Chapter 3 — Itô's Formula

Guiding question

If $f$ is a smooth function and $W_t$ is Brownian motion, how does $f(W_t)$ change over time? The ordinary chain rule says $d(f\circ W) = f'\,dW$. Why is this wrong, and what replaces it?

Itô's formula is the chain rule of stochastic calculus. It is the single most important computational tool in the entire theory. Every SDE that can be solved explicitly is solved via Itô's formula. Every partial differential equation that arises in option pricing is derived via Itô's formula.

The reason the ordinary chain rule fails is the quadratic variation of Brownian motion: $(dW_t)^2 = dt$. This creates an additional drift term proportional to the second derivative $f''$.

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3.1 Why the ordinary chain rule fails

For a deterministic differentiable function $g:[0,T]\to\mathbb R$ and a smooth function $f:\mathbb R\to\mathbb R$, the chain rule gives

$$f(g(T)) - f(g(0)) = \int_0^T f'(g(t))\,g'(t)\,dt.$$

In infinitesimal notation: $d(f\circ g) = f'(g)\,dg$. The second derivative plays no role because $(dg)^2\sim (g'(t))^2(dt)^2 = O(dt^2)$ is negligible.

For stochastic processes, increments of order $\sqrt{dt}$ change everything. Write the second-order Taylor expansion:

$$f(W_{t+h}) - f(W_t) = f'(W_t)(W_{t+h}-W_t) + \tfrac12 f''(W_t)(W_{t+h}-W_t)^2 + O(|W_{t+h}-W_t|^3).$$

Summing over a partition:

$$f(W_T)-f(W_0) \approx \sum_k f'(W_{t_k})\Delta W_k + \tfrac12\sum_k f''(W_{t_k})(\Delta W_k)^2.$$

As the partition refines, the first sum converges to the Itô integral $\int_0^T f'(W_t)\,dW_t$. The second sum involves $(\Delta W_k)^2\approx\Delta t_k$, so it converges to $\frac12\int_0^T f''(W_t)\,dt$.

The third-order remainder is negligible because $(\Delta W_k)^3 \sim (\Delta t_k)^{3/2}$ and $\sum(\Delta t_k)^{3/2} \le |\Pi|^{1/2}\cdot T\to 0$.

Why the second derivative survives

In classical calculus, the chain rule is first-order: only $f'$ appears because $(dg)^2 = O(dt^2)\to 0$.

For Brownian motion, $(dW)^2 = O(dt)$, which is the same order as $dt$ itself. So the second derivative is not negligible: it contributes a term of order $dt$, exactly the same magnitude as the drift.

This is the geometric content of Itô's formula: Brownian paths are so rough that second derivatives must be included in the linearization.

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3.2 Itô's formula: one-dimensional, time-independent

Itô's formula (basic version)

Let $f:\mathbb R\to\mathbb R$ be twice continuously differentiable. Then

$$f(W_T) - f(W_0) = \int_0^T f'(W_t)\,dW_t + \frac12\int_0^T f''(W_t)\,dt.$$

In differential notation:

$$df(W_t) = f'(W_t)\,dW_t + \tfrac12 f''(W_t)\,dt.$$

Proof via partition and Taylor expansion

Let $\Pi = \{0=t_0<t_1<\cdots<t_n=T\}$. By the Taylor expansion to second order:

$$f(W_{t_{k+1}})-f(W_{t_k}) = f'(W_{t_k})\Delta W_k + \tfrac12 f''(W_{t_k})(\Delta W_k)^2 + R_k,$$

where $R_k = \frac12[f''(\xi_k)-f''(W_{t_k})](\Delta W_k)^2$ for some $\xi_k$ between $W_{t_k}$ and $W_{t_{k+1}}$.

Step 1. Summing and telescoping:

$$f(W_T)-f(W_0) = \sum_k f'(W_{t_k})\Delta W_k + \tfrac12\sum_k f''(W_{t_k})(\Delta W_k)^2 + \sum_k R_k.$$

Step 2. The first sum converges in $L^2$ to $\int_0^T f'(W_t)\,dW_t$ by definition of the Itô integral.

Step 3. For the second sum, write:

$$\sum_k f''(W_{t_k})(\Delta W_k)^2 = \sum_k f''(W_{t_k})\Delta t_k + \sum_k f''(W_{t_k})[(\Delta W_k)^2-\Delta t_k].$$

The first part converges to $\int_0^T f''(W_t)\,dt$. For the second part, its $L^2$ norm vanishes as $|\Pi|\to 0$ (the increments $(\Delta W_k)^2-\Delta t_k$ are centered and orthogonal, so their sum's variance is $2\sum_k(\Delta t_k)^2\le 2|\Pi|\cdot T\to 0$).

Step 4. The remainder terms $\sum_k R_k\to 0$ in $L^2$ by continuity of $f''$ and the fact that $\sum_k(\Delta W_k)^2\to T$.

Combining, the result holds in $L^2$.

Itô formula: f(W_t) decomposes into martingale + drift correction

Function:

Itô: d(W²) = 2W dW + dt. Taking expectations: E[W_t²] = t. Without the correction term dt, classical calculus would predict E[W_t²] = 0 (wrong).

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3.3 The symbolic rule $(dW_t)^2 = dt$

Itô's formula is efficiently remembered via the multiplication table:

$\times$	$dW_t$	$dt$
$dW_t$	$dt$	$0$
$dt$	$0$	$0$

This says: $(dW_t)^2 = dt$, $dW_t\cdot dt = 0$, $(dt)^2 = 0$. The table encodes the order-of-magnitude argument:

$$dW_t\sim\sqrt{dt},\quad (dW_t)^2\sim dt,\quad dW_t\cdot dt\sim dt^{3/2}\to 0,\quad (dt)^2\sim dt^2\to 0.$$

Using this table, the chain rule $df = f'dW + \frac12 f''(dW)^2 = f'dW + \frac12 f''dt$ follows immediately from the formal second-order Taylor expansion.

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3.4 General Itô process

The formula extends to any Itô process, not just Brownian motion.

Itô's formula for Itô processes

Let $X_t$ satisfy

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

where $b$ and $\sigma$ are adapted and satisfy integrability conditions. Let $f:\mathbb R\to\mathbb R$ be $C^2$. Then

$$df(X_t) = f'(X_t)\,dX_t + \tfrac12 f''(X_t)\,d[X]_t,$$

where $d[X]_t = \sigma_t^2\,dt$. Expanding:

$$df(X_t) = f'(X_t)b_t\,dt + f'(X_t)\sigma_t\,dW_t + \tfrac12 f''(X_t)\sigma_t^2\,dt.$$

The proof follows the same structure, using $(dX_t)^2 = \sigma_t^2\,dt$ from the Itô multiplication table.

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3.5 Time-dependent Itô formula

Itô's formula with time dependence

Let $f = f(t,x)$ be $C^{1,2}$ (once in $t$, twice in $x$). Then

$$df(t,X_t) = \partial_t f\,dt + \partial_x f\,dX_t + \tfrac12\partial_{xx}f\,(dX_t)^2,$$

which expands to

$$df(t,X_t) = \left[\partial_t f + b_t\,\partial_x f + \tfrac12\sigma_t^2\,\partial_{xx}f\right]dt + \sigma_t\,\partial_x f\,dW_t.$$

Three sources of change

The function $f(t,X_t)$ changes due to three effects:

1. Time drift $\partial_t f\,dt$: direct time-dependence of $f$.
2. First-order $X$ term $\partial_x f\cdot dX_t$: classical chain rule contribution.
3. Curvature correction $\frac12\partial_{xx}f\cdot\sigma^2\,dt$: Brownian quadratic variation creates an extra drift proportional to the second derivative. This is the purely stochastic term.

In the language of differential geometry, the second term is the gradient dotted with the velocity, and the third term is a Laplacian-like correction from the rough geometry of Brownian paths.

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3.6 Key examples

Example 1: $d(W_t^2)$

Take $f(x) = x^2$, so $f' = 2x$, $f'' = 2$:

$$d(W_t^2) = 2W_t\,dW_t + \tfrac12\cdot 2\,dt = 2W_t\,dW_t + dt.$$

Integrating:

$$W_T^2 = 2\int_0^T W_t\,dW_t + T.$$

This recovers the computation of Section 2.10 from a different angle.

Example 2: $d(e^{W_t})$

Take $f(x) = e^x$, so $f'=f''=e^x$:

$$d(e^{W_t}) = e^{W_t}\,dW_t + \tfrac12 e^{W_t}\,dt.$$

Taking expectations: $\frac{d}{dt}\mathbb E[e^{W_t}] = \tfrac12\mathbb E[e^{W_t}]$, which gives $\mathbb E[e^{W_t}] = e^{t/2}$. This is the standard moment generating function of $N(0,t)$.

Example 3: $d(e^{\theta W_t - \theta^2 t/2})$ — the exponential martingale

Take $f(t,x) = e^{\theta x - \theta^2 t/2}$. Then

$$\partial_t f = -\tfrac12\theta^2 f,\quad \partial_x f = \theta f,\quad \partial_{xx}f = \theta^2 f.$$

Itô's formula gives

$$df = \left(-\tfrac12\theta^2 f + \tfrac12\theta^2 f\right)dt + \theta f\,dW_t = \theta f\,dW_t.$$

Since the drift vanishes, $\mathcal E_t(\theta) = e^{\theta W_t - \theta^2 t/2}$ is a local martingale (and under integrability conditions, a true martingale). This is the Doléans-Dade exponential or stochastic exponential, central to Girsanov's theorem.

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3.7 Geometric Brownian Motion

The most important SDE in finance is:

$$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t, \quad S_0 > 0.$$

Strategy: apply Itô's formula to $\log S_t$.

Let $f(x) = \log x$ (for $x > 0$). Then $f'(x) = 1/x$, $f''(x) = -1/x^2$. Itô's formula gives:

$$d(\log S_t) = \frac{1}{S_t}\,dS_t + \tfrac12\left(-\frac{1}{S_t^2}\right)(dS_t)^2.$$

Now $(dS_t)^2 = \sigma^2 S_t^2\,dt$, so:

$$d(\log S_t) = \frac{\mu S_t\,dt + \sigma S_t\,dW_t}{S_t} - \tfrac12\sigma^2\,dt = \left(\mu - \tfrac12\sigma^2\right)dt + \sigma\,dW_t.$$

The right side is a Brownian motion with constant coefficients. Integrating:

$$\log S_t = \log S_0 + \left(\mu - \tfrac12\sigma^2\right)t + \sigma W_t,$$

so

$$\boxed{S_t = S_0\exp\!\left[\left(\mu - \tfrac12\sigma^2\right)t + \sigma W_t\right].}$$

Geometric Brownian Motion: S_t = S₀ exp((μ − σ²/2)t + σW_t)

μ = 0.08σ = 0.25Log scale

GBM log-returns are normal: log(S_t/S_0) ~ N((μ−σ²/2)t, σ²t). Notice that higher σ widens the band but also drags the median below the mean — Jensen's inequality in action.

Log-normal returns and the volatility drag

The expected value is $\mathbb E[S_t] = S_0 e^{\mu t}$ (the drift $\mu$ drives the mean). But the median of $S_t$ is $S_0 e^{(\mu-\sigma^2/2)t}$, which is smaller. The term $-\sigma^2/2$ is the volatility drag or Itô correction: randomness moves the median below the mean due to Jensen's inequality applied to the convex exponential function. Higher volatility means slower median growth, even with the same expected return.

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3.8 Multidimensional Itô formula

Let $\mathbf X_t = (X_t^1,\ldots,X_t^d)$ with $dX_t^i = b_t^i\,dt + \sigma_t^{ij}\,dW_t^j$ (summed over $j$). For $f\in C^{1,2}$,

$$df(t,\mathbf X_t) = \partial_t f\,dt + \sum_i \partial_i f\,dX_t^i + \tfrac12\sum_{i,j}\partial_{ij}f\,d[X^i,X^j]_t.$$

where $d[X^i,X^j]_t = \sum_k\sigma_t^{ik}\sigma_t^{jk}\,dt$. The details are in Chapter 8.

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3.9 Itô's product rule

As a special case, for two Itô processes $X_t$ and $Y_t$:

$$d(X_t Y_t) = X_t\,dY_t + Y_t\,dX_t + d[X,Y]_t.$$

The last term is the stochastic correction. Compare with the deterministic product rule (Leibniz rule), which has no such correction.

Derivation from the bilinear identity

Apply Itô's formula to $f(x,y)=xy$: then $\partial_{xy}f = 1$, all other second derivatives vanish. The formula gives $d(XY) = X\,dY + Y\,dX + d[X,Y]$.

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3.10 Problems

Problem 1: d(W_t^3)

Use Itô's formula to compute $d(W_t^3)$ and integrate to find $\int_0^T W_t^2\,dW_t$ in terms of $W_T$ and $T$.

Problem 2: Exponential process

Let $X_t = e^{at + bW_t}$ for constants $a,b$. Find the SDE satisfied by $X$.

Problem 3: Itô's formula for $f(t,x) = x^2 - t$

Show directly that $f(t,W_t) = W_t^2 - t$ is a local martingale. What is its SDE?

Problem 4: Integration by parts

Use the Itô product rule to show that

$$W_T^2 = 2\int_0^T W_t\,dW_t + T.$$

Problem 5: Lognormal moments

For GBM with $S_t = S_0\exp((\mu-\sigma^2/2)t + \sigma W_t)$, compute $\mathbb E[S_t^n]$ for integer $n\ge 1$.

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3.11 Selected solutions

Solution 1

With $f(x) = x^3$, $f'=3x^2$, $f''=6x$:

$$d(W_t^3) = 3W_t^2\,dW_t + \tfrac12\cdot 6W_t\,dt = 3W_t^2\,dW_t + 3W_t\,dt.$$

Integrating: $W_T^3 = 3\int_0^T W_t^2\,dW_t + 3\int_0^T W_t\,dt$, so

$$\int_0^T W_t^2\,dW_t = \tfrac13 W_T^3 - \int_0^T W_t\,dt.$$

The remaining integral $\int_0^T W_t\,dt$ is a Gaussian with mean $0$ and variance $T^3/3$.

Solution 2

Let $f(t,x) = e^{at+bx}$. Then $\partial_t f = af$, $\partial_x f = bf$, $\partial_{xx}f = b^2f$. Itô's formula:

$$dX_t = af\,dt + bf\,dW_t + \tfrac12 b^2 f\,dt = (a+\tfrac12 b^2)X_t\,dt + bX_t\,dW_t.$$

Solution 3

With $f(t,x)=x^2-t$: $\partial_t f = -1$, $\partial_x f=2x$, $\partial_{xx}f=2$. Itô:

$$d(W_t^2-t) = (-1+\tfrac12\cdot 2)\,dt + 2W_t\,dW_t = 2W_t\,dW_t.$$

The drift vanishes: $W_t^2-t$ is a local martingale with zero mean.

Solution 5

Since $S_t^n = S_0^n\exp(n(\mu-\sigma^2/2)t + n\sigma W_t)$, and $e^{n\sigma W_t}$ has moment generating function $e^{n^2\sigma^2 t/2}$:

$$\mathbb E[S_t^n] = S_0^n\exp\!\left[\left(n\mu - \tfrac{n}{2}\sigma^2 + \tfrac{n^2}{2}\sigma^2\right)t\right] = S_0^n\exp\!\left[\left(n\mu + \tfrac{n(n-1)}{2}\sigma^2\right)t\right].$$

For $n=1$: $\mathbb E[S_t] = S_0 e^{\mu t}$. For $n=2$: $\mathbb E[S_t^2] = S_0^2 e^{(2\mu+\sigma^2)t}$.

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3.12 What this chapter gives you

Itô's formula is the engine of stochastic calculus:

$$df(t,X_t) = \underbrace{\partial_t f\,dt}_{\text{time}} + \underbrace{\partial_x f\,dX_t}_{\text{first order}} + \underbrace{\tfrac12\partial_{xx}f\,d[X]_t}_{\text{Itô correction}}.$$

The third term is unique to stochastic calculus and arises from $(dW)^2 = dt$. It:

• creates the volatility drag in GBM;
• is the source of the $\frac12\sigma^2$ term in the Black–Scholes PDE;
• is what makes stochastic calculus fundamentally different from classical calculus.

The next chapter uses Itô's formula as the main tool for stating and solving stochastic differential equations.