Girsanov's theorem
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Result that changes of equivalent probability measure transform Brownian motion by an added drift, leaving its 'Brownian-ness' under the new measure. If dW_t^mathbbP is Brownian under mathbbP and the Radon-Nikodým density is left.dfracdmathbbQdmathbbPright|_mathcalF_t = exp!left(-int_0^t θ_s,dW_s^mathbbP - 1/2int_0^t θ_s^2,dsright) (Doléans exponential), then dW_t^mathbbQ = dW_t^mathbbP + θ_t,dt is Brownian under mathbbQ. The engine of risk-neutral pricing: choosing θ = (μ - r)/σ kills the drift of discounted GBM and produces mathbbQ.
Girsanov's theorem in plain English
Girsanov's theorem is a mathematical trick for switching between two 'versions of reality' that agree on what's *possible* but disagree on how *likely* each outcome is. It says you can re-weight the probabilities of a random process and, remarkably, the randomness stays the same clean, well-behaved kind — the only thing that changes is the average direction it drifts. In finance this is the secret engine behind pricing options: by choosing the re-weighting just right (picking θ = (μ - r)/σ), you cancel out the tricky question of how much return investors demand for risk, leaving a simplified 'risk-neutral' world where prices are easy to compute. It's like watching a game through betting-adjusted odds instead of raw gut feelings — same players, same possible plays, but now the numbers line up so fair prices pop right out.
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