Two-scale realised volatility: Definition & Examples

Two-scale realised volatility (TSRV) estimates an asset's accumulated return variance from noisy high-frequency prices by combining a slow-sampled, averaged squared-return sum with a correction from a faster sampling scale. It helps a quant avoid treating trading noise as movement in the underlying efficient price. Despite its name, the estimator below produces variance, not its square root.

The target in the continuous-price model is integrated variance,

IV_T=\int_0^T\sigma_t^2\,dt,

where

\sigma_t

is the efficient log-price's instantaneous volatility. This is the continuous process's quadratic variation. It is accumulated over the stated interval, with no annualisation built in.

Why more observations can create more bias

Write the observed log-price at observation $i$ as

Y_i=X_{t_i}+\varepsilon_i.

Here

X

is the latent efficient log-price, while

\varepsilon_i

represents observation noise from market microstructure. For the basic construction, assume the noise is independent and identically distributed, has mean zero and common variance

\eta^2

, and is independent of the entire efficient-price process. The model makes a distinction between noise in log-price levels and noise in returns:

Y_i-Y_{i-1}=X_{t_i}-X_{t_{i-1}}+\varepsilon_i-\varepsilon_{i-1}.

Because two independent level errors enter each return,

\mathbb{E}[(\varepsilon_i-\varepsilon_{i-1})^2]=2\eta^2.

Conditional on the efficient path, the cross term has zero expectation. Thus the usual uncentred realised variance, computed from $n+1$ log-price observations and $n$ returns, satisfies

\begin{aligned} RV_{\mathrm{fast}}(Y)&=\sum_{i=1}^n(Y_i-Y_{i-1})^2,\\ \mathbb{E}[RV_{\mathrm{fast}}(Y)\mid X] &=RV_{\mathrm{fast}}(X)+2n\eta^2. \end{aligned}

Refining observations over the same horizon adds noise contributions. Adjacent observed returns also share a level error with opposite signs; independent level noise does not mean independent return noise.

The Expectation, Variance and Covariance lesson, available in full with a free signed-in account, develops the variance and covariance identities used in this expansion.

Two scales, one cancellation

Choose an integer

2\le K\le n

. Take every

K

th observation, repeating the calculation for each starting offset

r=0,\ldots,K-1

. A subgrid contains indices

r,r+K,r+2K,\ldots

up to

n

. Let

n_r

count its returns, not its price observations, and define

\overline n=\frac1K\sum_{r=0}^{K-1}n_r, \qquad RV_{\mathrm{slow}}(Y)=\frac1K\sum_{r=0}^{K-1}RV_r(Y).

For this partition of the observations, $\overline n=(n-K+1)/K$ . It is approximately $n/K$ in a large sample, but those are different finite-sample quantities.

Each slow-grid return still contains two level errors. Its expected noise contribution is therefore $2n_r\eta^2$ , and the average's is $2\overline n\eta^2$ . Subtracting $\overline n/n$ times the fast measure gives

\boxed{\widehat{IV}_{\mathrm{TS}} =RV_{\mathrm{slow}}(Y) -\frac{\overline n}{n}RV_{\mathrm{fast}}(Y).}

The noise terms cancel because

2\overline n\eta^2 -\frac{\overline n}{n}(2n\eta^2)=0.

This is Zhang, Mykland and Aït-Sahalia's equation (55). Averaging slow grids alone reduces the noise contribution relative to the fast sum; it does not eliminate it. The subtraction supplies the correction.

Noise cancellation is more precise than a claim of exact unbiasedness for $IV_T$ . Conditional on $X$ , the remaining expectation is

\mathbb{E}[\widehat{IV}_{\mathrm{TS}}\mid X] =RV_{\mathrm{slow}}(X) -\frac{\overline n}{n}RV_{\mathrm{fast}}(X),

which still involves discretely sampled efficient returns and different boundary coverage. Equation (64) of the same paper gives the small-sample rescaling

\widehat{IV}_{\mathrm{TS,adj}} =\frac{\widehat{IV}_{\mathrm{TS}}}{1-\overline n/n}.

Its motivation is that, when both efficient-price sums approximate the same integrated variance, the subtraction also removes approximately $\overline n/n$ of the signal. The rescaling addresses that attenuation; it does not universally repair finite-grid or boundary errors.

Seven observations, fully counted

Consider these fictional, equally spaced observed log-prices, measured in decimal log points:

(Y_0,\ldots,Y_6)=(0,\ 0.012,\ 0.020,\ 0.033,\ 0.040,\ 0.055,\ 0.060).

There are six adjacent returns. With $K=2$ , the two offset grids produce:

Grid	Observation indices	Log returns	Sum of squared returns
Fast	0, 1, 2, 3, 4, 5, 6	0.012, 0.008, 0.013, 0.007, 0.015, 0.005	0.000676
Slow offset 0	0, 2, 4, 6	0.020, 0.020, 0.020	0.001200
Slow offset 1	1, 3, 5	0.021, 0.022	0.000925

Seven fictional observed log-prices yield six fast returns. The even offset has three slow returns and RV 0.001200; the odd offset has two and RV 0.000925. The odd grid spans observations 1–5, so mean count is 2.5 returns and boundary coverage differs.

Original fictional equal-interval example. RV uses squared decimal log returns. The odd grid's shorter coverage is retained in the calculation.

The slow estimate is an equal average of the two grid sums, not a pooled average of their five squared returns:

RV_{\mathrm{slow}}=\frac{0.001200+0.000925}{2}=0.0010625, \qquad \overline n=\frac{3+2}{2}=2.5.

Consequently,

\begin{aligned} \widehat{IV}_{\mathrm{TS}} &=0.0010625-\frac{2.5}{6}(0.000676)\\ &=0.0007808333\ldots,\\ \widehat{IV}_{\mathrm{TS,adj}} &=\frac{0.0007808333\ldots}{1-2.5/6}\\ &=0.0013385714\ldots. \end{aligned}

All these estimates are in squared log-return units over the observed window. No latent price path or true integrated variance was supplied, so these calculations establish the estimator's arithmetic, not its estimation accuracy. The adjusted number exceeding the fast estimate is possible; a noise correction does not impose a ranking for every observed path.

What the correction does not establish

The original consistency result is an infill result: the observation horizon stays fixed while the observation mesh shrinks. Under its regularity conditions, regularly allocated subgrids and

K

of order

n^{2/3}

balance noise and discretisation errors. Both

K

and the number of observations per slow grid grow. Keeping

K=2

forever is not that asymptotic regime. See the original paper, §4.1 and Theorem 4.

The iid, common-variance, exogenous-noise assumptions are substantive. For stationary dependent noise with lag-

h

covariance

\gamma_h

, expanding the same square instead gives

2(\eta^2-\gamma_h)

. The fast lag and slow lag can then have different noise variances, breaking the simple cancellation above. Time-varying noise variance and dependence on the efficient price also require separate treatment. The authors' dependent-noise extension, §4, changes the fast scale rather than assuming the iid formula is automatically robust.

TSRV is not a jump-removal procedure. The integrated-variance interpretation here uses a continuous efficient-price model. With jumps, squared jump sizes enter quadratic variation; isolating the continuous component is a separate task, as in bipower variation, with its own assumptions about noise and sampling.

Finally, a difference of two nonnegative sums can be negative. For example, the observed log-price sequence $(0,0.01,0,0.01,0,0.01,0)$ has zero slow-grid variation at $K=2$ , but fast RV of $0.0006$ . Its raw TSRV is $-0.00025$ . A negative estimate has no real-valued volatility square root. Truncating it to zero changes the estimator and its statistical properties. When an estimate is nonnegative, taking its square root gives a volatility measure over that window; an annualised quotation additionally needs a stated time convention.

Sources

Zhang, Mykland and Aït-Sahalia, A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data (2005), Journal of the American Statistical Association 100(472), 1394–1411. §2.1–2.2 state the observation model; §3.2 defines grids; equations (55) and (64) define the two estimators; §4.1 gives the limiting result.
The authors' 4 September 2004 manuscript, equations (55) and (64). Its overview in §1.2.5 displays an apparent extra factor of two in the subtraction. The formula used here follows equation (55), the published paper and the explicit noise-cancellation calculation above.
Aït-Sahalia, Mykland and Zhang, Ultra High Frequency Volatility Estimation with Dependent Microstructure Noise, author manuscript dated 2 September 2009, introduction and §4.1, §4.5–4.6: extensions for serial dependence and the limits of small-sample adjustments.

Why more observations can create more bias

Two scales, one cancellation

Seven observations, fully counted

What the correction does not establish

Sources

Related terms

Bipower variation

Market microstructure

Quadratic variation

Realised volatility