Itô's Lemma
Itô's lemma adds a second-order correction to the chain rule, and that correction is what turns stochastic dynamics into pricing PDEs.
Ordinary calculus differentiates functions of time; stochastic calculus must handle functions of random paths. Because Brownian motion has nonzero quadratic variation, a Taylor expansion picks up an extra second-order term, and Itô's lemma says that term survives: the drift of a composed function gains one half of its second derivative times the variance. That correction is why option prices obey a PDE rather than simple drift arithmetic, and it powers the replication argument used next. Every closed form in this track is one lemma application away from a stochastic differential equation, which is why the lemma, not the formula, is the real engine.