Itô's lemma
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
The chain rule of stochastic calculus. For X_t satisfying dX_t = μ_t,dt + σ_t,dW_t and f in C^1,2: df(t, X_t) = left((partial f)/(partial t) + μ_t (partial f)/(partial x) + 1/2σ_t^2 (partial^2 f)/(partial x^2)right)dt + σ_t (partial f)/(partial x),dW_t. The extra 1/2σ^2 partial_xx f term comes from (dW)^2 = dt (Brownian quadratic variation), absent from ordinary calculus. Applied to f = ln S under GBM yields the famous drift correction dln S = (μ - 1/2σ^2)dt + σ,dW.
Itô's lemma in plain English
In ordinary math, if you know how one thing changes, the chain rule tells you how something built from it changes. Itô's lemma is that same idea but for quantities that jitter randomly every instant — like a stock price that never sits still. The twist is that pure randomness adds an extra correction term that normal calculus doesn't have: because tiny random wiggles don't cancel out, their squared size actually adds up to real, predictable movement (written (dW)^2 = dt). A famous payoff is the growth of a stock: its typical compounded return isn't quite its average return but a bit lower, (μ - 1/2σ^2) — the more it jiggles (σ), the bigger that drag, which is why volatility quietly eats into long-run growth.
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