Itô's Lemma

How to take derivatives of functions of stochastic processes.

Brownian motion is too rough for ordinary calculus to work. Itô's Lemma is the chain rule for stochastic processes. For $X_t$ satisfying $dX_t = \mu\, dt + \sigma\, dW_t$ and a smooth function $f(t, x)$ :

df(t, X_t) = \left(\frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2 \frac{\partial^2 f}{\partial x^2}\right) dt + \sigma \frac{\partial f}{\partial x}\, dW_t

The extra term $\frac{1}{2}\sigma^2 f_{xx}$ is what makes stochastic calculus different from ordinary calculus. It comes from the fact that $(dW)^2 = dt$ — Brownian increments aren't negligible at second order.

Itô's Lemma is the workhorse of derivative pricing. Apply it to $f(S) = \log S$ where $S$ follows GBM, and you get the SDE for log-returns. Apply it to an option payoff and you get the Black-Scholes PDE. Without Itô, modern continuous-time finance doesn't function.