Structural credit and the Merton model

Definition A structural credit model describes default as an event determined by the value of a firm’s assets relative to its liabilities. The Merton model is the structural credit model with the following hypotheses.

1. The firm’s assets have a market value $A_t$ that can be traded or replicated in a frictionless market with a constant continuously compounded rate $r$, so claims on the assets are priced as discounted $\mathbb{Q}$-expectations.
2. Under $\mathbb{Q}$ the asset value follows the geometric Brownian motion $dA_t = r A_t\,dt + \sigma A_t\,dW^{\mathbb{Q}}_t$ with $A_0 > 0$ and a constant $\sigma > 0$, and under $\mathbb{P}$ it follows the same equation with a constant drift $\mu$ in place of $r$ and a $\mathbb{P}$-Brownian motion $W_t$.
3. The only liability is one zero-coupon debt with face $F > 0$ due at $T > 0$, and there are no asset payouts, coupons or dividends before $T$.
4. Limited liability and absolute priority hold, and there are no bankruptcy costs or taxes.
5. Default can happen only at $T$, and it happens exactly when $A_T < F$.

Under these hypotheses equity and debt pay

at $T$, and $E_T + D_T = A_T$ in every state. The cards below derive the values

On this page $E_t$ is the equity value and $D_t$ the debt value, while $\mathbb{E}$ keeps its meaning as an expectation.

Definition The firm asset value $A_t$ is the market value at time $t$ of the cash flows of the firm’s assets. Under the hypotheses of the Merton model it equals the total value $E_t + D_t$ of the claims on the firm. It is a modeled market value, not the asset total of a balance sheet, and it is not directly observed. Under $\mathbb{Q}$ the geometric Brownian motion has the exact solution

where $Z = W^{\mathbb{Q}}_T/\sqrt{T}$ is standard normal under $\mathbb{Q}$. The volatility $\sigma$ is per square-root year and $T$ is in years, so $\sigma\sqrt{T}$ has no unit. Before maturity the equity value is a smooth function $E_t = E(t, A_t)$ with $\partial E/\partial A = \Phi(d_1)$, where $d_1$ uses the remaining time $T - t$, and Itô’s lemma gives the instantaneous equity volatility

It changes with $A_t$ and $t$, so equity is not itself a geometric Brownian motion. Inferring $A_0$ and $\sigma$ from an observed equity value and equity volatility means solving this relation together with the equity formula, a nonlinear inverse problem that is stated here and not developed. Even a good fit leaves the physical drift $\mu$ undetermined.

Definition The default boundary of a structural model is the level of the asset value at which default is declared. In the Merton model the boundary is the face $F$, checked only at maturity, so the default event is

and on this event default happens at time $T$. The tie $A_T = F$ has $\mathbb{Q}$-probability zero under the lognormal law, and in that state the creditors are paid in full anyway, so it affects no value. A path that falls below $F$ before $T$ and ends above it is not a default. A first-passage model instead declares default at the first time the asset value reaches a boundary before $T$. That changes the default event and leads to a barrier problem in place of the terminal option problem of this lesson, and first-passage credit models are outside this lesson.

Definition Under the hypotheses of the Merton model, equity pays $E_T = (A_T - F)^{+}$ at $T$, the payoff of a European call on the asset value with strike $F$. Its value is

where

To derive it, write the payoff as $A_T \mathbf{1}_{\{A_T > F\}} - F \mathbf{1}_{\{A_T > F\}}$. Solving $\ln A_T > \ln F$ for $Z$ gives $\{A_T > F\} = \{Z > -d_2\}$, so the second term is worth $F e^{-rT}\Phi(d_2)$. For the first term put $v = \sigma\sqrt{T}$, so that $e^{-rT} A_T = A_0 e^{vZ - v^2/2}$. Completing the square gives $e^{vz - v^2/2}\varphi(z) = \varphi(z - v)$, and the substitution $y = z - v$ gives

With $A_0$, $F$, $r$ and $T$ held fixed, the equity value is strictly increasing in $\sigma$, because $\partial E_0/\partial \sigma = A_0 \varphi(d_1)\sqrt{T} > 0$.

Definition Under the hypotheses of the Merton model, debt pays

the face minus the payoff of a European put on the asset value with strike $F$. Its value is

where $\mathrm{Put}_0 = F e^{-rT}\Phi(-d_2) - A_0\Phi(-d_1)$, and the same number is $A_0 - E_0$. The term $A_0\Phi(-d_1) = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}\bigl[A_T \mathbf{1}_{\{A_T < F\}}\bigr]$ is the value of what the creditors take in the default states, and $F e^{-rT}\Phi(d_2)$ is the value of the full face in the survival states. The payoff $\min(A_T, F)$ is below $A_T$ when $A_T > F$ and below $F$ when $A_T < F$, and both events have positive probability, so $0 < D_0 < \min(A_0, F e^{-rT})$. The continuously compounded yield $y$ solves $D_0 = F e^{-yT}$, and the credit spread is

The debt value depends on how much is missing in each default state, so a default probability alone does not determine it.

Definition Under the hypotheses of the Merton model, the risk-neutral default probability is

and $\Phi(d_2)$ is the $\mathbb{Q}$-probability of survival, so a claim paying $1$ at $T$ if the firm survives costs $e^{-rT}\Phi(d_2)$. Under $\mathbb{P}$, with drift $\mu$, the same event has the probability

The two laws are linked by Girsanov’s theorem. Take the constant market price of risk $\theta = (\mu - r)/\sigma$ and the density $d\mathbb{Q}/d\mathbb{P} = \exp(-\theta W_T - \tfrac12\theta^2 T)$ on $\mathcal{F}_T$, a true martingale density because Novikov’s condition holds for a constant $\theta$. Then $W^{\mathbb{Q}}_t = W_t + \theta t$ is a $\mathbb{Q}$-Brownian motion, and substituting $dW_t = dW^{\mathbb{Q}}_t - \theta\,dt$ turns the drift $\mu$ into $r$. If $\mu > r$, then $d_2^{\mathbb{P}} > d_2$ and the physical default probability is the smaller of the two. The boundary and the payoffs are the same under both measures, and only the probability law changes.