Itô's lemma

Itô's lemma is the chain rule of stochastic calculus, and the single most-used formula in derivatives pricing. It says: if X is a process driven by Brownian motion, then any smooth function f(X) has a differential that is X's differential plus a second-order correction proportional to the quadratic variation of X. This second-order term is the entire source of the Black-Scholes machinery.

Statement (one-dimensional)

Let X(t) satisfy dX = μ dt + σ dW (general Itô process). Let f(t, x) be twice continuously differentiable. Then:

df = (∂f/∂t + μ ∂f/∂x + (σ²/2) ∂²f/∂x²) dt + σ ∂f/∂x dW

Compared to ordinary calculus's df = (∂f/∂t + ∂f/∂x · dx/dt) dt for a deterministic x(t), the new term is (σ²/2) ∂²f/∂x² · dt — the Itô correction.

Heuristic derivation

Taylor-expand f(t + dt, x + dx) to second order:

df ≈ ∂f/∂t dt + ∂f/∂x dx + (1/2) ∂²f/∂x² (dx)² + ...

Now substitute dx = μ dt + σ dW. Compute (dx)² = (μ dt + σ dW)² = μ² (dt)² + 2μσ dt·dW + σ² (dW)². Using (dt)² ≈ 0, dt · dW ≈ 0, (dW)² = dt: (dx)² = σ² dt. Substituting:

df = (∂f/∂t + μ ∂f/∂x + (σ²/2) ∂²f/∂x²) dt + σ ∂f/∂x dW

This 'multiplication-table' approach — (dt)² = 0, dt · dW = 0, (dW)² = dt — is the working tool. Memorise it, apply it, and Itô's lemma derivations become routine.

Itô's lemma for GBM

Let S satisfy dS = μ S dt + σ S dW. Compute d(log S).

f(x) = log x.
∂f/∂x = 1/x.   ∂²f/∂x² = -1/x².
d(log S) = (μ S · 1/S + (σ² S²/2)(-1/S²)) dt + σ S (1/S) dW
         = (μ - σ²/2) dt + σ dW

log S has constant drift (μ - σ²/2) and constant diffusion σ. Integrating: log S(t) = log S(0) + (μ - σ²/2) t + σ W(t). Therefore S(t) = S(0) exp((μ - σ²/2) t + σ W(t)) — the classic GBM expression.

Itô's lemma in higher dimensions

For X(t) = (X_1, ..., X_n) with dX_i = μ_i dt + Σ_j σ_{ij} dW_j and f(t, X):

df = ∂f/∂t dt + Σ_i ∂f/∂x_i dX_i + (1/2) Σ_{i,j} (∂²f/∂x_i ∂x_j) (Σ_k σ_{ik} σ_{jk}) dt

The second-order Hessian-style term involves the covariation Σ_k σ_{ik}σ_{jk} = (σ σᵀ)_{ij}. For multi-asset derivatives (basket, exchange options) this is the multi-variate Itô that drives the pricing PDE.

Products and quotients

Itô product rule: d(XY) = X dY + Y dX + dX·dY. The cross-term dX·dY is the quadratic covariation, computed using the same multiplication table.

Itô quotient: d(X/Y) more cumbersome — usually do this via log. d(log(X/Y)) = d log X - d log Y, then exponentiate.

Why Itô's lemma matters

• Stock price dynamics derived from log-price BM with drift.
• Black-Scholes PDE derived in two lines via Itô on V(S, t).
• Forward price dynamics under change of measure (Module 6).
• Stochastic volatility models (Heston: vol itself follows an Itô process).
• Term-structure models — Vasicek and CIR (Module 10).