Girsanov's Theorem

Girsanov's theorem is the machine that makes risk-neutral pricing work. It answers a precise question: can we tilt the probabilities so that a drifting process becomes driftless, a Brownian motion, without changing anything else? The answer is yes, and the recipe is explicit. It is how the real-world drift $\mu$ gets swapped for the risk-free rate $r$ while the volatility $\sigma$ stays put. Every no-arbitrage price is, at bottom, an expectation under a Girsanov-transformed measure.

The problem: removing a drift

Under the real-world measure $\mathbb{P}$, suppose a Brownian motion $W_t^{\mathbb{P}}$ drives an asset whose discounted price has a drift we want gone. Concretely, consider the process

$\tilde W_t = W_t^{\mathbb{P}} + \int_0^t \lambda_s\,ds,$

which is $W^{\mathbb{P}}$ plus a drift $\lambda$. Under $\mathbb{P}$, $\tilde W_t$ is clearly not a martingale, it has a systematic upward pull. Girsanov says: there is an equivalent measure $\mathbb{Q}$ under which $\tilde W_t$ is a standard Brownian motion. We do not change the paths; we change the probabilities assigned to them so that the average drift washes out.

The Radon-Nikodym derivative

The reweighting is defined by the Radon-Nikodym derivative $\frac{d\mathbb{Q}}{d\mathbb{P}}$, and Girsanov gives its exact form as a Doléans-Dade exponential martingale:

$Z_t = \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_t} = \exp\!\left(-\int_0^t \lambda_s\,dW_s^{\mathbb{P}} - \frac12\int_0^t \lambda_s^2\,ds\right).$

$Z_t$ is a positive $\mathbb{P}$-martingale with $\mathbb{E}^{\mathbb{P}}[Z_t] = 1$ (it is the exponential martingale of $-\int\lambda\,dW$, so by Itô it is driftless). It reweights outcomes: paths where $W$ went up (which produced the unwanted positive drift) are down-weighted by the $-\int\lambda\,dW$ term, exactly enough to neutralize the drift. The $-\tfrac12\int\lambda^2$ term is the Itô normalizer that keeps $\mathbb{E}[Z_t] = 1$ so $\mathbb{Q}$ is a genuine probability measure.

The statement

Girsanov's theorem. Let $\lambda_t$ be an adapted process satisfying the Novikov condition

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

which guarantees $Z_t$ is a true martingale (not just a local one). Define $\mathbb{Q}$ by $d\mathbb{Q}/d\mathbb{P} = Z_T$. Then under $\mathbb{Q}$,

$\tilde W_t = W_t^{\mathbb{P}} + \int_0^t \lambda_s\,ds$

is a standard Brownian motion. Equivalently, $dW_t^{\mathbb{P}} = d\tilde W_t - \lambda_t\,dt$: passing to $\mathbb{Q}$ adds $-\lambda\,dt$ to the drift of anything driven by $W^{\mathbb{P}}$.

Volatility is invariant, drift is not

The single most important structural fact: Girsanov changes only the drift, never the diffusion coefficient. Take $dX_t = \mu_t\,dt + \sigma_t\,dW_t^{\mathbb{P}}$ and substitute $dW^{\mathbb{P}} = d\tilde W - \lambda\,dt$:

$dX_t = (\mu_t - \sigma_t\lambda_t)\,dt + \sigma_t\,d\tilde W_t.$

The drift shifted by $-\sigma\lambda$; the $\sigma$ multiplying the Brownian term is untouched. This is why:

• Volatility is the same under $\mathbb{P}$ and $\mathbb{Q}$, so implied vol (a $\mathbb{Q}$-quantity) is directly comparable to realized vol (a $\mathbb{P}$-quantity), and the whole vol-trading enterprise makes sense.
• The real-world drift $\mu$ is unobservable from option prices, pricing lives under $\mathbb{Q}$, which erases $\mu$, so options tell you nothing about expected returns.

The engine of risk-neutral pricing

To turn $\mathbb{P}$ into the risk-neutral measure, choose $\lambda$ to make the discounted stock driftless. With $dS = \mu S\,dt + \sigma S\,dW^{\mathbb{P}}$, we need the new drift of $S$ to be $r$:

$\mu - \sigma\lambda = r \;\Longrightarrow\; \boxed{\,\lambda = \frac{\mu - r}{\sigma}.\,}$

This $\lambda$ is the market price of risk, the excess return per unit of volatility (the Sharpe ratio of the asset). Girsanov with exactly this $\lambda$ produces the measure $\mathbb{Q}$ under which every asset drifts at $r$ and discounted prices are martingales. The abstract fundamental theorem ("an equivalent martingale measure exists") is constructive here: Girsanov builds it, and the Radon-Nikodym density $Z_T = \exp(-\int\lambda\,dW - \tfrac12\int\lambda^2)$ is the pricing kernel that encodes the risk premium.

Worked example

A stock has $\mu = 12\%$, $\sigma = 20\%$, $r = 4\%$. The market price of risk is $\lambda = (0.12 - 0.04)/0.20 = 0.4$, its Sharpe ratio. Girsanov defines $\tilde W_t = W_t^{\mathbb{P}} + 0.4\,t$, and under the resulting $\mathbb{Q}$ the stock follows $dS = 0.04\,S\,dt + 0.20\,S\,d\tilde W$. The change of measure has stripped 8% of drift (the risk premium) out of the dynamics while leaving the 20% volatility exactly intact, precisely the ingredients that go into Black-Scholes.

What breaks in practice

• Novikov failures / non-uniqueness. If $\lambda$ grows too fast (e.g. certain stochastic-vol or non-linear models), $Z_t$ may be only a local martingale, a strict supermartingale, and the "measure change" fails or introduces bubbles. Checking Novikov (or Kazamaki) is not a formality.
• Incompleteness fixes $\lambda$ ambiguously. With more risk factors than tradeable assets (stochastic vol, jumps), $\lambda$ is not unique: there is a market price of volatility risk the market must set, and different $\lambda$'s give different (all arbitrage-free) $\mathbb{Q}$'s.
• Jumps need a different Girsanov. For jump processes the measure change reweights both the diffusion drift and the jump intensity/size distribution (Esscher transform); the pure-diffusion statement above does not cover them.
• $\mathbb{Q}\ne\mathbb{P}$, always. Practitioners sometimes slip and read $\mathbb{Q}$-drifts as forecasts. Girsanov's whole point is that they are not, only volatility survives the crossing.

In interviews

State the theorem in words: there is an equivalent measure under which a drifted Brownian motion becomes driftless, with Radon-Nikodym derivative the exponential martingale $Z_t = \exp(-\int\lambda\,dW - \tfrac12\int\lambda^2)$. The two facts they want: (1) the change of measure removes drift but preserves volatility, show $dX = (\mu-\sigma\lambda)dt + \sigma\,d\tilde W$; and (2) choosing $\lambda = (\mu-r)/\sigma$, the market price of risk, produces the risk-neutral measure, this is how drift becomes $r$. Mention the Novikov condition as what makes $Z$ a true martingale. A favorite: "why can't you infer expected stock returns from option prices?", because Girsanov erases $\mu$; options see only $\sigma$.