Itô's Lemma
Itô’s lemma is the chain rule for stochastic calculus: a smooth function of an Itô process picks up a second-order dt term from (dW)² = dt.
Definition
If dX = μ dt + σ dW and f is C², then df = (f_t + μ f_x + ½ σ² f_xx) dt + σ f_x dW. The extra ½ σ² f_xx is why option PDEs have a gamma term and why naive calculus mis-prices convex claims.
Why it matters
Every derivation of Black–Scholes, every delta-hedge P&L identity, goes through Itô. If you treat dS² as zero, you will not see the gamma P&L.
Case
A long option that is delta-hedged earns realized variance times gamma. That is Itô, not a trading anecdote.
How to read it
State the process (diffusion vs jump). Jumps need an extra compensator term — ‘Itô’ is not a license to ignore them.
Key papers
Landmark references with real publisher or arXiv links — not token-overlap noise.
- On Stochastic Differential EquationsItô · 1951 · Memoirs of the American Mathematical SocietyItô calculus.