Bitcoin's 20-minute variance is 7% higher after falls than equal rises

Bitcoin's variance is about 7% higher over the 20 minutes following a fall than following an equal rise. The gap appears in 31 of 33 ten-week epochs since January 2020. Whether it can help anyone size risk is beyond the question the paper takes on.

Measuring the response

The leverage effect describes volatility rising more after a fall than after an equal rise. Its namesake explanation, a falling share price increasing a firm's debt-to-equity ratio, cannot apply to a coin without a balance sheet. The authors ask whether crypto shows the asymmetry at all, and how quickly it unfolds. Their crypto data come entirely from Coinbase: 933.8M trades for BTC and 747.8M for ETH between 2020-01-20 and 2026-05-18, with SOL added from mid-2021. The comparison uses AAPL, TSLA, MSFT and SPY from the consolidated US tape during regular hours only.

At each instant, they measure log realized variance over a 20-minute window. They match it to the slope of log price over another 20-minute window, moving the lag from -120 to +600 minutes. Negative lags put the trend after the volatility. A genuine response to the trend should disappear there.

They group samples by trend within each epoch and lag. Separate straight lines fit log variance against trend for falls and rises, the two "wings". The asymmetry is half the difference between those slopes. Scaling it by the epoch's typical trend size gives Δs, the log-variance gap between a typical fall and a typical rise. The authors then separate a T-odd component, which changes sign under time reversal and alone carries an arrow of time, from a T-even component, which keeps its sign. A response occurring only after the trend makes these components equal.

At the 20-minute lag, the log channel gives Δs of 0.071±0.008 for BTC and 0.066±0.008 for ETH. The gap peaks at this first lag, then halves by 89 and 90 minutes. Solana's 0.022±0.009 is about a third as large.

What does a 7% variance gap buy?

Our arithmetic puts exp(0.071) at about 1.074. Taking the square root gives a volatility gap near 3.6%. In the linear channel, which divides variance by the wings' baseline, Bitcoin's amplitude roughly doubles to 0.130±0.026. The authors favour the log channel because the most volatile windows dominate raw variance. They describe the symmetric component, volatility increasing after moves of either sign, as much larger. They leave it uninterpreted because large moves cluster in volatile periods.

A sizing decision needs a sign term to improve a forecast that already incorporates recent variance. This paper measures the conditional gap in-sample. It attaches no forecast comparison to Δs.

BTC shows no sign of decay between its sample halves: they return 0.076±0.014 and 0.066±0.011. Solana moves the other way, from 0.002±0.011 in the first half to 0.049±0.011 in the second.

Volatility mostly follows the fall

BTC's T-odd component is 0.032±0.006 at 20 minutes, a 5.1σ result. ETH's is 0.026±0.007, or 4.0σ. For BTC, the anti-causal constant is -0.005±0.037. Its band ratio is 0.85. The 68% interval, 0.53 to 1.24, includes one, corresponding to a purely one-sided response, and excludes zero, corresponding to a mirror.

The simulated market helps interpret this ratio. The authors build in a purely one-sided response with τ=90 minutes, yet recover 0.65±0.05 at the first lag and roughly 0.5 over the band. The simulation also generates an opposite-sign negative side. The response skews the trend distribution, and the wing fits read that skew backwards. For a like-for-like comparison, BTC at the first lag has T-even 0.038±0.005 and T-odd 0.032±0.006, giving a ratio of about 0.9 for the 20-minute window. Its band ratio of 0.85 also exceeds the control's 0.5. The statistic therefore reads no more skew in Bitcoin than in the simulated market.

The abstract reports a range of 0.4 to 0.9 at 10 and 20 minutes. The authors also acknowledge that their reported 20-minute window produces the highest value. BTC's first-lag ratios are 0.5, 0.9, 0.4 and 0.6 for windows of 10, 20, 40 and 60 minutes. At 40 and 60 minutes, the T-odd component is within 1.7σ of zero and its error roughly doubles. Only the 10 and 20 minutes estimates measure the ratio reliably. The control covers only 20 minutes.

Removing the daily volatility cycle reduces BTC's T-odd component to 0.025±0.007. The authors only "suspect the trend might cause it". They acknowledge that news moving price before volatility would produce the same profile.

SPY's mirror depends on the fit range

SPY looks mirrored when the straight-line fit spans ±2.5 trend scales. The T-odd component is -0.025±0.015; the anti-causal constant is -0.36±0.14. Narrowing the fit to ±1.5 scales raises first-lag Δs from 0.026±0.015 to 0.110±0.026. With a quadratic term added to each wing, it reaches 0.30±0.07 and T-odd reaches +0.21±0.06. The same trend cuts move BTC and ETH by at most 0.004±0.011.

The abstract concedes this dependence. The authors argue that SPY responds more to a move's sign than its size, while retaining the 2.5-scale line as the common comparison instrument. They concede the consequence as well. Within the inner 1.5 scales, SPY's estimates cannot distinguish a one-sided response from a mirror. Their equity claim holds only "for that fit".

Ordinary correlation gives a different picture. For SPY, the correlation between a 20-minute return and log variance 20 minutes later is about three times larger in magnitude than for BTC: -0.11±0.01 versus -0.037±0.006. Reverse-direction correlations are small for both, at +0.013±0.005 for SPY and +0.008±0.002 for BTC. Reading both directions together, the authors judge the fund's arrow as evident as Bitcoin's. Using one-minute returns to estimate variance also reduces the crypto-to-SPY amplitude ratio from 2.7 to 2.1.

The crypto result is the stable half of this paper.

Two-second marks, and the limits of our check

The variance calculation uses log prices marked at least two seconds apart. Trend uses marks ten seconds apart. A single exchange supplies the crypto trades to avoid offsets between venues. For equities, the authors remove corrections, non-updating sale conditions and one-trade spikes. That cleaning matters: Apple's asymmetry over 20 to 180 minutes changes sign, from -0.022±0.035 to +0.069±0.037.

We could not reproduce this. Our one-minute bars lack the within-minute sums of squared two-second returns needed to construct the grid. We also have no Coinbase-specific trade series. Estimating variance from minute returns would measure something different. The paper's own one-minute check lifts BTC's Δs to 0.087±0.011, suggesting bars might retain the sign. Someone with the tape still has to demonstrate it.

The 90-minute half-life should be treated as a research target. In the simulated control, the measured half-decay lag is short: 69 to 74 minutes against an implied 82. The exponential fit estimates BTC τ=95 minutes (64 to 136). But the pre-specified second estimator gives 248 minutes and fails its own checks. There are roughly 3.5 effectively independent causal lags out of 30, leaving too little shape information to separate an exponential from a power law. To turn this fact into an edge, a sign term would need to lower out-of-sample variance forecast error over the next 20 to 90 minutes, after giving the symmetric component its due.