Monte Carlo simulation of stock price paths

Interactive simulation comparing real-world and risk-neutral stock price paths, with theoretical confidence bands

The lognormal model generates stock price paths by simulating in discrete steps (see Hull 2023, chap. 10):

\[ S_{t+\Delta t} = S_t \exp\!\left[\left(\mu - \frac{\sigma^2}{2}\right)\Delta t + \varepsilon\,\sigma\sqrt{\Delta t}\right] \]

where \(\varepsilon\) is a standard normal random variable. Toggle between real-world and risk-neutral paths to see Girsanov’s theorem in action: only the drift changes, not the volatility.

Tip

How to experiment

1. Toggle between real-world and risk-neutral paths — notice the fan of paths shifts up or down but has the same spread.
2. Increase \(\sigma\) to see paths become more volatile.
3. Add more paths for a smoother picture; check whether simulated quantiles match the theoretical bands.

viewof mcS0 = Inputs.range([10, 200], {
  label: "Current stock price S₀ ($)",
  step: 5,
  value: 50
})

viewof mcMu = Inputs.range([0.01, 0.25], {
  label: "Real-world expected return μ (per year)",
  step: 0.01,
  value: 0.10
})

viewof mcR = Inputs.range([0.01, 0.10], {
  label: "Risk-free rate r (per year)",
  step: 0.01,
  value: 0.03
})

viewof mcSigma = Inputs.range([0.05, 0.60], {
  label: "Volatility σ (per year)",
  step: 0.01,
  value: 0.25
})

viewof mcT = Inputs.range([0.5, 5], {
  label: "Time horizon T (years)",
  step: 0.5,
  value: 2
})

viewof mcNPaths = Inputs.range([50, 2000], {
  label: "Number of paths",
  step: 50,
  value: 500
})

viewof mcWorld = Inputs.radio(["Real world (μ)", "Risk-neutral (r)"], {
  label: "World",
  value: "Real world (μ)"
})

mutable mcResimClicks = 0

viewof mcResimButton = {
  const btn = html`<button style="background: #2f71d5; color: white; border: none; border-radius: 6px; padding: 0.35rem 1rem; font-weight: 500; font-size: 0.9rem; cursor: pointer;">New simulation</button>`
  btn.onclick = () => { mutable mcResimClicks++ }
  return btn
}

mulberry32 = seed => {
  return function () {
    let t = seed += 0x6D2B79F5
    t = Math.imul(t ^ t >>> 15, t | 1)
    t ^= t + Math.imul(t ^ t >>> 7, t | 61)
    return ((t ^ t >>> 14) >>> 0) / 4294967296
  }
}

randomNormal = rng => {
  let u = 0, v = 0
  while (u === 0) u = rng()
  while (v === 0) v = rng()
  return Math.sqrt(-2.0 * Math.log(u)) * Math.cos(2.0 * Math.PI * v)
}

mcSeed = 20260213 + mcResimClicks
mcDrift = mcWorld === "Real world (μ)" ? mcMu : mcR
mcStepsPerYear = 12  // monthly steps
mcNSteps = Math.round(mcT * mcStepsPerYear)
mcDt = mcT / mcNSteps

mcPaths = {
  const rng = mulberry32(mcSeed)
  const paths = []
  const finalPrices = []

  for (let p = 0; p < mcNPaths; p++) {
    let S = mcS0
    const path = [{ time: 0, price: S, trial: p }]
    for (let step = 1; step <= mcNSteps; step++) {
      const eps = randomNormal(rng)
      S = S * Math.exp((mcDrift - mcSigma ** 2 / 2) * mcDt + eps * mcSigma * Math.sqrt(mcDt))
      path.push({ time: step * mcDt, price: S, trial: p })
    }
    paths.push(path)
    finalPrices.push(S)
  }

  finalPrices.sort((a, b) => a - b)

  return { paths, finalPrices }
}

mcBandP5 = {
  const pts = []
  for (let step = 0; step <= mcNSteps; step++) {
    const t = step * mcDt
    const meanLog = Math.log(mcS0) + (mcDrift - mcSigma ** 2 / 2) * t
    const sdLog = mcSigma * Math.sqrt(Math.max(t, 1e-10))
    pts.push({ time: t, price: Math.exp(meanLog - 1.6449 * sdLog) })
  }
  return pts
}

mcBandMedian = {
  const pts = []
  for (let step = 0; step <= mcNSteps; step++) {
    const t = step * mcDt
    const meanLog = Math.log(mcS0) + (mcDrift - mcSigma ** 2 / 2) * t
    pts.push({ time: t, price: Math.exp(meanLog) })
  }
  return pts
}

mcBandP95 = {
  const pts = []
  for (let step = 0; step <= mcNSteps; step++) {
    const t = step * mcDt
    const meanLog = Math.log(mcS0) + (mcDrift - mcSigma ** 2 / 2) * t
    const sdLog = mcSigma * Math.sqrt(Math.max(t, 1e-10))
    pts.push({ time: t, price: Math.exp(meanLog + 1.6449 * sdLog) })
  }
  return pts
}

// Flatten paths for plotting
mcPathData = mcPaths.paths.flat()

mcSimMedian = mcPaths.finalPrices[Math.floor(mcPaths.finalPrices.length * 0.5)]
mcSimP5 = mcPaths.finalPrices[Math.floor(mcPaths.finalPrices.length * 0.05)]
mcSimP95 = mcPaths.finalPrices[Math.floor(mcPaths.finalPrices.length * 0.95)]
mcTheoMedian = Math.exp(Math.log(mcS0) + (mcDrift - mcSigma ** 2 / 2) * mcT)
mcTheoP5 = Math.exp(Math.log(mcS0) + (mcDrift - mcSigma ** 2 / 2) * mcT - 1.6449 * mcSigma * Math.sqrt(mcT))
mcTheoP95 = Math.exp(Math.log(mcS0) + (mcDrift - mcSigma ** 2 / 2) * mcT + 1.6449 * mcSigma * Math.sqrt(mcT))

fmtDollar = v => v.toFixed(2)

Simulated paths

Plot.plot({
  height: 360,
  marginLeft: 55,
  marginRight: 20,
  x: { label: "Time (years)", grid: false },
  y: { label: "Stock price ($)", grid: true },
  marks: [
    Plot.line(mcPathData, {
      x: "time", y: "price", z: "trial",
      stroke: "#4682b4", strokeWidth: 0.4, strokeOpacity: 0.15
    }),
    Plot.line(mcBandP5, {
      x: "time", y: "price",
      stroke: "#2ca02c", strokeWidth: 2, strokeDasharray: "6 3"
    }),
    Plot.line(mcBandMedian, {
      x: "time", y: "price",
      stroke: "#d62728", strokeWidth: 2, strokeDasharray: "6 3"
    }),
    Plot.line(mcBandP95, {
      x: "time", y: "price",
      stroke: "#2ca02c", strokeWidth: 2, strokeDasharray: "6 3"
    }),
    Plot.text([{ time: mcT, price: mcBandP95[mcBandP95.length - 1].price, label: "95th" }], {
      x: "time", y: "price", text: "label",
      fill: "#2ca02c", fontSize: 10, fontWeight: "bold", dx: 5, textAnchor: "start"
    }),
    Plot.text([{ time: mcT, price: mcBandMedian[mcBandMedian.length - 1].price, label: "Median" }], {
      x: "time", y: "price", text: "label",
      fill: "#d62728", fontSize: 10, fontWeight: "bold", dx: 5, textAnchor: "start"
    }),
    Plot.text([{ time: mcT, price: mcBandP5[mcBandP5.length - 1].price, label: "5th" }], {
      x: "time", y: "price", text: "label",
      fill: "#2ca02c", fontSize: 10, fontWeight: "bold", dx: 5, textAnchor: "start"
    })
  ]
})

html`<p style="color:#666; font-size:0.85rem;">Showing ${mcNPaths} paths in the <strong>${mcWorld}</strong> (drift = ${(mcDrift * 100).toFixed(1)}% per year). Dashed lines show the theoretical 5th percentile, median, and 95th percentile from the analytical formula.</p>`

Simulated vs theoretical

html`<table class="table" style="width:100%;">
<thead><tr><th>Statistic at T = ${mcT} years</th><th>Simulated (${mcNPaths} paths)</th><th>Theoretical</th></tr></thead>
<tbody>
<tr><td style="font-weight:500;">5th percentile</td><td>$${fmtDollar(mcSimP5)}</td><td>$${fmtDollar(mcTheoP5)}</td></tr>
<tr><td style="font-weight:500;">Median</td><td>$${fmtDollar(mcSimMedian)}</td><td>$${fmtDollar(mcTheoMedian)}</td></tr>
<tr><td style="font-weight:500;">95th percentile</td><td>$${fmtDollar(mcSimP95)}</td><td>$${fmtDollar(mcTheoP95)}</td></tr>
<tr><td style="font-weight:500;">World</td><td colspan="2">${mcWorld} — drift = ${(mcDrift * 100).toFixed(1)}%, volatility = ${(mcSigma * 100).toFixed(1)}%</td></tr>
</tbody></table>
<p style="margin-top:0.5rem; color:#666; font-size:0.85rem;">Toggle between worlds to verify Girsanov's theorem: only the drift changes. With more paths, the simulated quantiles converge to the theoretical values.</p>`

References

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