Bipower Variation

Definition

Realized bipower variation is a high-frequency measure of the continuous (diffusive) part of the quadratic variation of a price process, constructed from products of adjacent absolute intraday returns. Introduced by Barndorff-Nielsen and Shephard (2004), it is robust to rare jumps and consistently estimates the integrated variance even when the price contains jumps — in contrast to realized variance, which estimates the full quadratic variation (continuous part plus squared jumps).

Key Ideas

• Realized power variation. The generalization of realized variance to other powers: $\sum_j |y_j|^{p}$ (scaled), which converges to an integrated power of the spot volatility. The case $p=2$ is realized variance.
• Bipower variation. The $\{1,1\}$ bipower variation multiplies neighboring absolute returns: $$\{y\}^{[1,1]}_\delta = \sum_{j=2}^{M} |y_j|\,|y_{j-1}|, \qquad \mu_1^{-2}\{y\}^{[1,1]}_\delta \xrightarrow{p} \int_0^t \sigma^2(u)\,du,$$ with $\mu_1 = E|Z| = \sqrt{2/\pi}$. It consistently estimates integrated variance — a model-free alternative to realized variance.
• Jump robustness. A single jump appears in exactly one return; because each bipower term pairs two adjacent returns, the jump's contribution vanishes as the sampling interval $\delta\to 0$. Realized variance instead absorbs each squared jump.
• Continuous/jump decomposition. Realized variance $\to$ integrated variance $+\sum_i \kappa_i^2$; bipower variation $\to$ integrated variance. Hence the difference estimates the jump quadratic variation: $$[y]_\delta - \mu_1^{-2}\{y\}^{[1,1]}_\delta \xrightarrow{p} \sum_i \kappa_i^2.$$ This is the first nonparametric separation of quadratic variation into its continuous and jump components.

How It Works

Given $M$ intraday returns per day, compute realized variance $RV=\sum_j y_j^2$ and realized bipower variation $BV=\mu_1^{-2}\sum_{j\ge 2}|y_j||y_{j-1}|$. Under a stochastic-volatility-plus-jumps semimartingale, $RV$ targets total quadratic variation while $BV$ targets integrated variance. The relative jump measure $(RV-BV)/RV$ estimates the fraction of total variation due to jumps; formal jump tests (Barndorff-Nielsen–Shephard 2006; Huang–Tauchen; Andersen–Bollerslev–Diebold) compare $RV-BV$ to its (quarticity-scaled) null standard error to detect days with jumps. Robustness to market-microstructure noise typically requires staggered/skip-one variants and coarser sampling.

Why It Matters

• It converts the abstract "quadratic variation = continuous + jumps" identity into estimable pieces from ordinary intraday returns, enabling day-by-day jump detection.
• Forecasting improves when the smooth integrated-variance component (highly persistent) and the jump component (much less persistent) are modeled separately — the basis of HAR-type continuous/jump volatility models.
• It is the empirical foundation for testing continuous-time asset-pricing models that do or do not include a jump component.

Open Questions

• Microstructure noise. At the highest frequencies, noise dominates; how best to combine bipower variation with noise-robust estimators (pre-averaging, realized kernels) remains active.
• Finite-sample jump-test size/power. The asymptotic jump tests can over-reject in finite samples; small-sample corrections and the treatment of many small jumps vs. few large jumps are debated.

Related

• Quadratic Variation
• Realized Volatility
• Stochastic Volatility
• Jump-Diffusion Model
• Time-Changed Lévy Process
• Barndorff-Nielsen-Shephard (2004)
• Barndorff-Nielsen and Shephard (2006)
• Ole E. Barndorff-Nielsen
• Neil Shephard