Merton Structural Credit Model

At a company's debt maturity, there are only two outcomes for shareholders: if the firm's assets are worth more than what's owed to debtholders, shareholders pay off the debt and keep everything left over; if assets are worth less, shareholders walk away, hand the keys to the debtholders, and lose nothing further than what they already put in (limited liability). Look closely at that payoff and it's identical in shape to a call option: shareholders get $\max(\text{Assets} - \text{Debt}, 0)$, exactly a call struck at the face value of the debt, with the firm's total asset value playing the role of the underlying stock price. Merton's model takes this observation completely literally and prices equity using the Black-Scholes formula, treating the firm's assets as the underlying.

Equity as a call option

$$E_0 = V_0\, N(d_1) - D\, e^{-rT}\, N(d_2), \qquad d_1 = \frac{\ln(V_0/D) + (r + \tfrac12\sigma_V^2)T}{\sigma_V \sqrt{T}}, \qquad d_2 = d_1 - \sigma_V\sqrt{T}.$$

In plain English: $E_0$ is today's equity value, $V_0$ the firm's total asset value today, $D$ the face value of debt due at time $T$, $r$ the risk-free rate, and $\sigma_V$ the volatility of the firm's assets (not its equity, which is more volatile since equity is a leveraged claim). This is literally the Black-Scholes call formula with the stock price replaced by asset value and the strike replaced by the debt's face value — and $N(d_2)$, the same term that's the risk-neutral probability of the call finishing in the money in ordinary Black-Scholes, becomes here the model's implied risk-neutral probability of the firm surviving past $T$.

Worked example 1 — pricing equity and reading off default risk

A firm has asset value V_0 = \120$m, debt face value$D = $100$m due in$T=1$year, asset volatility$\sigma_V = 25%$, risk-free rate$r = 4%$. Compute$d_1 = \frac{\ln(120/100) + (0.04 + 0.03125)}{0.25} = \frac{0.1823 + 0.07125}{0.25} = \frac{0.2536}{0.25} = 1.014$, and$d_2 = 1.014 - 0.25 = 0.764$. Using$N(1.014) \approx 0.8446$and$N(0.764) \approx 0.7776$:$E_0 = 120 \times 0.8446 - 100 \times e^{-0.04} \times 0.7776 = 101.35 - 96.08 \times 0.7776 = 101.35 - 74.71 = $26.64$m. Implied risk-neutral default probability:$1 - N(d_2) = 1 - 0.7776 = 22.24%$ over the next year — high, reflecting the firm's asset value sitting only 20% above its debt with meaningful volatility.

Worked example 2 — leverage changes the default probability sharply

Keep everything the same but raise debt to D = \115$m (higher leverage).$d_1 = \frac{\ln(120/115) + 0.07125}{0.25} = \frac{0.0426 + 0.07125}{0.25} = \frac{0.1139}{0.25} = 0.4554$,$d_2 = 0.4554 - 0.25 = 0.2054$. Using$N(0.2054) \approx 0.5814$: implied default probability jumps to$1 - 0.5814 = 41.86% — nearly double the first case, from a debt increase of only \15m (12.5% of assets). Default probability in this model is extremely sensitive to leverage precisely because $d_2$ depends on $\ln(V_0/D)$, and small percentage changes in that ratio move the option deep toward or away from the money.

Set the strike above to represent the debt's face value and imagine the firm's asset value as the underlying — equity's payoff at maturity is exactly this call shape, with debtholders effectively owning everything below the strike (a risk-free bond minus a put they've implicitly sold).

What this means in practice

Merton's model is the theoretical backbone of "distance to default" style credit risk tools (like Moody's KMV), which infer a firm's unobservable asset value and volatility from its observable equity price and volatility, then compute an implied default probability without needing any bond or CDS prices at all — useful for private or thinly-traded credits where market credit spreads don't exist.