Barndorff-Nielsen-Shephard (2004) Power and Bipower Variation with Stochastic Volatility and Jumps

Summary

Barndorff-Nielsen and Shephard introduce realized bipower variation (an extension of realized power variation) and show that, unlike realized variance, it is robust to rare jumps. In stochastic volatility models with jumps, realized bipower variation consistently estimates the integrated variance (the continuous part of quadratic variation), while realized variance estimates the full quadratic variation (continuous plus jumps). Consequently the difference between realized variance and realized bipower variation consistently estimates the quadratic variation of the jump component — the first method able to separate quadratic variation into its continuous and jump parts. Various extensions and proofs of special cases are given.

Key Claims

• Realized power variation. For intraday returns $y_j$ over $M$ intervals, the realized power variation $\sum_j |y_j|^{p}$ (suitably scaled) converges to an integrated power of volatility; the choice $p=2$ recovers realized variance / quadratic variation.
• Realized bipower variation. The $\{1,1\}$ realized bipower variation $$\{y\}^{[1,1]}_\delta = \sum_{j=2}^{M} |y_{j}|\,|y_{j-1}|,\qquad \mu_1^{-2}\{y\}^{[1,1]}_\delta \xrightarrow{p} \int_0^t \sigma^2(u)\,du,$$ with $\mu_1=\sqrt{2/\pi}=E|Z|$, consistently estimates integrated variance — a model-free, consistent alternative to realized variance.
• Jump robustness. Because each term multiplies two adjacent returns and a jump lands in only one of them, an isolated jump contributes negligibly to bipower variation as $\delta\to 0$. Realized bipower variation is therefore robust to a finite number of jumps, whereas realized variance absorbs the squared jumps.
• Separating continuous and jump variation. With a stochastic-volatility-plus-jumps process, realized variance $\to$ integrated variance $+\sum \kappa_i^2$ (squared jumps), while bipower variation $\to$ integrated variance. Hence $$[y]_\delta - \mu_1^{-2}\{y\}^{[1,1]}_\delta \xrightarrow{p} \sum_{i}\kappa_i^2,$$ estimating the jump contribution to quadratic variation. "This seems to be the first method that can separate quadratic variation into its continuous and jump components."
• Foundation for jump tests and decompositions. The continuous/jump split gives a nonparametric basis for testing whether jumps are present and for modeling/forecasting the two volatility components separately.

Concepts Introduced or Extended

• Bipower Variation
• Quadratic Variation
• Realized Volatility

Entities Mentioned

• Ole E. Barndorff-Nielsen
• Neil Shephard

Quotes

"This article shows that realized power variation and its extension, realized bipower variation, which we introduce here, are somewhat robust to rare jumps."

"This seems to be the first method that can separate quadratic variation into its continuous and jump components."

My Take

If BNS (2002) made realized variance a measured quantity with a known error, this paper solved the next problem: realized variance measures everything — the smooth diffusive part and the violent jumps together — and for most purposes you want them apart. Bipower variation is a beautifully simple trick: multiply adjacent absolute returns so that any single jump, which shows up in exactly one return, gets diluted to nothing in the limit, leaving only the continuous integrated variance. The estimator $[y]-BPV$ for the jump quadratic variation is what launched the high-frequency jump-testing literature (Huang–Tauchen; Andersen–Bollerslev–Diebold's "Roughing It Up") and the practice of forecasting continuous and jump volatility with separate dynamics. It sits directly on top of the quadratic-variation and realized-volatility machinery and is the bridge to jump-aware volatility models.