Itô's lemma: where the half-sigma-squared comes from
Every SDE derivation in a quant interview — Black-Scholes, GBM, CIR — uses this rule. The stochastic chain rule with one extra term: that $-\tfrac{1}{2}\sigma^2$ you see everywhere comes from here.
Itô's lemma is the chain rule for stochastic processes. It looks like the ordinary calculus chain rule plus one extra term — the Itô correction $\tfrac{1}{2}\sigma^2 f_{xx}\,dt$ — and that extra term is the source of every $-\tfrac{1}{2}\sigma^2$ you have ever seen in option pricing. The reason it appears is that Brownian motion has non-zero quadratic variation: even though $dW$ has mean zero, $(dW)^2$ has mean $dt$, so when you Taylor-expand a smooth function to second order the squared-Brownian piece refuses to vanish. This tutorial walks through the rule on its smallest non-trivial example, then derives the lognormal solution to geometric Brownian motion and the famous $\mu - \tfrac{1}{2}\sigma^2$ drift.
✓ Intro · expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine — being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
Reflection
The Itô correction depends only on $\sigma^2 f_{xx}$, not on the drift $\mu$. Why is the variance, not the drift, what creates the extra term, and what does that say about which information about a path is 'truly stochastic'?