What Is Girsanov's Theorem? Change the Drift, Not the Vol

Girsanov’s theorem is usually presented as a technical result about Brownian motion under a change of measure, and then applied without comment. That skips the part worth understanding.

Here is the whole thing in one sentence: **you can change the drift of a diffusion to anything you like, and you cannot touch the volatility at all.**

That asymmetry is not a limitation of the theorem. It is the reason risk-neutral pricing works, the reason option prices don’t depend on expected returns, and the reason volatility is the only parameter you can actually calibrate from market prices.

The statement

Take a Brownian motion $W_t$ under a measure $\mathbb{P}$ , and pick any adapted process $\theta_t$ . Define

$Z_T = \exp\left(-\int_0^T \theta_t\,dW_t - \frac{1}{2}\int_0^T \theta_t^2\,dt\right)$

Then $Z_T$ is a valid Radon-Nikodym derivative, and under the measure $\mathbb{Q}$ it defines,

$W_t^{\mathbb{Q}} = W_t + \int_0^t \theta_s\,ds$

is a Brownian motion.

Read that last line carefully. The process $W_t^{\mathbb{Q}}$ has a drift under $\mathbb{P}$ — it is $W_t$ plus a drift term. Under $\mathbb{Q}$ it has none. **The same process, drifting under one measure and driftless under another.** Nothing about the paths changed; only how much weight each one carries.

What it does to a stock

Under $\mathbb{P}$ , geometric Brownian motion with real-world drift:

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$

Choose $\theta = \dfrac{\mu - r}{\sigma}$ , the **market price of risk** — excess return per unit of volatility. Substituting $dW_t = dW_t^{\mathbb{Q}} - \theta\,dt$ :

$dS_t = \mu S_t\,dt + \sigma S_t\left(dW_t^{\mathbb{Q}} - \frac{\mu-r}{\sigma}dt\right) = r S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}$

The $\mu$ cancels. Same stock, same volatility, drift now $r$ .

Simulating 200,000 paths under $\mathbb{P}$ with $\mu = 12\%$ , $r = 4\%$ , $\sigma = 20\%$ , then reweighting rather than resimulating:

	Simulated	Theory
$\mathbb{E}^{\mathbb{P}}[S_T]$	112.7370	$S_0e^{\mu T} = 112.7497$
$\mathbb{E}^{\mathbb{Q}}[S_T]$	104.0945	$S_0e^{rT} = 104.0811$

Why the volatility survives

This is the part that is usually asserted and rarely explained, and it has a clean reason.

Quadratic variation is a **pathwise** quantity. You compute $\sum (\Delta \log S)^2$ from the path itself — no probabilities enter the calculation anywhere. Each path already carries its own quadratic variation before you have decided which measure to use.

A change of measure reweights paths. It cannot alter a number that each path already holds.

Measuring it on those same 200,000 paths:

	Value
$[\log S]_T$ per path, mean	0.040003
$\sigma^2 T$	0.040000
$\mathbb{E}^{\mathbb{P}}\big[[\log S]_T\big]$	0.040003
$\mathbb{E}^{\mathbb{Q}}\big[[\log S]_T\big]$	0.040010

A relative difference of $1.8 \times 10^{-4}$ , which is Monte Carlo noise. The drift moved by 800 basis points and the volatility did not move at all.

So the division of labour is sharp: **drift is a property of the measure, volatility is a property of the path.** One is a modelling choice, the other is a fact about the trajectory.

Three consequences that follow immediately

**Option prices don’t depend on expected returns.** Everyone learns this and few people are told why. It is because $\mu$ is entirely absorbed into $\theta$ , which is entirely absorbed into the weights. By the time you have changed measure, $\mu$ has cancelled algebraically. Your view on whether the stock is going up is irrelevant to the price of the option — not because the market disagrees with you, but because the drift is not in the formula.

**Volatility is the only thing you can calibrate.** Since $\sigma$ is the same under both measures, an implied volatility extracted from option prices is directly comparable with a realised volatility estimated from history. That comparison is meaningful. There is no equivalent for drift, because the $\mathbb{P}$ -drift and the $\mathbb{Q}$ -drift are different objects by construction.

**Stochastic volatility models change fewer parameters than people expect.** In Heston, the vol-of-vol $\xi$ and the correlation $\rho$ are diffusion coefficients, so Girsanov cannot touch them — they are identical under $\mathbb{P}$ and $\mathbb{Q}$ . Only the variance drift shifts, and under Heston’s own specification that leaves exactly one free parameter. If a calibration produces a $\mathbb{Q}$ -measure $\rho$ far from the historical one, that is not a market phenomenon, it is a misspecification somewhere.

The condition, and what it is protecting you from

Girsanov needs $Z_T$ to be a true martingale with $\mathbb{E}^{\mathbb{P}}[Z_T] = 1$ , and that is not automatic. The usual sufficient condition is **Novikov**:

$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\frac{1}{2}\int_0^T \theta_t^2\,dt\right)\right] < \infty$

With constant $\theta$ this always holds, which is why textbook examples never run into trouble. With stochastic $\theta$ it can fail, and when it does, $Z_T$ is only a local martingale, $\mathbb{E}[Z_T] < 1$ , and $\mathbb{Q}$ is not a probability measure at all. Everything downstream is then meaningless while looking perfectly normal.

And the failure that arrives long before Novikov does

There is a practical limit that bites well before the mathematical one, and it is worth seeing.

Girsanov says you can shift the drift to anything. That is true and it is a statement about existence. Numerically, the weights spread out as $\theta$ grows, and past a point the reweighting stops working. Same 200,000 paths, pushing the target drift further and further:

Target drift	$\theta$	$\mathbb{E}[Z]$	Largest weight	Weight in top 1% of paths
6%	0.10	1.0001	2×	1.3%
12%	0.40	1.0003	5×	2.7%
25%	1.05	1.0015	54×	10.1%
50%	2.30	0.9935	1,475×	48.3%
100%	4.80	0.3526	10,165×	98.1%

At a target drift of 100%, the weights sum to 0.35 instead of 1, and one percent of the paths carry ninety-eight percent of the total weight. The measure exists — Novikov is satisfied, constant $\theta$ — but your effective sample size has collapsed to a few hundred paths out of two hundred thousand. The estimator is worthless.

This is why long-dated exposure engines simulate directly under the measure they need instead of simulating once and reweighting. The change of measure is exact in theory and unusable past a certain distance. If you ever meet a Monte Carlo with wild variance and no obvious bug, an over-stretched Girsanov transformation is a good first place to look.

Code

import numpy as np

S0, mu, r, sig, T, N, M = 100., 0.12, 0.04, 0.20, 1.0, 2520, 200_000
dt = T/N
rng = np.random.default_rng(4)

logS = np.full(M, np.log(S0)); W = np.zeros(M); QV = np.zeros(M)
for _ in range(N):
    dW = np.sqrt(dt)*rng.standard_normal(M)
    dlog = (mu - 0.5*sig**2)*dt + sig*dW
    logS += dlog; W += dW; QV += dlog**2       # QV built from the path alone
ST = np.exp(logS)

theta = (mu - r)/sig
Z = np.exp(-theta*W - 0.5*theta**2*T)          # Girsanov density

print(f"E[Z]      = {Z.mean():.5f}")           # must be 1
print(f"E^P[S_T]  = {ST.mean():.4f}")          # drifts at mu
print(f"E^Q[S_T]  = {(ST*Z).mean():.4f}")      # drifts at r
print(f"E^P[QV]   = {QV.mean():.6f}")          # sigma^2 T
print(f"E^Q[QV]   = {(QV*Z).mean():.6f}")      # the same

Two checks worth building into any implementation. The density must average to one — if it doesn’t, the $-\frac{1}{2}\int\theta^2$ term is wrong or missing. And the quadratic variation must be identical under both measures; if your reweighting changes it, you have altered the volatility, which Girsanov cannot do and your code should not either.

The takeaway

Girsanov’s theorem lets you set the drift to whatever you need and leaves the volatility exactly where it was. The drift lives in the measure and can be reweighted away; the volatility lives in the path and cannot.

That single asymmetry explains why $\mu$ vanishes from option prices, why implied and realised volatility are comparable quantities while $\mathbb{P}$ – and $\mathbb{Q}$ -drifts are not, and why the whole apparatus of risk-neutral pricing is possible in the first place.

This completes a chain worth reading in order: what a martingale is, the Radon-Nikodym derivative that reweights the paths, and the risk-neutral measure that Girsanov constructs.

The measure change, the martingale machinery underneath it and the numerical checks that catch a broken implementation are worked through from the ground up in the course modules.

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What Is Girsanov’s Theorem?