default probabilities and the credit spread
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- Distance-to-default and the KMV/EDF methodology
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- Extensions and limitations
Before you start
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