Change of Measure, Girsanov, and Risk-Neutral Pricing — Overview

Every derivative pricing argument bottoms out in a single question: under which probability measure should we compute expectations? In the real world — the physical measure $\mathbb{P}$ — investors require a risk premium and assets earn more than the risk-free rate. But pricing formulas become dramatically simpler under the risk-neutral measure $\mathbb{Q}$, where all assets grow at the risk-free rate $r$ and fair value is just $e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[V_T]$.

The bridge between $\mathbb{P}$ and $\mathbb{Q}$ is Girsanov's theorem: under a change of probability measure specified by an exponential martingale (the Radon-Nikodym derivative $d\mathbb{Q}/d\mathbb{P}$), a $\mathbb{P}$-Brownian motion with drift becomes a $\mathbb{Q}$-Brownian motion with no drift. This single fact justifies the entire risk-neutral pricing framework: it says changing the drift of a GBM (from $\mu$ to $r$) is the same as changing the probability measure.