Itô's Lemma

Itô's lemma is the chain rule for functions of Brownian motion. Ordinary calculus says $df = f'\,dx$; Itô's lemma adds a second-order term that does not vanish, because Brownian motion has non-zero quadratic variation. It is the workhorse of derivatives pricing, the Black-Scholes PDE, the drift correction in geometric Brownian motion, and every change-of-variable in a model all come out of one application of this lemma.

The setup

Let $X_t$ be an Itô process, a diffusion driven by a single Brownian motion:

$dX_t = \mu_t\,dt + \sigma_t\,dW_t,$

where $\mu_t$ (the drift) and $\sigma_t$ (the diffusion coefficient) may depend on $t$ and $X_t$. Let $f(t, x)$ be twice continuously differentiable in $x$ and once in $t$. We want the dynamics of the transformed process $Y_t = f(t, X_t)$.

Statement

Itô's lemma states

$\boxed{\,df = \Big(f_t + \mu_t f_x + \tfrac12 \sigma_t^2 f_{xx}\Big)dt + \sigma_t f_x\,dW_t.\,}$

The first bracket is the drift of $Y$; the coefficient on $dW$ is its diffusion. Compared with the deterministic chain rule $df = f_t\,dt + f_x\,dx$, the new piece is the Itô term $\tfrac12 \sigma_t^2 f_{xx}\,dt$. It is a genuine first-order (in $dt$) contribution, not a rounding error.

Why the second-order term survives

The mechanism is the quadratic variation of $W$. Taylor-expand $f(t+dt,\, X_t + dX)$ to second order:

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

Now substitute $dX = \mu\,dt + \sigma\,dW$ and apply the Itô multiplication rules, which encode $(dW)^2 = dt$:

$dt\cdot dt = 0, \qquad dt\cdot dW = 0, \qquad dW\cdot dW = dt.$

Then

$(dX)^2 = \mu^2\,(dt)^2 + 2\mu\sigma\,dt\,dW + \sigma^2\,(dW)^2 = \sigma^2\,dt,$

because only the $(dW)^2 = dt$ term survives; every term with a $(dt)^2$ or $dt\,dW$ factor is of higher order and drops. Substituting back,

$df = f_t\,dt + f_x(\mu\,dt + \sigma\,dW) + \tfrac12 f_{xx}\,\sigma^2\,dt,$

which is precisely the boxed formula after collecting $dt$ and $dW$ terms. The rigorous version replaces this heuristic with an $L^2$ limit of Riemann sums, but the algebra is identical, the key input is that $\sum(\Delta W_i)^2 \to \int \sigma^2\,dt$ rather than to zero.

The multidimensional and product forms

For two Itô processes driven by correlated Brownian motions with $dW^{(1)}dW^{(2)} = \rho\,dt$, the cross term gives the Itô product rule:

$d(X_t Y_t) = X_t\,dY_t + Y_t\,dX_t + dX_t\,dY_t, \qquad dX_t\,dY_t = \rho\,\sigma^X_t\sigma^Y_t\,dt.$

The extra covariation term $dX\,dY$ is what distinguishes it from the ordinary product rule and is essential in stochastic-volatility and multi-asset models.

Application: geometric Brownian motion

Take the stock SDE $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$ (so the "$\mu$" and "$\sigma$" of the lemma are $\mu S$ and $\sigma S$) and apply Itô to $f(S) = \ln S$. Here $f_S = 1/S$, $f_{SS} = -1/S^2$, $f_t = 0$:

$d(\ln S_t) = \frac{1}{S}\,dS_t - \frac12 \frac{1}{S^2}(dS_t)^2 = \Big(\mu - \tfrac12\sigma^2\Big)dt + \sigma\,dW_t,$

using $(dS)^2 = \sigma^2 S^2\,dt$. This is a driftless-plus-constant SDE for $\ln S$, so it integrates directly:

$\ln S_t = \ln S_0 + \Big(\mu - \tfrac12\sigma^2\Big)t + \sigma W_t \;\Longrightarrow\; S_t = S_0\exp\!\Big(\big(\mu - \tfrac12\sigma^2\big)t + \sigma W_t\Big).$

Two things to notice. First, $S_t$ is lognormal, guaranteed positive, the fix that makes GBM a viable price model. Second, the $-\tfrac12\sigma^2$ correction on the drift is pure Itô: the log-return has drift $\mu - \tfrac12\sigma^2$ even though the arithmetic return has drift $\mu$. This gap, the difference between $\mathbb{E}[\ln S]$ and $\ln \mathbb{E}[S]$, is Jensen's inequality made dynamic, and it is exactly the term that appears in $d_2$ of the Black-Scholes formula.

Worked example: expected value of the stock

Using the solution above, $\mathbb{E}[S_t] = S_0 e^{(\mu - \frac12\sigma^2)t}\,\mathbb{E}[e^{\sigma W_t}]$. Since $W_t \sim \mathcal{N}(0,t)$, the moment-generating function gives $\mathbb{E}[e^{\sigma W_t}] = e^{\frac12\sigma^2 t}$, so

$\mathbb{E}[S_t] = S_0 e^{(\mu - \frac12\sigma^2)t}\,e^{\frac12\sigma^2 t} = S_0 e^{\mu t}.$

The two $\tfrac12\sigma^2 t$ terms cancel exactly: the expected price grows at $\mu$, even though the expected log-price grows only at $\mu - \tfrac12\sigma^2$. Getting this cancellation right is the essence of the Itô drift correction.

What breaks in practice

• Non-differentiable payoffs. Itô requires $f \in C^{2}$. Option payoffs like $(S-K)^+$ have a kink, so the lemma applies only away from the strike; at the kink one needs the Itô-Tanaka formula and local time. This is why gamma is a delta function at expiry for a digital.
• Jumps. If $X$ has jumps, the pure-diffusion Itô lemma is wrong, you need the Itô formula for semimartingales with a jump-compensation term. Markets gap, so this matters.
• Estimating $\sigma$. The lemma is exact given the model, but the whole edifice rests on knowing the diffusion coefficient. In practice $\sigma$ is uncertain and time-varying, which is what implied vol and stochastic vol confront.

In interviews

The canonical ask is "state and sketch Itô's lemma, then apply it to $\ln S$ for geometric Brownian motion." Nail the drift correction: $d\ln S = (\mu - \tfrac12\sigma^2)dt + \sigma\,dW$, and be ready to explain that the $-\tfrac12\sigma^2$ comes from the $\tfrac12 f_{xx}\sigma^2$ Itô term with $f = \ln$, $f_{xx} = -1/S^2$. A frequent trap: "$\mathbb{E}[S_t]$, is it $S_0 e^{\mu t}$ or $S_0 e^{(\mu-\frac12\sigma^2)t}$?" The answer is $S_0 e^{\mu t}$ for the price and $e^{(\mu-\frac12\sigma^2)t}$ for the median. Interviewers also probe whether you know why $(dW)^2 = dt$, the quadratic variation argument, because a candidate who treats it as a memorized rule cannot generalize to the multi-factor case.