Two worlds: risk-neutral vs real-world

Visual comparison of real-world and risk-neutral stock price distributions, illustrating Girsanov’s theorem

A confusing aspect of risk management is that valuation and scenario analysis require different assumptions about how asset prices behave (see Hull 2023, chap. 10). The same stock price model applies in both the real world and the risk-neutral world — but the expected return \(\mu\) differs. In the risk-neutral world, \(\mu = r\) (the risk-free rate). In the real world, \(\mu\) includes a risk premium.

Girsanov’s theorem tells us that when moving between worlds with different risk preferences, the volatility \(\sigma\) of market variables stays the same — only the expected growth rate changes.

The chart below overlays both distributions for the same stock, so you can see how the real-world distribution (blue) is shifted to the right of the risk-neutral distribution (orange) when \(\mu > r\).

Tip

How to experiment

Increase the gap between \(\mu\) and \(r\) to see the distributions separate.
Increase \(T\) to amplify the difference.
Notice that both curves have the same spread (volatility) — only the center shifts.

viewof twS0 = Inputs.range([10, 200], {
  label: "Current stock price S₀ ($)",
  step: 5,
  value: 50
})

viewof twMu = Inputs.range([0.01, 0.25], {
  label: "Real-world expected return μ (per year)",
  step: 0.01,
  value: 0.12
})

viewof twR = Inputs.range([0.01, 0.15], {
  label: "Risk-free rate r (per year)",
  step: 0.01,
  value: 0.03
})

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

viewof twT = Inputs.range([0.25, 5], {
  label: "Time horizon T (years)",
  step: 0.25,
  value: 1
})

// Compute lognormal PDF for given parameters
lognormalPDF = (S0, mu, sigma, T, nPts) => {
  const meanLog = Math.log(S0) + (mu - sigma ** 2 / 2) * T
  const sdLog = sigma * Math.sqrt(T)
  const meanST = S0 * Math.exp(mu * T)
  const sdST = meanST * Math.sqrt(Math.exp(sigma ** 2 * T) - 1)
  const xMax = meanST + 4 * sdST
  const xMin = Math.max(0.01, meanST - 3.5 * sdST)
  const dx = (xMax - xMin) / nPts
  const pts = []
  for (let i = 0; i <= nPts; i++) {
    const x = xMin + i * dx
    if (x <= 0) continue
    const logx = Math.log(x)
    const pdf = (1 / (x * sdLog * Math.sqrt(2 * Math.PI))) *
      Math.exp(-0.5 * ((logx - meanLog) / sdLog) ** 2)
    pts.push({ x, pdf })
  }
  return pts
}

twRealPDF = lognormalPDF(twS0, twMu, twSigma, twT, 400).map(d => ({ ...d, world: "Real world (μ)" }))
twRnPDF = lognormalPDF(twS0, twR, twSigma, twT, 400).map(d => ({ ...d, world: "Risk-neutral (r)" }))
twBothPDF = [...twRealPDF, ...twRnPDF]

twERealST = twS0 * Math.exp(twMu * twT)
twERnST = twS0 * Math.exp(twR * twT)
twMedianReal = Math.exp(Math.log(twS0) + (twMu - twSigma ** 2 / 2) * twT)
twMedianRn = Math.exp(Math.log(twS0) + (twR - twSigma ** 2 / 2) * twT)
twSdLogReal = twSigma * Math.sqrt(twT)
twSdLogRn = twSigma * Math.sqrt(twT)

fmtDollar = v => v.toFixed(2)

Plot.plot({
  height: 340,
  marginLeft: 50,
  marginRight: 20,
  color: { legend: true, domain: ["Real world (μ)", "Risk-neutral (r)"], range: ["#4682b4", "#ff7f0e"] },
  x: { label: "Stock price at time T ($)", grid: false },
  y: { label: "Density", grid: true },
  marks: [
    Plot.areaY(twRealPDF, {
      x: "x", y: "pdf",
      fill: "#4682b4", fillOpacity: 0.12,
      curve: "natural"
    }),
    Plot.areaY(twRnPDF, {
      x: "x", y: "pdf",
      fill: "#ff7f0e", fillOpacity: 0.12,
      curve: "natural"
    }),
    Plot.line(twBothPDF, {
      x: "x", y: "pdf", stroke: "world",
      strokeWidth: 2.5, curve: "natural"
    }),
    Plot.ruleX([twERealST], { stroke: "#4682b4", strokeDash: [4, 4], strokeOpacity: 0.6 }),
    Plot.ruleX([twERnST], { stroke: "#ff7f0e", strokeDash: [4, 4], strokeOpacity: 0.6 })
  ]
})

html`<table class="table" style="width:100%;">
<thead><tr><th>Statistic (T = ${twT} years)</th><th>Real world (μ = ${(twMu * 100).toFixed(0)}%)</th><th>Risk-neutral (r = ${(twR * 100).toFixed(0)}%)</th></tr></thead>
<tbody>
<tr><td style="font-weight:500;">Expected return</td><td>${(twMu * 100).toFixed(1)}%</td><td>${(twR * 100).toFixed(1)}%</td></tr>
<tr><td style="font-weight:500;">Volatility</td><td>${(twSigma * 100).toFixed(1)}%</td><td>${(twSigma * 100).toFixed(1)}%</td></tr>
<tr><td style="font-weight:500;">E(S<sub>T</sub>)</td><td>$${fmtDollar(twERealST)}</td><td>$${fmtDollar(twERnST)}</td></tr>
<tr><td style="font-weight:500;">Median of S<sub>T</sub></td><td>$${fmtDollar(twMedianReal)}</td><td>$${fmtDollar(twMedianRn)}</td></tr>
<tr><td style="font-weight:500;">Std. dev. of ln S<sub>T</sub></td><td>${twSdLogReal.toFixed(4)}</td><td>${twSdLogRn.toFixed(4)}</td></tr>
</tbody></table>
<p style="margin-top:0.5rem; color:#666; font-size:0.85rem;">The volatility (std. dev. of ln S<sub>T</sub>) is identical in both worlds --- this is <strong>Girsanov's theorem</strong>.</p>`

Note

Girsanov’s theorem in practice

When moving from the risk-neutral world to the real world (or vice versa), only the expected growth rate changes. The volatility stays the same. This means a volatility estimated from historical data can be used directly in risk-neutral pricing, and vice versa. The theorem is what makes it practical to switch between the two worlds.

References

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