Girsanov's Theorem
Reweight the scenarios and a drift appears — but the volatility cannot be touched.
The idea
Theorem (Girsanov's theorem).
Let $W$ be a standard Brownian motion under $\mathbb{P}$ and fix a constant $\theta$. Reweight each path by $Z_{t} = \exp\!\left(-\theta W_{t} - \tfrac{1}{2}\theta^{2} t\right).$ Under the resulting measure $Q$, the process $W_{t} + \theta t$ is a standard Brownian motion.
Girsanov's theorem describes what changing the probability measure does to a Brownian motion. The question comes from risk-neutral pricing in finance, which values a claim as a discounted average under an invented probability — one under which every asset drifts at the risk-free rate — and leaves open both what justifies inventing a probability and what the invention is allowed to change.
Changing measure supplies the first answer. Keep every scenario that was possible and the value it produces; change only how much each scenario counts. Nothing is added and nothing is deleted, so the whole operation is described by one number per scenario: the factor by which its weight is multiplied.
Girsanov's theorem supplies the second answer, in continuous time. Reweighting the paths of a Brownian motion shifts its drift, and choosing $\theta$ places the drift wherever we need it. Equally important is what the theorem does not permit: the reweighted process is still a standard Brownian motion, with the same volatility as before. This asymmetry is what makes risk-neutral pricing work — the drift depends on which measure we price under, while the volatility is a property of the paths themselves, and no reweighting of paths can alter it.
Ways to work on it
- Walkthrough. Reweighting a binomial tree, the factor dQ/dP, and Girsanov turning real-world drift into r.
- Proof. Prove the drift change by tilting a random walk's coin probabilities and taking the limit.
- Practice. Compute reweighting factors on a tree, check their normalization, and find the market price of risk.
- Hardest. Price a digital option: the same event under P and under Q, and why only one of the two numbers is a price.
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