The Merton 1974 model

Merton (1974) treats firm equity as a European call option on the firm’s underlying assets.

Setup

A firm has total asset value \(A_t\) that follows geometric Brownian motion under the risk-neutral measure:

\[ \mathrm{d}A_t = (r - q)\, A_t\, \mathrm{d}t + \sigma_A A_t\, \mathrm{d}W_t. \]

At maturity \(T\) the firm owes a single liability \(D\).

  • If \(A_T \geq D\), equity holders receive \(A_T - D\) and bondholders are paid in full.

  • If \(A_T < D\), equity holders walk away (limited liability); the firm defaults; bondholders receive \(A_T\).

Equity therefore has the payoff \(E_T = \max(A_T - D, 0)\) — a European call.

Equity value via Black-Scholes-Merton

\[ E_0 = A_0\, e^{-qT} \Phi(d_1) - D\, e^{-rT} \Phi(d_2) \]

with

\[ d_1 = \frac{\ln(A_0 / D) + (r - q + \tfrac{1}{2}\sigma_A^2)\, T}{\sigma_A \sqrt{T}}, \qquad d_2 = d_1 - \sigma_A \sqrt{T}. \]

Distance to default and PD

Define the distance to default as \(d_2\):

\[ \mathrm{DD} = d_2 = \frac{\ln(A_0 / D) + (r - q - \tfrac{1}{2}\sigma_A^2)\, T}{\sigma_A \sqrt{T}}. \]

The risk-neutral probability of default is

\[ \mathrm{PD}_Q = \Phi(-d_2) = \Phi(-\mathrm{DD}). \]

Converting to the physical measure uses the asset-class Sharpe ratio \(\lambda = (\mu - r)/\sigma\):

\[ \mathrm{DD}_P = \mathrm{DD}_Q + \lambda \sigma_A \sqrt{T}, \qquad \mathrm{PD}_P = \Phi(-\mathrm{DD}_P). \]

Implied credit spread

For a horizon \(T\) and loss-given-default \(\mathrm{LGD}\), the implied continuous-compounding spread is

\[ s = -\frac{1}{T}\ln\!\bigl(1 - \mathrm{PD}\cdot\mathrm{LGD}\bigr). \]

Inferring \(A_0\) and \(\sigma_A\)

Both are unobservable. The classic Jones-Mason-Rosenfeld system uses the observed equity value \(E_0\) and equity volatility \(\sigma_E\) together with Itô’s lemma to solve

\[ E_0 = A_0 e^{-qT} \Phi(d_1) - D e^{-rT} \Phi(d_2), \qquad \sigma_E E_0 = e^{-qT}\Phi(d_1)\sigma_A A_0, \]

for the two unknowns \(A_0\) and \(\sigma_A\). Vassalou-Xing extend the procedure to use the full equity time series under maximum likelihood; the snapshot case collapses back to the JMR two-equation system.

Limitations

  • A single debt maturity is assumed (Geske handles compound debt structures).

  • The default barrier is reached only at \(T\) (Black-Cox allows touch-default at any \(t \leq T\)).

  • The asset process is pure Brownian (Zhou’s jump-diffusion adds jumps).

  • Interest rates are constant (Longstaff-Schwartz adds stochastic rates).

  • All values are assumed to be lognormal (CreditGrades, Leland-Toft relax this).

All of these refinements ship under merton.extensions.