Quadratic Variation

Definition

The quadratic variation of a price (or log-price) process is the theoretical object that realized volatility estimates. For a semimartingale $p_t$ , the quadratic variation over $[t-1,t]$ is the almost-sure limit of sums of squared increments as the partition mesh shrinks:

$[p,p]_t - [p,p]_{t-1} = \lim_{m\to\infty} \sum_{j=1}^{m}\bigl(p_{t-1+j/m}-p_{t-1+(j-1)/m}\bigr)^2 .$

For a continuous (no-jump) price $dp_t = \mu_t\,dt + \sigma_t\,dW_t$ , the quadratic variation equals the integrated variance (a.k.a. integrated volatility):

$[p,p]_t - [p,p]_{t-1} = \int_{t-1}^{t}\sigma_s^2\,ds .$

The multivariate analogue is the quadratic covariation matrix, whose $(i,j)$ element is $[p^i,p^j]$ .

Key Ideas

The drift does not matter. Over a short interval the drift contributes $O(\text{d}t)$ while the diffusion contributes $O(\sqrt{\text{d}t})$ ; squaring makes the drift's contribution to quadratic variation vanish. Quadratic variation is driven entirely by the martingale (diffusion + jump) component of the return.
Quadratic variation is the right notion of "actual" volatility. Under an arbitrage-free special-semimartingale price, the conditional return covariance is governed by the expected quadratic variation; the ex-post realized quadratic variation is the object of interest for measurement, and is (nearly) observable via realized volatility.
Continuous plus jump decomposition. For a general semimartingale, quadratic variation splits into the integrated variance of the continuous part plus the sum of squared jumps: $[p,p]_t = \int_0^t \sigma_s^2\,ds + \sum_{0<s\le t}(\Delta p_s)^2$ . Separating these two pieces is the central problem of the jump-testing literature.
Makes volatility observable. Because quadratic variation is estimable model-free from high-frequency data, latent volatility becomes an (almost) observed series that can be modeled with ordinary time-series tools — the conceptual pivot of ABDL (2003).

How It Works

From prices to integrated variance (ABDL 2003)

Let the $n$ -vector log-price follow a continuous-time arbitrage-free process, so it is a special semimartingale $p_t = p_0 + \int_0^t \mu_s\,ds + M_t$ with $M$ a local martingale. The unique quadratic variation process $[M,M]_t$ satisfies:

$\operatorname{Var}(r_{t}\mid \mathcal{F})$ is governed by the increment of $[M,M]$ — i.e. the integrated (co)variance is the conditional covariance matrix of returns;
for a continuous $M$ , $[M,M]_t=\int_0^t \Sigma_s\,ds$ where $\Sigma_s=\sigma_s\sigma_s'$ is the instantaneous (spot) covariance;
realized volatility converges to it: $\sum_j r_{t,j}r_{t,j}' \xrightarrow{a.s.} \int_{t-1}^{t}\Sigma_s\,ds$ as the sampling interval $\to 0$ .

Mixture-of-normals for returns

Conditioning on the volatility path, the daily return is Gaussian with covariance equal to the integrated variance: $r_t \mid \{\sigma_{t+s}\}_{s\in[0,1]} \sim \mathcal{N}\!\Bigl(0,\ \textstyle\int_0^1 \Sigma_{t+s}\,ds\Bigr).$ Thus the heavy tails of the unconditional return distribution are a normal scale mixture with mixing variate the integrated variance — the theoretical basis for the "standardize by $\sqrt{RV}$ ⇒ Gaussian" regularity and for lognormal-normal mixture VaR forecasts.

Jumps

When the price has jumps, quadratic variation over-states integrated variance by the squared-jump sum. Realized volatility estimates total quadratic variation; jump-robust measures (bipower variation, truncated/quantile RV) estimate the continuous part, and their difference identifies jump variation.

Why It Matters

Quadratic variation is the bridge between continuous-time finance and discrete-time volatility measurement. It gives realized volatility its theoretical license: RV is not just a heuristic proxy but a consistent nonparametric estimator of a well-defined population quantity. This underwrites treating volatility as observable, the whole realized-volatility modeling programme, model-free jump detection, and the scaling of volatility measurement to large covariance matrices.

Open Questions

Microstructure noise biases the naive RV estimator of quadratic variation at the highest frequencies; the optimal-sampling / noise-robust literature addresses how best to recover quadratic variation.
Separating the continuous and jump components of quadratic variation in finite samples has limited power, especially for small or frequent jumps.
Non-synchronous trading complicates estimation of the off-diagonal quadratic-covariation (the Epps effect).