Lognormal stock price distribution

Interactive exploration of the lognormal model for stock prices and exceedance probabilities

A common assumption in risk management is that the stock price \(S_T\) at time \(T\) follows a lognormal distribution (see Hull 2023, chap. 10):

\[ \ln S_T \sim \phi\!\left[\ln S_0 + \left(\mu - \frac{\sigma^2}{2}\right)T,\; \sigma^2 T\right] \]

where \(S_0\) is the current price, \(\mu\) is the expected return per year, and \(\sigma\) is the volatility per year. The probability that \(S_T\) exceeds a threshold \(V\) is \(N(d_2)\), where:

\[ d_2 = \frac{\ln(S_0/V) + (\mu - \sigma^2/2)\,T}{\sigma\sqrt{T}} \]

Adjust the parameters below to explore how the distribution changes shape.

Tip

How to experiment

1. Increase \(\sigma\) to see the distribution fan out and become more right-skewed.
2. Increase \(T\) for a similar effect — uncertainty grows with time.
3. Move the threshold \(V\) to read off exceedance probabilities directly.

viewof lnS0 = Inputs.range([10, 200], {
  label: "Current stock price S₀ ($)",
  step: 5,
  value: 50
})

viewof lnMu = Inputs.range([0, 0.20], {
  label: "Expected return μ (per year)",
  step: 0.01,
  value: 0.10
})

viewof lnSigma = Inputs.range([0.05, 0.60], {
  label: "Volatility σ (per year)",
  step: 0.01,
  value: 0.20
})

viewof lnT = Inputs.range([0.25, 5], {
  label: "Time horizon T (years)",
  step: 0.25,
  value: 2
})

viewof lnV = Inputs.range([5, 400], {
  label: "Threshold V ($)",
  step: 5,
  value: 70
})

// --- Standard normal CDF (Abramowitz & Stegun approximation) ---
normcdf = 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 t = 1.0 / (1.0 + p * Math.abs(x))
  const y = 1 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-x * x / 2)
  return 0.5 * (1 + sign * y)
}

lnMeanLog = Math.log(lnS0) + (lnMu - lnSigma ** 2 / 2) * lnT
lnSdLog = lnSigma * Math.sqrt(lnT)

// Lognormal PDF curve
lnPDF = {
  const meanST = lnS0 * Math.exp(lnMu * lnT)
  const sdST = meanST * Math.sqrt(Math.exp(lnSigma ** 2 * lnT) - 1)
  const xMax = Math.max(meanST + 3.5 * sdST, lnV * 1.3)
  const xMin = Math.max(0.01, meanST - 3 * sdST)
  const n = 400
  const dx = (xMax - xMin) / n
  const pts = []
  for (let i = 0; i <= n; i++) {
    const x = xMin + i * dx
    if (x <= 0) continue
    const logx = Math.log(x)
    const pdf = (1 / (x * lnSdLog * Math.sqrt(2 * Math.PI))) *
      Math.exp(-0.5 * ((logx - lnMeanLog) / lnSdLog) ** 2)
    pts.push({ x, pdf, aboveV: x >= lnV })
  }
  return pts
}

lnD2 = (Math.log(lnS0 / lnV) + (lnMu - lnSigma ** 2 / 2) * lnT) / (lnSigma * Math.sqrt(lnT))
lnProbExceed = normcdf(lnD2)

lnMedian = Math.exp(lnMeanLog)
lnP5 = Math.exp(lnMeanLog - 1.6449 * lnSdLog)
lnP95 = Math.exp(lnMeanLog + 1.6449 * lnSdLog)
lnExpected = lnS0 * Math.exp(lnMu * lnT)

Plot.plot({
  height: 320,
  marginLeft: 50,
  marginRight: 20,
  x: { label: "Stock price at time T ($)", grid: false },
  y: { label: "Density", grid: true },
  marks: [
    Plot.areaY(lnPDF.filter(d => d.x >= lnV), {
      x: "x", y: "pdf",
      fill: "#d62728", fillOpacity: 0.2,
      curve: "natural"
    }),
    Plot.line(lnPDF, {
      x: "x", y: "pdf",
      stroke: "#4682b4", strokeWidth: 2.5,
      curve: "natural"
    }),
    Plot.ruleX([lnV], { stroke: "#d62728", strokeWidth: 2, strokeDash: [6, 3] }),
    Plot.text([{ x: lnV, label: `V = $${lnV}` }], {
      x: "x",
      y: lnPDF.reduce((m, d) => Math.max(m, d.pdf), 0) * 0.95,
      text: "label", fill: "#d62728", fontWeight: "bold", fontSize: 11, dx: 5, textAnchor: "start"
    }),
    Plot.ruleX([lnP5], { stroke: "#2ca02c", strokeDash: [3, 3], strokeOpacity: 0.6 }),
    Plot.ruleX([lnP95], { stroke: "#2ca02c", strokeDash: [3, 3], strokeOpacity: 0.6 }),
    Plot.text([{ x: lnP5, label: "5th" }], {
      x: "x",
      y: lnPDF.reduce((m, d) => Math.max(m, d.pdf), 0) * 0.7,
      text: "label", fill: "#2ca02c", fontSize: 9, dx: -5, textAnchor: "end"
    }),
    Plot.text([{ x: lnP95, label: "95th" }], {
      x: "x",
      y: lnPDF.reduce((m, d) => Math.max(m, d.pdf), 0) * 0.7,
      text: "label", fill: "#2ca02c", fontSize: 9, dx: 5, textAnchor: "start"
    })
  ]
})

fmtPct = v => (v * 100).toFixed(2)
fmtDollar = v => v.toFixed(2)

html`<table class="table" style="width:100%;">
<thead><tr><th colspan="2">Stock price distribution at T = ${lnT} years</th></tr></thead>
<tbody>
<tr><td style="font-weight:500;">Expected value E(S<sub>T</sub>)</td><td>$${fmtDollar(lnExpected)}</td></tr>
<tr><td style="font-weight:500;">Median</td><td>$${fmtDollar(lnMedian)}</td></tr>
<tr><td style="font-weight:500;">5th percentile</td><td>$${fmtDollar(lnP5)}</td></tr>
<tr><td style="font-weight:500;">95th percentile</td><td>$${fmtDollar(lnP95)}</td></tr>
<tr style="border-top: 2px solid #888;"><td style="font-weight:700;">Prob(S<sub>T</sub> > $${lnV})</td><td style="font-weight:700;">${fmtPct(lnProbExceed)}%</td></tr>
</tbody></table>`

Note

Lognormal vs normal

The lognormal distribution ensures that stock prices remain positive — a realistic property that a normal distribution cannot guarantee. Notice that the distribution is right-skewed: the mean exceeds the median, and the right tail extends further than the left. This skew increases with volatility and time horizon.

References

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