Two-Scale Realized Volatility

Weigh yourself on a bathroom scale that jitters by half a pound between readings, and one measurement is noisy — but a thousand rapid readings, each treated as a "real" weight change, doesn't help either; it just injects a thousand jitters' worth of noise. There's a way to combine fast noisy readings with slow stable ones to cancel the jitter out. That's the problem two-scale realized volatility (TSRV) solves for prices sampled at very high frequency.

Why naive high-frequency RV breaks

Plain realized volatility (see realized volatility estimators) sums squared returns, and in theory, sampling more often should only help. In practice, at very short intervals, observed prices bounce between bid and ask even when the "true" price hasn't moved — bid-ask bounce adds spurious squared returns that inflate RV. Plot RV against sampling frequency and you get a signature plot: RV rises sharply as frequency increases past a point, revealing noise contamination rather than signal.

The two-scale estimator, defined

$$TSRV = RV^{(\text{slow})} - \frac{\bar{n}}{n} RV^{(\text{all})}$$

In words: compute $RV^{(\text{all})}$ using every tick (fast, noisy, biased up), and separately $RV^{(\text{slow})}$ by averaging RV over several subsampled grids at a coarser frequency (slower, much less noisy, but throws away most data). $n$ is ticks used in the fast estimate, $\bar{n}$ the average used per slow subsample. The correction term estimates and subtracts the noise bias baked into the fast estimate, since pure noise contributes a predictable amount to $RV^{(\text{all})}$ that grows with $n$ in a specific way, while true variance doesn't.

Worked example 1 — the signature plot pattern

Suppose a stock's RV, computed at different sampling intervals over the same day, comes out as: 1-second bars give $RV = 0.00210$; 1-minute bars give $RV = 0.00095$; 5-minute bars give $RV = 0.00082$; 30-minute bars give $RV = 0.00079$. As sampling gets coarser, RV falls and flattens around 0.00080 — that flat region is roughly where microstructure noise stops dominating, classic evidence that the 1-second estimate is badly contaminated, over double the "true" flattened level.

Worked example 2 — applying the TSRV correction

Suppose the fast (all-tick) estimate uses $n = 20{,}000$ ticks and gives $RV^{(\text{all})} = 0.00210$. A slow subsampled estimate, averaging over grids of about $\bar{n} = 390$ observations each (roughly 1-minute sampling), gives $RV^{(\text{slow})} = 0.00095$. The correction term: $\frac{390}{20{,}000} \times 0.00210 = 0.0000410$. So $TSRV = 0.00095 - 0.0000410 = 0.000909$ — close to the flattened signature-plot level from example 1 (0.00079-0.00082), and noticeably closer to that "true" region than either raw estimate alone.

What this means in practice

TSRV (and related noise-robust estimators like realized kernels) let researchers use the full richness of tick-by-tick data without being misled by microstructure noise, rather than the older brute-force fix of just sampling at a fixed "safe" interval like 5 minutes and throwing away everything finer. It's standard in high-frequency volatility forecasting research and in building the realized-vol inputs that feed models like HAR-RV.

Practice

1. A signature plot shows RV falling from 0.0040 at 1-second sampling to a flat 0.0012 by 10-minute sampling. What does the flattening tell you about where microstructure noise stops dominating?
2. If the fast all-tick RV is 0.0035 with $n = 15{,}000$ and the slow subsampled RV is 0.0012 with $\bar{n} = 300$, compute the TSRV correction term and the resulting TSRV estimate.