Financial Mathematics · Lecture 8 of 12

Itô's Lemma: The Chain Rule for Randomness

Taylor-expand a function of Brownian motion, apply (dW)² = dt, and an extra term appears. The single most important formula in quantitative finance, derived and used.

November 15, 2026By Diganta Bhattacharya5 min read

Derivation in four lines

Let $dX=a\,dt+b\,dW$ and let $f(t,x)$ be smooth. Taylor-expand (Lecture 4):

$$df=f_t\,dt+f_x\,dX+\tfrac12f_{xx}(dX)^2+f_{tx}\,dt\,dX+\tfrac12 f_{tt}(dt)^2+\cdots$$

Now apply the Brownian multiplication table from Lecture 6: $(dW)^2=dt$, $dW\,dt=0$, $(dt)^2=0$. So $(dX)^2=b^2(dW)^2=b^2\,dt$ and every other second-order term vanishes:

Itô's lemma$$df(t,X_t)=\Big(f_t+a\,f_x+\tfrac12b^2f_{xx}\Big)dt+b\,f_x\,dW_t.$$

Compared with the ordinary chain rule there is one extra term, $\tfrac12b^2f_{xx}\,dt$. It exists because Brownian motion wiggles so much that its squared increments add up to something of size $dt$.

The last line of the figure's output is a classic example. Ordinary calculus would say $\int_0^1W\,dW=\tfrac12W_1^2$. Itô's lemma applied to $f(x)=x^2$ gives $d(W^2)=2W\,dW+dt$, so

$$\int_0^1 W\,dW=\tfrac12W_1^2-\tfrac12,$$

and the simulated left-point sums agree.

Solving geometric Brownian motion

Take $f(S)=\ln S$ with $dS=\mu S\,dt+\sigma S\,dW$. Then $f_S=1/S$, $f_{SS}=-1/S^2$, $b=\sigma S$:

$$d\ln S=\Big(\mu S\cdot\frac1S+\frac12\sigma^2S^2\cdot\Big(-\frac1{S^2}\Big)\Big)dt+\sigma S\cdot\frac1S\,dW=\Big(\mu-\frac{\sigma^2}{2}\Big)dt+\sigma\,dW.$$

The right-hand side has constant coefficients, so integrate directly:

$$S_T=S_0\exp\!\Big(\big(\mu-\tfrac12\sigma^2\big)T+\sigma W_T\Big).$$

This is the lognormal law from Lecture 2, now derived. The −σ²/2 that looked like a convention is the Itô correction term.

Itô for products and several variables

For two Itô processes the product rule gains a cross term:

$$d(XY)=X\,dY+Y\,dX+dX\,dY,\qquad dW^{(1)}dW^{(2)}=\rho\,dt.$$

Example: discounting. With $B_t=e^{rt}$, $d(e^{-rt}S_t)=e^{-rt}(dS_t-rS_t\,dt)=e^{-rt}S_t\big((\mu-r)dt+\sigma\,dW_t\big)$. The discounted stock has drift $\mu-r$. Lecture 9 changes measure to kill exactly this drift.

Itô applied to an option

Let $V(t,S)$ be the value of an option. Itô's lemma with $a=\mu S$, $b=\sigma S$:

$$dV=\Big(V_t+\mu SV_S+\tfrac12\sigma^2S^2V_{SS}\Big)dt+\sigma SV_S\,dW.$$

The random part is $\sigma S V_S\,dW$, and the stock's random part is $\sigma S\,dW$. So holding $V_S$ shares against a short option cancels the randomness exactly over each instant. That is delta hedging, and it gives the Black–Scholes PDE in Lecture 11.

Common mistakeForgetting the ½b²f_xx term. A quick check: apply the lemma to $f=W^2$. If you get $d(W^2)=2W\,dW$, the expectation would be zero, but $\mathbb E W_t^2=t$. The missing $dt$ is the Itô term.

Key takeaways

Itô's lemma: df = (f_t + a f_x + ½ b² f_xx) dt + b f_x dW, which is Taylor plus (dW)² = dt.
Applied to ln S it solves GBM: S_T = S₀ exp((μ − σ²/2)T + σW_T).
Applied to V(t, S), the dW term is σS V_S dW, so holding V_S shares cancels risk. That is delta hedging.

Exercises

Compute d(W_t³) and use it to find E[W_t³] and E[W_t⁴] (hint: take expectations and integrate).
If dX = κ(θ − X)dt + σdW, apply Itô to e^{κt}X_t and solve the OU process.
Let Y = S² with S GBM. Show Y is also GBM and find its drift and volatility.