Credit Models · Module 6 · July 14, 2026

Structural Models: Merton and His Descendants

Equity is a call option on the firm's assets. Take that sentence seriously and you get default probabilities from market prices — the Merton model, KMV's EDF, Black-Cox, and Leland.

The models in Module 3 find default by pattern-matching on accounting ratios. In 1974, Robert Merton proposed something more ambitious: a mechanism. A firm defaults when the value of what it owns falls below what it owes. If you can model the assets, default probability stops being a regression output and becomes a theorem.

The setup

Take the simplest possible firm. Its assets are worth $V_t$ , evolving randomly through time. Its entire debt is one zero-coupon bond with face value $D$ due at date $T$ . No coupons, no covenants, no renegotiation. At maturity, one of two things happens:

$V_T > D$ — shareholders pay the debt, keep the difference $V_T - D$ .
$V_T \le D$ — shareholders hand over the keys. They have limited liability, so they get zero, not a negative number. Creditors recover $V_T$ .

Now look at the equity payoff: $\max(V_T - D,\, 0)$ . That is exactly the payoff of a European call option on $V$ with strike $D$ . This is not an analogy — it is an identity:

The formula

Assume assets follow geometric Brownian motion with volatility $\sigma_V$ , and price the call with Black-Scholes (which had been published just one year earlier):

E = V\,N(d_1) - D\,e^{-rT}N(d_2)

d_1 = \frac{\ln(V/D) + (r + \sigma_V^2/2)\,T}{\sigma_V\sqrt{T}}, \qquad d_2 = d_1 - \sigma_V\sqrt{T}

The quantity to stare at is $N(d_2)$ : the risk-neutral probability that $V_T > D$ , i.e. that the firm survives. Which means:

\text{PD}^{\mathbb{Q}} = N(-d_2)

Unpack $d_2$ and it reads like a credit officer’s checklist: PD rises with leverage ( $D/V$ ), rises with asset volatility, and depends on horizon. The numerator of $d_2$ measures how far assets sit above the default point; the denominator scales that distance in standard deviations. That is why $d_2$ is called the distance to default — “the firm is 2.3 sigmas from failing” is a sentence this model makes precise.

Play with the mechanics — watch what leverage and volatility each do to the PD, the equity value, and the spread:

Two behaviors worth reproducing by hand. Push volatility up with everything else fixed: equity gains value while debt loses it. Shareholders of a distressed firm rationally love risk — they own the upside, creditors own the downside. This is the asset substitution problem, and it is why loan covenants restrict what borrowers can do with the money. Second: push leverage toward 100% and watch equity value refuse to hit zero. Even deeply underwater equity retains option value as long as there is time and volatility left. That’s why the stock of visibly insolvent companies still trades above zero.

The problem KMV solved

Merton’s model prices a firm whose asset value and asset volatility you know. You never know either — assets aren’t traded; equity is. KMV (Kealhofer, McQuown, Vasicek — later bought by Moody’s) turned the model into a production system with three moves:

1. Back out $V$ and $\sigma_V$ from the equity market. You observe the equity value $E$ and equity volatility $\sigma_E$ . The model gives two equations linking them to the unobservables:

E = V N(d_1) - De^{-rT}N(d_2), \qquad \sigma_E E = N(d_1)\,\sigma_V V

Two equations, two unknowns — solve numerically. (The downloadable notebook implements exactly this with a few lines of scipy.)

2. Fix the default point. Real firms have a maturity ladder, not one zero-coupon bond, and they don’t default the moment book assets dip below total debt — short-term debt forces default, long-term debt gives breathing room. KMV’s empirical compromise: default point ≈ short-term liabilities + half of long-term liabilities.

3. Don’t trust the normal distribution — map empirically. The Gaussian tail says a firm 4σ from default has a PD around 0.003%. Reality is fatter-tailed. So KMV computes the distance to default within the model, then throws away $N(\cdot)$ and maps DD to default frequency using its historical database of realized defaults. The output — EDF, Expected Default Frequency — is a structural model wearing an empirical calibration. That hybrid design is the honest admission that the model’s ordering of firms is better than its tail probabilities.

The payoff of the whole construction: EDFs move with the stock market, daily. When a firm’s equity slides and its volatility spikes, its EDF deteriorates months before an accounting ratio or a rating action shows anything. Speed is the structural model’s killer feature — at the cost of inheriting every mood swing of the equity market, including the false alarms.

Black-Cox: default before maturity

Merton’s firm can only default on the maturity date — asset value can visit zero at $t = T/2$ and recover, and no default occurs. Black and Cox (1976) fixed this with a first-passage model: default happens the first time assets touch a barrier, at any time. This matches how covenants and cross-default clauses actually work, and it strictly increases PD relative to Merton for the same parameters (more chances to die). The mathematics upgrades from a terminal-value distribution to a first-hitting-time problem, but stays closed-form for a constant or exponential barrier.

Leland: making the capital structure endogenous

Everything so far takes the debt level as given. Leland (1994) closes the loop: let shareholders choose the default barrier and the leverage, trading off the tax shield of debt against bankruptcy costs. Out comes an optimal capital structure, an endogenous default trigger (shareholders keep injecting equity while the option value of continuing exceeds the cost — then stop), and credit spreads that reflect strategic behavior rather than a mechanical barrier. It’s the bridge between credit risk and corporate finance proper: the same model prices the debt and explains why the firm issued that much of it.

What structural models are for

A fair scorecard, sixty years in:

Best at: ranking public firms by credit quality, fast early warning, pricing intuition (every spread is a put premium), and providing the conceptual chassis for portfolio models — the Vasicek/Basel IRB formula in Module 10 is Merton applied to a whole portfolio.
Worst at: absolute PD levels straight from $N(-d_2)$ (calibrate empirically, as KMV did), private firms with no traded equity (back to Module 3’s ratios), short-horizon spreads (real one-month spreads are never near zero — a diffusion can’t jump, so the model says they should be), and anything where the “single debt barrier” story is too coarse — financials especially.

That short-spread failure is not a detail: it’s the observation that motivates the entire next family of models, where default arrives as a jump governed by an intensity, calibrated directly to market spreads — Module 7: reduced-form models.

Where this connects

The scorecards of Module 2 and ratios of Module 3 are statistical; Merton is the first model in the track where PD follows from an economic argument.
Bank supervisors’ stress tests (Module 9) ask “what happens to capital if asset values fall X%?” — a Merton question asked with regulatory vocabulary.
The single-factor version of Merton, applied to a portfolio of many borrowers, is the Basel IRB capital formula — that derivation is the centerpiece of Module 10.

Try it yourself

The notebook below builds everything in this post from scratch in Python: Black-Scholes pricing of the firm’s equity, the risk-neutral PD, the KMV-style numerical solve for unobservable $V$ and $\sigma_V$ from observable equity data, and a distance-to-default calculation on a worked example — with plots of how PD responds to leverage and volatility. Every function is a few lines; nothing is imported from a black box.

Merton Model from Scratch (Jupyter notebook)

Free download — no signup required.

Download

#merton#structural-models#kmv#distance-to-default

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