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Bipower variation

Estimate continuous integrated variance with adjacent absolute returns, compare it with realised variance, and understand jump robustness and testing limits.

Bipower variation estimates the integrated variance of a log-price's continuous component by adding products of neighbouring absolute returns and applying a normalising constant. Its practical use is to separate ordinary continuous price variation from the squared-jump contribution included in realised variance, under an appropriate high-frequency model.
The distinction matters after a large market move: a squared-return sum records the move, but cannot by itself say how much variation came from diffusion and how much from jumps. Bipower variation supplies a second measure with different sensitivity to an isolated jump. The construction originates in Barndorff-Nielsen and Shephard (2004), §1.4 and §2.3.

Multiply neighbours instead of squaring each return

Observe a positive price at n+1n+1 equally spaced times over a fixed window [0,T][0,T]. Let Xt=log⁡PtX_t=\log P_t and ri=XiT/n−X(i−1)T/nr_i=X_{iT/n}-X_{(i-1)T/n}. We use the following uncorrected finite-sample convention:
RVn=∑i=1nri2,BVn=μ1−2∑i=2n∣ri∣∣ri−1∣,μ1=E[∣Z∣]=2π,Z∼N(0,1).RV_n=\sum_{i=1}^{n}r_i^2, \qquad BV_n=\mu_1^{-2}\sum_{i=2}^{n}\lvert r_i\rvert\lvert r_{i-1}\rvert, \qquad \mu_1=\mathbb{E}[\lvert Z\rvert]=\sqrt{\frac{2}{\pi}}, \quad Z\sim\mathcal{N}(0,1).

Thus μ1−2=π/2\mu_1^{-2}=\pi/2. There are nn squared returns but n−1n-1 adjacent products. Absolute values prevent a positive return followed by a negative return from cancelling. These are uncentred measures; no sample mean is subtracted.

The constant has a simple calibration. Under a zero-drift, constant-volatility Brownian model, ri=σT/n Zir_i=\sigma\sqrt{T/n}\,Z_i with independent standard normal ZiZ_i. Each product has expectation μ12σ2T/n\mu_1^2\sigma^2T/n, so

E[BVn]=n−1nσ2T.\mathbb{E}[BV_n]=\frac{n-1}{n}\sigma^2T.
Multiplying by n/(n−1)n/(n-1) removes this boundary shortfall in that particular model. Some authors use this finite-sample adjustment, including Andersen and co-authors (2006), equation (5). Others report the uncorrected statistic used here. Both multipliers approach one as nn grows; reports must state the convention. This adjustment does not remove every finite-sample bias under time-varying volatility, drift or observation noise.
For the Gaussian prerequisite, the Normal Distribution lesson explains Z∼N(0,1)Z\sim\mathcal{N}(0,1), standardisation and the distinction between variance and standard deviation. The full lesson is available with a free signed-in account; its preview is public.

Five returns: where the large observation enters

Take the fictional intraday decimal log returns

(r1,r2,r3,r4,r5)=(0.001, 0.001, 0.10, 0.001, 0.001).(r_1,r_2,r_3,r_4,r_5)=(0.001,\ 0.001,\ 0.10,\ 0.001,\ 0.001).

The middle return is 10% in log-return units. These five observations illustrate arithmetic, not an empirical price process or a statistically significant jump.

CalculationContributionsSum
Squared returns0.000001, 0.000001, 0.01, 0.000001, 0.0000010.000001,\ 0.000001,\ 0.01,\ 0.000001,\ 0.0000010.0100040.010004
Adjacent absolute products0.000001, 0.0001, 0.0001, 0.0000010.000001,\ 0.0001,\ 0.0001,\ 0.0000010.0002020.000202

Consequently,

BV5=π2(0.000202)≈0.000317300858,RV5−BV5≈0.009686699142.BV_5=\frac{\pi}{2}(0.000202)\approx0.000317300858, \qquad RV_5-BV_5\approx0.009686699142.
Five fictional intraday log returns are 0.1%, 0.1%, 10%, 0.1% and 0.1%. Their four adjacent absolute products are 1, 100, 100 and 1 millionths of squared log points. RV is 0.010004 and uncorrected BV is 0.000317301.
Five fictional intraday log returns are 0.1%, 0.1%, 10%, 0.1% and 0.1%. Their four adjacent absolute products are 1, 100, 100 and 1 millionths of squared log points. RV is 0.010004 and uncorrected BV is 0.000317301.
Original fictional equal-interval example. Return bars use percent; product bars use millionths of squared log points. BV uses the uncorrected convention, with no n/(n−1) multiplier.

The middle observation contributes 0.010.01 when squared, but appears in two products of 0.10×0.001=0.00010.10\times0.001=0.0001. Its influence on BV is reduced, not eliminated. If the finite-sample multiplier were included, BV would instead be approximately 0.0003966260730.000396626073. Neither convention makes this five-observation example an exact decomposition of latent continuous and jump variance.

Both statistics have squared decimal log-return units over the sampled window. BV estimates integrated variance, not volatility. A square root changes the units; annualisation additionally requires a stated window length and clock. No annualisation is applied here.

What “jump robust” actually promises

For a noise-free log-price with a regular continuous Brownian-semimartingale component and a finite-activity jump process, write

Xt=X0+∫0tas ds+∫0tσs dWs+Jt.X_t=X_0+\int_0^t a_s\,ds+\int_0^t\sigma_s\,dW_s+J_t.
As the equally spaced observation intervals shrink within the same fixed window, the relevant probability limits are
RVn→p∫0Tσs2 ds+∑0<s≤T(ΔJs)2,BVn→p∫0Tσs2 ds.\begin{aligned} RV_n&\xrightarrow{p}\int_0^T\sigma_s^2\,ds+\sum_{0<s\le T}(\Delta J_s)^2,\\ BV_n&\xrightarrow{p}\int_0^T\sigma_s^2\,ds. \end{aligned}
This is infill asymptotics, rather than observing more days at unchanged frequency. The assumptions include adapted, suitably integrable coefficients, a regular volatility path (typically càdlàg), finite jump activity on the window and observations of the efficient price without measurement noise. The general limit theory, §2–3, treats continuous Brownian semimartingales and allows leverage effects under its stated conditions; early proofs used stronger independence assumptions.
The finite-jump intuition is local: eventually separate jumps occupy nonadjacent intervals. Each jump then enters a product with a neighbouring continuous return that shrinks toward zero. Its square in RV survives. With only finitely many affected products, their contribution to BV vanishes. This reasoning does not give immunity to two large adjacent returns on a coarse grid, or a blanket guarantee for every infinite-activity jump model. Barndorff-Nielsen and Shephard's testing paper, §2, distinguishes these limits from their discrete estimates.

A difference is not yet a jump test

The limiting difference is the nonnegative sum of squared jumps. The finite-sample difference can be negative. For example, five identical fictional returns of 0.0010.001 give
RV5=0.000005,BV5=π2(0.000004)≈0.000006283185.RV_5=0.000005, \qquad BV_5=\frac{\pi}{2}(0.000004)\approx0.000006283185.

Hence RV5−BV5≈−0.000001283185RV_5-BV_5\approx-0.000001283185. This is estimator behaviour, not negative jump variance. Truncating the difference at zero creates a different reported estimator; it does not establish statistical significance.

A jump test must compare the difference, or an appropriate ratio, with its sampling uncertainty under a continuous-path null. The Barndorff-Nielsen–Shephard tests, §3, scale and standardise the statistic using a variance estimate involving integrated quarticity, ∫0Tσs4 ds\int_0^T\sigma_s^4\,ds, and specify a rejection rule. Raw subtraction supplies neither a standard error nor a p-value. Five fictional returns cannot support that inference.
Finally, jump robustness is not noise robustness. Market microstructure noise, including bid–ask bounce and price discreteness, changes observed returns and adjacent products. Sampling faster can therefore worsen measurement. Andersen and co-authors, §4, examine spacing returns to reduce particular noise dependencies; this is a modified estimator with its own assumptions, not a universal repair. A usable measurement needs cleaned prices, an explicit sampling design and uncertainty analysis before the RV–BV gap is interpreted as evidence about jumps.

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  • bipower variation
  • realised variance
  • integrated variance
  • jump detection
  • high frequency data
  • volatility estimation
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