Girsanov's Theorem

Girsanov's theorem is the drift-ectomy. A stock's real-world dynamics carry an awkward drift \mu — every investor's private, unknowable guess about the future. Girsanov is the surgical tool that removes it: reweight the probabilities of the paths by exactly the right exponential factor, and a Brownian motion with drift becomes a Brownian motion without one. No path is bent, no wiggle is smoothed — the operation touches only the weights — and yet the world you wake up in after surgery is the risk-neutral one where everything grows at the riskless rate and options have prices.

The equivalent martingale measure was promised but never built. Girsanov's theorem is the construction. It answers a precise question: if I reweight my probabilities by a clever density, what happens to a Brownian motion? The astonishing answer is that changing measure changes the drift, and nothing else. The randomness — the Brownian wiggle — survives untouched; only the tilt of the path shifts.

Concretely, let W be a Brownian motion under \mathbb{P}, fix an adapted process \theta_t, and use as the density the exponential martingale

Z_T = \frac{d\mathbb{Q}}{d\mathbb{P}} = \exp\!\left(-\int_0^T \theta_s\, dW_s - \tfrac{1}{2}\int_0^T \theta_s^2\, ds\right).

Then under the new measure \mathbb{Q} the shifted process

\tilde{W}_t = W_t + \int_0^t \theta_s\, ds

is itself a Brownian motion. The drift \int_0^t \theta_s\,ds that \mathbb{Q} grafts on under \mathbb{P} is exactly the drift it strips off going the other way: under \mathbb{Q}, \tilde{W} is driftless and clean. In the workhorse case \theta is a constant and everything collapses to three memorable lines: Z_t = e^{-\theta W_t - \theta^2 t/2}, then d\mathbb{Q} = Z_T\,d\mathbb{P}, then \tilde{W}_t = W_t + \theta t is \mathbb{Q}-Brownian. That constant-\theta version is the one we now put to work, twice: first on a bare drifting Brownian motion, then on the stock itself.

The simplest surgery: killing a constant drift

Before the stock, operate on the simplest patient. Under \mathbb{P}, let

X_t = \mu t + \sigma W_t

— a Brownian motion scaled by \sigma and tilted by a constant drift \mu. We want a measure under which X is driftless.

Choose the dose. Take \theta = \mu/\sigma. Then the Girsanov-shifted process is \tilde{W}_t = W_t + \theta t = W_t + (\mu/\sigma)\,t, and substituting W_t = \tilde{W}_t - (\mu/\sigma)t into X:

X_t = \mu t + \sigma\!\left(\tilde{W}_t - \tfrac{\mu}{\sigma}t\right) = \mu t - \mu t + \sigma \tilde{W}_t = \sigma \tilde{W}_t.

Under \mathbb{Q}, X is a pure scaled Brownian motion — the drift has been excised completely. But this only counts if \tilde{W} genuinely is a Brownian motion under \mathbb{Q}. Check the axioms:

The key computation — complete the square. Under \mathbb{P}, W_T \sim N(0, T) with density \varphi(w) = \tfrac{1}{\sqrt{2\pi T}}e^{-w^2/2T}. Under \mathbb{Q}, every outcome is reweighted by Z_T = e^{-\theta w - \theta^2 T/2}, so the \mathbb{Q}-density of W_T is the product:

e^{-\theta w - \frac{\theta^2 T}{2}}\cdot\frac{1}{\sqrt{2\pi T}}\,e^{-\frac{w^2}{2T}} = \frac{1}{\sqrt{2\pi T}}\,\exp\!\left(-\frac{w^2 + 2\theta T w + \theta^2 T^2}{2T}\right) = \frac{1}{\sqrt{2\pi T}}\,e^{-\frac{(w + \theta T)^2}{2T}}.

That is exactly the N(-\theta T,\, T) density: under \mathbb{Q}, W_T is centred at -\theta T — so the shifted variable \tilde{W}_T = W_T + \theta T \sim N(0, T), precisely what a Brownian motion demands at time T. The exponential weight e^{-\theta w} boosts the outcomes where W came out low and shrinks the ones where it came out high — recentring the bell curve by pure reweighting, with the variance (the width of the bell) left exactly where it was.

The centrepiece: sending the stock risk-neutral, line by line

Now the operation the whole theory was built for: turn the real-world stock dynamics into the risk-neutral dynamics. Start with geometric Brownian motion under \mathbb{P}, where the stock drifts at its real rate \mu:

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

Step 1 — invert the Girsanov shift. Take \theta constant. The relation \tilde{W}_t = W_t + \theta t differentiates to d\tilde{W}_t = dW_t + \theta\,dt, so the \mathbb{P}-Brownian increment is

dW_t = d\tilde{W}_t - \theta\, dt.

Step 2 — substitute into the SDE. Replace every dW_t with d\tilde{W}_t - \theta\,dt:

dS_t = \mu S_t\, dt + \sigma S_t\,\big(d\tilde{W}_t - \theta\, dt\big).

Step 3 — expand and collect the dt terms. Multiply out the bracket and gather the two drift pieces:

dS_t = \mu S_t\, dt - \sigma\theta S_t\, dt + \sigma S_t\, d\tilde{W}_t = (\mu - \sigma\theta)\,S_t\, dt + \sigma S_t\, d\tilde{W}_t.

The diffusion coefficient \sigma S_t is unchanged — only the drift moved, exactly as Girsanov advertised. The new drift is a knob we can turn by our choice of \theta.

Step 4 — choose \theta to hit the riskless rate. We want the stock to drift at r under \mathbb{Q}, so demand \mu - \sigma\theta = r and solve for \theta:

\theta = \frac{\mu - r}{\sigma}.

This special \theta is the market price of risk — the excess return \mu - r per unit of volatility \sigma. One Greek letter, and it is the same one for every asset driven by this Brownian motion: in an arbitrage-free market all such assets must offer the same compensation per unit of risk, or you could sell the stingy one and buy the generous one.

Step 5 — verify the drift collapses to r. Substitute \theta = (\mu - r)/\sigma back into the drift coefficient and watch the \sigma cancel:

\mu - \sigma\theta = \mu - \sigma\cdot\frac{\mu - r}{\sigma} = \mu - (\mu - r) = r.

Step 6 — read off the risk-neutral dynamics. The drift is now exactly the riskless rate:

\boxed{\,dS_t = r S_t\, dt + \sigma S_t\, d\tilde{W}_t.\,}

Under \mathbb{Q} the stock earns r on average — the real-world premium \mu has been reweighted clean away — and \tilde{W} is the new Brownian motion. The discounted price \tilde{S}_t = e^{-rt}S_t is then a \mathbb{Q}-martingale, so \mathbb{Q} is the equivalent martingale measure we were missing. Black–Scholes pricing is now a matter of computing one expectation under \mathbb{Q} — and notice what has quietly happened to \mu: it appears nowhere in the boxed SDE. The one parameter nobody can agree on has been surgically removed from the pricing problem. That, more than anything, is why option prices are objective while stock forecasts are opinions.

Let W be a \mathbb{P}-Brownian motion and \theta_t an adapted process (with \int_0^T \theta_s^2\,ds < \infty). Then:

The quantity \theta = (\mu - r)/\sigma is the market price of risk (or Sharpe ratio): how much extra expected return the market pays for each extra unit of volatility carried. Girsanov reweights probability until that price is exactly zero — under \mathbb{Q} risk is not compensated, which is why everything earns the same r.

One technical hazard: the density Z_T is only a local martingale in general, and could secretly lose mass (\mathbb{E}[Z_T] < 1), which would break the change of measure. Novikov's condition rules this out — if

\mathbb{E}_{\mathbb{P}}\!\left[\exp\!\left(\tfrac12\int_0^T \theta_s^2\, ds\right)\right] < \infty,

then Z_T is a genuine martingale with \mathbb{E}[Z_T] = 1 and Girsanov goes through. For constant \theta the integral is just \tfrac12\theta^2 T, finite always — so the textbook Black–Scholes case is safe.

Finally, recognise the density itself. With X_t = -\int_0^t \theta_s\,dW_s, its quadratic variation is \langle X\rangle_t = \int_0^t \theta_s^2\,ds, and Z_t = \exp\!\big(X_t - \tfrac12\langle X\rangle_t\big) is precisely the stochastic exponential of -\int\theta\,dW — the very exponential martingale \exp(\sigma W_t - \tfrac12\sigma^2 t) we built with Itô's lemma, now with the integrand -\theta in the role of \sigma. The -\tfrac12\int\theta^2 ds in the exponent is the same drift compensator, doing the same job: keeping the mean pinned at 1 so that Z_T is a legitimate density.

Why the -\tfrac12\theta^2 t in the exponent?

It is not decoration — it is the normalisation that makes Z_t a legitimate density. A density must average to 1: total probability is conserved, only shuffled between outcomes. But the naive exponential tilt e^{-\theta W_t} does not average to 1. Since -\theta W_t \sim N(0, \theta^2 t), the tilt is lognormal, and the lognormal mean formula \mathbb{E}[e^X] = e^{m + s^2/2} (for X \sim N(m, s^2)) gives

\mathbb{E}_{\mathbb{P}}\!\left[e^{-\theta W_t}\right] = e^{0 + \theta^2 t/2} = e^{\theta^2 t/2} > 1.

The convexity of e^x means the exponential of a symmetric wiggle gains more on the upside than it loses on the downside — Jensen's inequality made concrete. The tilt manufactures extra mass at the rate e^{\theta^2 t/2}, so we must divide it back out:

Z_t = \frac{e^{-\theta W_t}}{\mathbb{E}[e^{-\theta W_t}]} = e^{-\theta W_t}\cdot e^{-\theta^2 t/2} = e^{-\theta W_t - \theta^2 t/2}, \qquad \mathbb{E}_{\mathbb{P}}[Z_t] = 1.

You have met this correction before, wearing a different hat: it is the same -\tfrac12\sigma^2 t that appears in the GBM solution S_t = S_0 e^{(\mu - \sigma^2/2)t + \sigma W_t}, and the same +\sigma^2/2 in the lognormal mean e^{\mu + \sigma^2/2}. Wherever a Gaussian is fed through an exponential, the half-variance term shows up to balance the books — here it pins \mathbb{E}[Z_t] at exactly 1 for every t, which is what makes Z a martingale rather than a mere positive process.

Sanity check: reweighting on two outcomes

The formula d\mathbb{Q} = Z_T\,d\mathbb{P} is used through one identity — for any payoff X,

\mathbb{E}_{\mathbb{Q}}[X] = \mathbb{E}_{\mathbb{P}}[Z_T\, X].

Strip away the continuum and check it on a toy world with just two outcomes, up and down, each with \mathbb{P}-probability \tfrac12. Choose the density values Z(\text{up}) = 1.6 and Z(\text{down}) = 0.4 — note \mathbb{E}_{\mathbb{P}}[Z] = \tfrac12(1.6) + \tfrac12(0.4) = 1, the discrete stand-in for the -\theta^2t/2 normalisation. The new probabilities are the old ones times the density:

\mathbb{Q}(\text{up}) = \tfrac12 \times 1.6 = 0.8, \qquad \mathbb{Q}(\text{down}) = \tfrac12 \times 0.4 = 0.2.

Now price a payoff worth 10 on up and 0 on down, both ways:

\mathbb{E}_{\mathbb{Q}}[X] = 0.8 \times 10 + 0.2 \times 0 = 8, \qquad \mathbb{E}_{\mathbb{P}}[Z X] = \tfrac12(1.6 \times 10) + \tfrac12(0.4 \times 0) = 8.

Same answer, as it must be — computing a \mathbb{Q}-expectation is computing a Z-weighted \mathbb{P}-expectation. Girsanov's continuous version does exactly this, outcome by outcome, with Z_T = e^{-\theta W_T - \theta^2 T/2} playing the role of the numbers 1.6 and 0.4: paths where W wandered low get weight above 1, paths where it soared get weight below, and the Gaussian recentres.

Igor Vladimirovich Girsanov (1934–1967) was one of the brightest of the postwar Moscow probabilists — a student in the circle of Dynkin at Moscow State University, working on diffusion processes and stochastic differential equations. The theorem that carries his name appeared in his 1960 paper on transforming stochastic processes by an absolutely continuous substitution of measures — building on earlier work of Cameron and Martin, who had handled the deterministic-shift case in the 1940s. To Girsanov it was a clean piece of pure mathematics about Wiener space. Finance is mentioned nowhere.

He never learned what it would become. A passionate mountaineer and ski instructor, Girsanov died in March 1967, aged just 32, in an avalanche while leading a ski expedition in the Sayan Mountains. Black–Scholes–Merton was six years away; Harrison, Kreps and Pliska's martingale-pricing framework — the work that made "apply Girsanov, price under \mathbb{Q}" the daily grammar of quantitative finance — a decade beyond that. Today his theorem runs silently inside the pricing engines of derivative markets measured in the hundreds of trillions of dollars notional: arguably the largest gap ever between what a mathematician lived to see and what his theorem went on to do.

One outcome, two drifts

Here is Girsanov made visible. A single Brownian outcome \omega drives both curves: the \mathbb{P} stock climbing at its real drift \mu, and the \mathbb{Q} stock climbing at the riskless r. They share every wiggle — only the average tilt differs, the difference being \sigma\theta. Drag \theta to retune the market price of risk and watch the \mathbb{Q} path's drift slide; Refresh for a fresh \omega.

Two things to notice as you play. First, at \theta = 0 the curves coincide exactly — no reweighting, no surgery. Second, however far you push \theta, the two curves never differ in texture: every kink, every micro-wiggle, every burst of volatility is identical, because quadratic variation is a path property that no change of measure can touch. What you are steering is not the path but the exponential weight Z hanging invisibly on it — this one \omega is more probable under one measure than the other, and the drift you see is the population-level echo of that reweighting across all the \omegas you'd get by pressing Refresh forever.