Girsanov's Theorem and the Change of Measure

Girsanov's theorem changes the drift of a process while leaving its quadratic variation, and therefore its volatility, completely unchanged.

Girsanov's theorem states that changing from one probability measure to an equivalent one changes the drift of a stochastic process while leaving its volatility, and more precisely its quadratic variation, completely unchanged. This single fact is why option pricing can shift from the real world measure to the risk neutral measure without altering the volatility that appears in the Black-Scholes-Merton formula.

Let W be a standard Brownian motion under a probability measure P, and let theta be a suitably well-behaved process. Girsanov's theorem says that under a new measure Q, defined through a change of measure built from the exponential martingale involving theta, the process defined as W plus the integral of theta over time is itself a standard Brownian motion under Q. Equivalently, a process with drift mu and volatility sigma under P has drift mu plus sigma times theta under Q, while sigma itself is unchanged.

The change of measure is carried out through the Radon-Nikodym derivative, a random variable that reweights probabilities from P to Q without changing which events are possible, only how likely they are. Two measures related this way are called equivalent, meaning they agree on which outcomes have zero probability.

Volatility is a property of the path, encoded in the quadratic variation, which is a pathwise quantity computed from the realized increments of the process regardless of which measure is used to assign probabilities to those paths. Changing the measure reweights the likelihood of different paths but does not alter what happens along any individual path, so quadratic variation, and therefore volatility, is invariant. Drift is different. It is a statement about the average direction of movement, and averages depend entirely on the probability weights assigned to each path, which is exactly what the measure change alters.

This distinction is the reason an interviewer will ask, immediately after the statement of the theorem, why the volatility parameter in Black-Scholes-Merton is the same whether you compute it under the real world measure or the risk neutral measure, while the drift parameter is not. The mechanics of that formula are covered fully in The Black-Scholes-Merton Option Pricing Model.