Merton Model Explorer

Interactive Merton (1974) structural model: equity as a call option, debt as a risk-free bond short a put, risk-neutral default probability, and distance to default

Merton (1974) links equity, debt, and default probability through option pricing theory (Christoffersen 2012, chap. 12; Hull 2023, chap. 17). The firm holds assets \(A_t\) financed by a single zero-coupon bond with face value \(D\) maturing at \(t + T\) and residual equity.

At maturity the firm operates if \(A_{t+T} > D\) and otherwise defaults. The equity payoff is therefore the payoff of a call option on the firm’s assets:

\[ E_{t+T} = \max(A_{t+T} - D,\, 0) \]

Under geometric Brownian motion for assets with constant volatility \(\sigma_V\) and risk-free rate \(r\), the Black–Scholes–Merton formula gives the current equity value:

\[ E_0 = V_0\,\Phi(d_1) - D\,e^{-rT}\,\Phi(d_2), \qquad d_1 = \frac{\ln(V_0/D) + (r + \sigma_V^2/2)\,T}{\sigma_V\sqrt{T}}, \qquad d_2 = d_1 - \sigma_V\sqrt{T} \]

The corporate debt holders are long a risk-free bond with face \(D\) and short a put on firm assets with strike \(D\):

\[ D_0 = D\,e^{-rT}\,\Phi(d_2) + V_0\,\Phi(-d_1) \]

The risk-neutral default probability is the probability that the put is exercised, and the distance to default is how many standard deviations the log asset value sits above the default threshold:

\[ \Pr^{\mathbb{Q}}(A_{t+T} < D) = \Phi(-d_2), \qquad dd = d_2 = \frac{\ln(V_0/D) + (r - \sigma_V^2/2)\,T}{\sigma_V\sqrt{T}} \]

normalCDF = x => {
  const a1 = 0.254829592, a2 = -0.284496736, a3 = 1.421413741
  const a4 = -1.453152027, a5 = 1.061405429, p = 0.3275911
  const sign = x < 0 ? -1 : 1
  const z = Math.abs(x) / Math.sqrt(2)
  const t = 1.0 / (1.0 + p * z)
  const y = 1 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-z * z)
  return 0.5 * (1 + sign * y)
}

normalPDF = x => Math.exp(-x * x / 2) / Math.sqrt(2 * Math.PI)

fmt = (x, d) => x === undefined || isNaN(x) ? "N/A" : x.toFixed(d)

pctFmt = (x, d = 2) => (x * 100).toFixed(d) + "%"

Parameters

Tip

How to experiment

Start with the defaults and note the risk-neutral PD and distance to default. Increase \(\sigma_V\) while holding \(V_0\) fixed: equity value rises (long volatility), debt value falls (short volatility), and PD climbs. Push \(D\) above \(V_0\): equity becomes deep out of the money and the credit spread explodes. Shorten \(T\): PD drops sharply because the lognormal has less time to drift below \(D\).

viewof meV0 = Inputs.range([10, 200], {
  label: "V₀ asset value",
  step: 1,
  value: 100
})

viewof meD = Inputs.range([10, 200], {
  label: "D face value of debt",
  step: 1,
  value: 80
})

viewof meSigmaV = Inputs.range([0.05, 0.80], {
  label: "σ_V asset volatility (per year)",
  step: 0.01,
  value: 0.25
})

viewof meT = Inputs.range([0.25, 10], {
  label: "T horizon (years)",
  step: 0.25,
  value: 2
})

viewof meR = Inputs.range([0.0, 0.10], {
  label: "r risk-free rate",
  step: 0.001,
  value: 0.03
})

meCalc = {
  const V0 = meV0, D = meD, sV = meSigmaV, T = meT, r = meR
  const d1 = (Math.log(V0 / D) + (r + sV * sV / 2) * T) / (sV * Math.sqrt(T))
  const d2 = d1 - sV * Math.sqrt(T)
  const E0 = V0 * normalCDF(d1) - D * Math.exp(-r * T) * normalCDF(d2)
  const D0 = D * Math.exp(-r * T) * normalCDF(d2) + V0 * normalCDF(-d1)
  const D0_rf = D * Math.exp(-r * T)
  const putValue = D0_rf - D0
  const PD = normalCDF(-d2)
  const dd = d2
  const yCorp = -Math.log(D0 / D) / T
  const creditSpread = yCorp - r
  // Equity volatility via Ito: sE * E = N(d1) * sV * V
  const sE = normalCDF(d1) * sV * V0 / Math.max(E0, 1e-9)
  return { V0, D, sV, T, r, d1, d2, E0, D0, D0_rf, putValue, PD, dd, yCorp, creditSpread, sE }
}

mePayoff = {
  const D = meD
  const vals = []
  const maxV = Math.max(2 * D, 2 * meV0)
  for (let v = 0; v <= maxV; v += maxV / 200) {
    vals.push({
      V: v,
      equity: Math.max(v - D, 0),
      debt: Math.min(v, D),
      asset: v
    })
  }
  return vals
}

meDensity = {
  // Risk-neutral lognormal density of A_{t+T}
  const V0 = meV0, sV = meSigmaV, T = meT, r = meR, D = meD
  const mu = Math.log(V0) + (r - sV * sV / 2) * T
  const sd = sV * Math.sqrt(T)
  const lnLo = mu - 4 * sd
  const lnHi = mu + 4 * sd
  const data = []
  const N = 300
  for (let i = 0; i < N; i++) {
    const lnA = lnLo + (lnHi - lnLo) * i / (N - 1)
    const A = Math.exp(lnA)
    const dens = Math.exp(-0.5 * Math.pow((lnA - mu) / sd, 2)) / (A * sd * Math.sqrt(2 * Math.PI))
    data.push({ A, density: dens, below: A < D })
  }
  return { data, mu, sd, D }
}

{
  const r = meCalc
  return html`<div style="display:grid;grid-template-columns:repeat(auto-fit,minmax(220px,1fr));gap:12px;margin-top:8px;">
  <div style="padding:12px 16px;border-radius:8px;background:#e3f2fd;">
    <div style="font-size:0.8rem;color:#666;">Equity value E₀</div>
    <div style="font-size:1.4em;font-weight:700;color:#2f71d5;">${fmt(r.E0, 3)}</div>
    <div style="font-size:0.8rem;color:#666;">Implied equity σ_E = ${pctFmt(r.sE, 1)}</div>
  </div>
  <div style="padding:12px 16px;border-radius:8px;background:#fbe9e7;">
    <div style="font-size:0.8rem;color:#666;">Market value of debt D₀</div>
    <div style="font-size:1.4em;font-weight:700;color:#d62728;">${fmt(r.D0, 3)}</div>
    <div style="font-size:0.8rem;color:#666;">Risk-free value = ${fmt(r.D0_rf, 3)}; put = ${fmt(r.putValue, 3)}</div>
  </div>
  <div style="padding:12px 16px;border-radius:8px;background:#fff3e0;">
    <div style="font-size:0.8rem;color:#666;">Risk-neutral PD</div>
    <div style="font-size:1.4em;font-weight:700;color:#e67e22;">${pctFmt(r.PD, 3)}</div>
    <div style="font-size:0.8rem;color:#666;">Φ(−d₂), d₂ = ${fmt(r.d2, 3)}</div>
  </div>
  <div style="padding:12px 16px;border-radius:8px;background:#e8f5e9;">
    <div style="font-size:0.8rem;color:#666;">Distance to default</div>
    <div style="font-size:1.4em;font-weight:700;color:#2e7d32;">${fmt(r.dd, 3)}</div>
    <div style="font-size:0.8rem;color:#666;">d₁ = ${fmt(r.d1, 3)}</div>
  </div>
  <div style="padding:12px 16px;border-radius:8px;background:#ede7f6;">
    <div style="font-size:0.8rem;color:#666;">Corporate yield</div>
    <div style="font-size:1.4em;font-weight:700;color:#5e35b1;">${pctFmt(r.yCorp, 2)}</div>
    <div style="font-size:0.8rem;color:#666;">Credit spread = ${pctFmt(r.creditSpread, 2)}</div>
  </div>
</div>`
}

html`<div style="display:flex;gap:18px;flex-wrap:wrap;align-items:center;font-size:0.85rem;margin:6px 0 4px 0;">
  <span style="display:inline-flex;align-items:center;gap:6px;">
    <svg width="22" height="10"><line x1="0" y1="5" x2="22" y2="5" stroke="#2f71d5" stroke-width="2.5"/></svg>
    <span style="color:#2f71d5;font-weight:700;">Equity payoff</span>
  </span>
  <span style="display:inline-flex;align-items:center;gap:6px;">
    <svg width="22" height="10"><line x1="0" y1="5" x2="22" y2="5" stroke="#d62728" stroke-width="2.5"/></svg>
    <span style="color:#d62728;font-weight:700;">Debt payoff</span>
  </span>
  <span style="display:inline-flex;align-items:center;gap:6px;">
    <svg width="22" height="10"><line x1="0" y1="5" x2="22" y2="5" stroke="#aaa" stroke-width="1.5" stroke-dasharray="3 2"/></svg>
    <span style="color:#666;">Asset value (A = E + D)</span>
  </span>
  <span style="display:inline-flex;align-items:center;gap:6px;">
    <svg width="22" height="10"><line x1="11" y1="0" x2="11" y2="10" stroke="#888" stroke-width="1.5" stroke-dasharray="3 3"/></svg>
    <span style="color:#666;">Default threshold D = ${fmt(meD, 0)}</span>
  </span>
</div>`

{
  return Plot.plot({
    height: 360, marginLeft: 60, marginRight: 20,
    x: { label: "Asset value at maturity A_{t+T}", grid: true },
    y: { label: "Payoff", grid: true },
    marks: [
      Plot.ruleY([0]),
      Plot.ruleX([meD], { stroke: "#888", strokeDasharray: "3 3" }),
      Plot.line(mePayoff, { x: "V", y: "asset", stroke: "#aaa", strokeWidth: 1, strokeDasharray: "2 3" }),
      Plot.line(mePayoff, { x: "V", y: "debt", stroke: "#d62728", strokeWidth: 2.5 }),
      Plot.line(mePayoff, { x: "V", y: "equity", stroke: "#2f71d5", strokeWidth: 2.5 })
    ]
  })
}

html`<p style="color:#666;font-size:0.85rem;">Maturity payoffs. <span style="color:#2f71d5;font-weight:700;">Equity</span> is a call option on firm assets with strike D = ${fmt(meD, 0)}. <span style="color:#d62728;font-weight:700;">Debt</span> is min(A, D), equivalent to a risk-free bond short a put on firm assets. The grey dashed line is the asset value itself; equity plus debt always equals A.</p>`

html`<div style="display:flex;gap:18px;flex-wrap:wrap;align-items:center;font-size:0.85rem;margin:6px 0 4px 0;">
  <span style="display:inline-flex;align-items:center;gap:6px;">
    <svg width="14" height="10"><rect width="14" height="10" fill="#d62728" fill-opacity="0.45"/></svg>
    <span style="color:#d62728;font-weight:700;">Default region (A < D)</span>
  </span>
  <span style="display:inline-flex;align-items:center;gap:6px;">
    <svg width="14" height="10"><rect width="14" height="10" fill="#2f71d5" fill-opacity="0.25"/></svg>
    <span style="color:#2f71d5;font-weight:700;">Solvent region (A ≥ D)</span>
  </span>
  <span style="display:inline-flex;align-items:center;gap:6px;">
    <svg width="22" height="10"><line x1="11" y1="0" x2="11" y2="10" stroke="#d62728" stroke-width="2"/></svg>
    <span style="color:#d62728;">Default threshold D = ${fmt(meD, 0)}</span>
  </span>
</div>`

{
  const dens = meDensity.data
  const inside = dens.filter(d => d.below)
  const outside = dens.filter(d => !d.below)

  return Plot.plot({
    height: 360, marginLeft: 60, marginRight: 20,
    x: { label: "Asset value A_{t+T} (risk-neutral)", grid: true },
    y: { label: "Density", grid: true },
    marks: [
      Plot.ruleY([0]),
      Plot.areaY(inside, { x: "A", y: "density", fill: "#d62728", fillOpacity: 0.45 }),
      Plot.areaY(outside, { x: "A", y: "density", fill: "#2f71d5", fillOpacity: 0.25 }),
      Plot.line(dens, { x: "A", y: "density", stroke: "#333", strokeWidth: 1.5 }),
      Plot.ruleX([meD], { stroke: "#d62728", strokeWidth: 2 })
    ]
  })
}

{
  const r = meCalc
  return html`<p style="color:#666;font-size:0.85rem;">
Risk-neutral lognormal distribution of the asset value at T = ${fmt(meT, 2)} years. The <span style="color:#d62728;font-weight:700;">red shaded mass</span> is the risk-neutral default probability Φ(−d₂) = ${pctFmt(r.PD, 2)}. The distance to default is dd = ${fmt(r.dd, 3)} standard deviations.
</p>`
}

Comparative statics

viewof meSweep = Inputs.radio(["Asset volatility σ_V", "Face value D", "Horizon T", "Asset value V₀"], {
  label: "Sweep variable",
  value: "Asset volatility σ_V"
})

meSweepData = {
  const { V0, D, sV, T, r } = meCalc
  const out = []
  const pd = (V0_, D_, sV_, T_, r_) => {
    const d1 = (Math.log(V0_ / D_) + (r_ + sV_ * sV_ / 2) * T_) / (sV_ * Math.sqrt(T_))
    const d2 = d1 - sV_ * Math.sqrt(T_)
    return normalCDF(-d2)
  }
  const spread = (V0_, D_, sV_, T_, r_) => {
    const d1 = (Math.log(V0_ / D_) + (r_ + sV_ * sV_ / 2) * T_) / (sV_ * Math.sqrt(T_))
    const d2 = d1 - sV_ * Math.sqrt(T_)
    const D0 = D_ * Math.exp(-r_ * T_) * normalCDF(d2) + V0_ * normalCDF(-d1)
    return (-Math.log(D0 / D_) / T_) - r_
  }
  if (meSweep === "Asset volatility σ_V") {
    for (let x = 0.02; x <= 0.80; x += 0.005) out.push({ x, pd: pd(V0, D, x, T, r), spread: spread(V0, D, x, T, r) })
    return { rows: out, xlab: "σ_V", current: sV }
  }
  if (meSweep === "Face value D") {
    for (let x = 10; x <= 200; x += 1) out.push({ x, pd: pd(V0, x, sV, T, r), spread: spread(V0, x, sV, T, r) })
    return { rows: out, xlab: "D", current: D }
  }
  if (meSweep === "Horizon T") {
    for (let x = 0.1; x <= 10; x += 0.05) out.push({ x, pd: pd(V0, D, sV, x, r), spread: spread(V0, D, sV, x, r) })
    return { rows: out, xlab: "T", current: T }
  }
  for (let x = 10; x <= 200; x += 1) out.push({ x, pd: pd(x, D, sV, T, r), spread: spread(x, D, sV, T, r) })
  return { rows: out, xlab: "V₀", current: V0 }
}

{
  const { rows, xlab, current } = meSweepData
  const curPD = rows.reduce((best, r) => Math.abs(r.x - current) < Math.abs(best.x - current) ? r : best)
  return Plot.plot({
    height: 320, marginLeft: 60, marginRight: 60,
    x: { label: xlab, grid: true },
    y: { label: "Risk-neutral PD", grid: true, tickFormat: d => (d * 100).toFixed(1) + "%" },
    marks: [
      Plot.ruleY([0]),
      Plot.line(rows, { x: "x", y: "pd", stroke: "#e67e22", strokeWidth: 2.5 }),
      Plot.ruleX([current], { stroke: "#888", strokeDasharray: "3 3" }),
      Plot.dot([curPD], { x: "x", y: "pd", r: 5, fill: "#e67e22", stroke: "white", strokeWidth: 2 })
    ]
  })
}

{
  const { rows, xlab, current } = meSweepData
  const curS = rows.reduce((best, r) => Math.abs(r.x - current) < Math.abs(best.x - current) ? r : best)
  return Plot.plot({
    height: 320, marginLeft: 60, marginRight: 60,
    x: { label: xlab, grid: true },
    y: { label: "Credit spread", grid: true, tickFormat: d => (d * 100).toFixed(1) + "%" },
    marks: [
      Plot.ruleY([0]),
      Plot.line(rows, { x: "x", y: "spread", stroke: "#5e35b1", strokeWidth: 2.5 }),
      Plot.ruleX([current], { stroke: "#888", strokeDasharray: "3 3" }),
      Plot.dot([curS], { x: "x", y: "spread", r: 5, fill: "#5e35b1", stroke: "white", strokeWidth: 2 })
    ]
  })
}

html`<p style="color:#666;font-size:0.85rem;">Risk-neutral PD and credit spread as the selected parameter varies, holding the others at their current values. Volatility and leverage (D/V₀) drive PD monotonically upward; increasing V₀ or r reduces it.</p>`

References

Christoffersen, Peter F. 2012. Elements of Financial Risk Management. 2nd ed. Academic Press.

Hull, John. 2023. Risk Management and Financial Institutions. 6th ed. John Wiley & Sons.