Quadratic Variation

Motivation: why this matters in quant finance

The second derivative term in the Black-Scholes PDE exists because Brownian-driven prices have non-zero quadratic variation. A delta hedge removes the first-order exposure to

dS_t

, but the option's curvature still accumulates a deterministic order-

dt

contribution from squared price shocks.

For a smooth deterministic path, the sum of squared increments disappears as the trading grid gets finer. For Brownian motion, Lawler's calculation shows that the same sum converges to elapsed time. That contrast is the point: stochastic calculus keeps a second-order term that ordinary calculus throws away.

This lesson is deliberately narrow. It builds the Brownian quadratic-variation calculation that later supports stochastic integrals, Itô's lemma, and the product algebra used for discounted assets and hedging portfolios.

The informal idea

Quadratic variation measures accumulated squared movement along a path. It is not net displacement. A Brownian path can finish near zero and still spend a lot of "movement budget" along the way.

The scale is what matters. Over a small interval of length $\Delta t$ , a Brownian increment is typically of size $\sqrt{\Delta t}$ . Squaring it gives something of order $\Delta t$ . Summing roughly $T/\Delta t$ such terms produces something of order $T$ , not something that vanishes.

By contrast, a differentiable path moves by order $\Delta t$ . Its squared increment is order $(\Delta t)^2$ , and summing $T/\Delta t$ such terms still goes to zero. Quadratic variation is therefore a diagnostic for Brownian roughness: continuous paths can be too rough for ordinary finite-variation calculus.

Formal definitions

Let $\Pi=\{0=t_0<t_1<\cdots<t_n=t\}$ be a partition of $[0,t]$ , and let its mesh be

\lVert \Pi \rVert=\max_{1\leq j\leq n}(t_j-t_{j-1}).

For a process $X$ , the quadratic variation along a refining sequence of partitions is the limit, when it exists,

\langle X\rangle_t=\lim_{\lVert \Pi \rVert\to 0}\sum_{j=1}^{n}\left(X_{t_j}-X_{t_{j-1}}\right)^2.

The mode of convergence matters. Lawler first computes the limit for deterministic partitions and proves convergence in probability; under a summability condition on the mesh, the convergence is almost sure for that fixed sequence of partitions.

For standard Brownian motion $W$ ,

\langle W\rangle_t=t.

For Brownian motion with drift $m$ and variance parameter $\sigma^2$ ,

X_t=\sigma W_t+mt \qquad\Longrightarrow\qquad \langle X\rangle_t=\sigma^2t.

The drift does not contribute to quadratic variation. It moves on the finite-variation scale; the Brownian term moves on the $\sqrt{dt}$ scale.

Key properties

Brownian quadratic variation is deterministic

For a partition $\Pi$ of $[0,t]$ , define

Q(t;\Pi)=\sum_{j=1}^{n}\left(W_{t_j}-W_{t_{j-1}}\right)^2.

Since $W_{t_j}-W_{t_{j-1}}\sim \mathcal{N}(0,t_j-t_{j-1})$ and the increments are independent,

\mathbb{E}[Q(t;\Pi)]=t,

and

\text{Var}(Q(t;\Pi))=2\sum_{j=1}^{n}(t_j-t_{j-1})^2 \leq 2t\,\lVert \Pi\rVert.

As the mesh goes to zero, the variance vanishes, so

Q(t;\Pi)

converges to

t

in probability. In finance, this is why realised variance over very fine Brownian-model observations targets the volatility clock rather than the signed return.

Drift does not appear

If $X_t=\sigma W_t+mt$ , expanding squared increments gives

\Delta X_j^2 =\sigma^2\Delta W_j^2+2\sigma m\,\Delta W_j\Delta t_j+m^2(\Delta t_j)^2.

The first term sums to

\sigma^2t

. The mixed and drift-only terms vanish in the limit. This is why realised quadratic variation identifies volatility, not expected return.

Finite-variation terms are invisible to quadratic variation

Lawler later uses the same idea for Itô processes. If

dX_t=R_t\,dt+A_t\,dW_t,

then the quadratic variation is

d\langle X\rangle_t=A_t^2\,dt,

or equivalently

\langle X\rangle_t=\int_0^t A_s^2\,ds.

The $R_t\,dt$ drift term contributes no quadratic variation. In delta hedging language, this separates predictable drift accumulation from stochastic variance accumulation.

Covariation records shared Brownian noise

For two processes driven by the same Brownian motion,

dX_t=H_t\,dt+A_t\,dW_t,\qquad dY_t=K_t\,dt+C_t\,dW_t,

their covariation is

d\langle X,Y\rangle_t=A_tC_t\,dt.

This is the quantity that appears in the Itô product rule. If two assets share the same Brownian shock, their product dynamics include the shared volatility term.

Worked examples

Example 1: Brownian quadratic variation on an equal grid

Let $t_j=jt/n$ and

Q_n(t)=\sum_{j=1}^{n}\left(W_{jt/n}-W_{(j-1)t/n}\right)^2.

Each increment has distribution $\mathcal{N}(0,t/n)$ . Write

W_{jt/n}-W_{(j-1)t/n}=\sqrt{\frac{t}{n}}\,Z_j,

where $Z_1,\ldots,Z_n$ are independent standard normals. Then

Q_n(t)=\frac{t}{n}\sum_{j=1}^{n}Z_j^2.

Since $\mathbb{E}[Z_j^2]=1$ and $\text{Var}(Z_j^2)=2$ ,

\mathbb{E}[Q_n(t)]=t, \qquad \text{Var}(Q_n(t))=\frac{2t^2}{n}.

Therefore $Q_n(t)\to t$ in $L^2$ , hence in probability. This is Lawler's basic calculation behind $\langle W\rangle_t=t$ .

Example 2: Brownian motion with drift and volatility

Let $X_t=\sigma W_t+mt$ . On a fine partition,

\sum_j(\Delta X_j)^2 =\sigma^2\sum_j(\Delta W_j)^2 +2\sigma m\sum_j\Delta W_j\Delta t_j +m^2\sum_j(\Delta t_j)^2.

The first term converges to $\sigma^2t$ . The last term is bounded by $m^2t\lVert\Pi\rVert$ , so it goes to zero. The mixed term is of smaller order because Brownian increments sum on the $\sqrt{dt}$ scale while the extra $\Delta t_j$ suppresses the contribution. Thus

\langle X\rangle_t=\sigma^2t.

For an asset model $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$ , this says the instantaneous quadratic-variation rate is $\sigma^2S_t^2$ , not $\mu^2S_t^2$ .

Example 3: quadratic variation of a stochastic integral

Z_t=\int_0^t A_s\,dW_s,

with adapted continuous or piecewise continuous $A$ , Lawler states

\langle Z\rangle_t=\int_0^t A_s^2\,ds.

For constant $A_s=\sigma$ , this reduces to $\langle \sigma W\rangle_t=\sigma^2t$ . For a trading gain $\int_0^t H_s\,dS_s$ in a Brownian model, the same structure says that the risk accumulated by the strategy depends on the squared exposure rate.

Common confusions and pitfalls

"Quadratic variation is the same thing as variance." Variance is a distributional second moment at a fixed time. Quadratic variation is accumulated squared path movement along a time interval. For Brownian motion they are both equal to $t$ in the simplest case, but they answer different questions.

"Continuity makes squared increments vanish." Smooth continuous paths have zero quadratic variation. Brownian paths are continuous but rough enough that squared increments accumulate to time.

"The drift should affect realised variation." In the Brownian scaling limit, drift contributes on the $dt$ scale and its square is too small to survive. The volatility coefficient controls quadratic variation.

"The convergence holds uniformly over every possible partition chosen after seeing the path." Lawler warns about the order of quantifiers. For fixed refining partition sequences, convergence holds as stated. If the partition is chosen path-dependently, the conclusion can fail.

" $(dW_t)^2=dt$ is literal algebra." It is shorthand for the quadratic-variation limit. The notation is useful, but the mathematical statement is about sums of squared increments.

Where this goes next

Stochastic Integrals: Builds Brownian integrals as limits whose second moments are controlled by squared integrands.
Itô Isometry: Turns the quadratic-variation intuition into an exact $L^2$ identity.
Itô's Lemma: Uses $\langle W\rangle_t=t$ to explain the second-derivative correction in stochastic chain rules.
Itô Product and Quotient Rules: Uses covariation to compute products, ratios, and discounted processes.
Stochastic Differential Equations: Interprets $dX_t=R_t\,dt+A_t\,dW_t$ through its drift and quadratic-variation rates.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 2 §2.8 (Quadratic variation), §2.9 (Multidimensional Brownian motion and covariation), Ch. 3 §3.2 (Stochastic integral), §3.4 (More versions of Itô's formula), §3.6 (Covariation and the product rule).