The Lab

Where the idea comes from

• Fat tails. Benoit Mandelbrot (1963) showed that price changes have far more big moves than a bell curve allows.
• Business time. Mandelbrot and Howard Taylor (1967) suggested that prices move on a clock of trading, not of hours. Peter Clark (1973) made it testable: if each trade's worth of time is ordinary, then hours with uneven amounts of trading give fat tails. Clark found returns much closer to a bell curve when counted by volume. Thierry Ané and Hélyette Geman (2000) went further using the number of trades.
• Bars on a volume clock. Marcos López de Prado (2018) set out volume and dollar bars for trading research.
• Our pages "First passage: the chance a price touches a level" and "The touch map in calm and wild markets: does a table for each help?" give the touch map's background. Our theory page "Quantum finance: what it adds, and what it doesn't" puts this idea in the wider story.

How we tested

Part 1, the shape.

• Bybit's 1-minute futures prices and volumes, October 2022 to October 2026.
• Three clocks, each with as many bars as there are hours (35,063):
  • hour bars: one per clock hour;
  • volume bars: a bar closes when a set number of coins has traded;
  • dollar bars: the same, counted in dollars.
• For each clock, from one bar's close to the next:
  • kurtosis, how heavy the tails are (0 for a bell curve);
  • the Jarque–Bera test of whether the shape is a bell curve;
  • the Hill index, how fast the biggest moves thin out (higher means thinner tails), on the largest 1%.
• "Closer to a bell curve" meant lower kurtosis and a higher Hill index than hour bars, over the four years and in each year.

Part 2, the touch map.

• The touch map's own hourly data and starts.
• (a) today's map: its distances in typical moves, the volatility of the moment times the square root of the hours.
• (v) the volume clock: the same, but with the square root of the trading expected over the coming hours. The expectation used the usual pattern of each hour of the week, from the first half only, and the current level of trading.
• Both versions were asked about the same lines (today's), fitted on one half and judged on the other, then swapped.
• Set in advance, the bar: (v) wins only if it scores better (Brier) than (a) on all three coins, at both the day and the week, each gain clear of luck (outside a 95% band from re-drawing the judged weeks 1,000 times).

The maths, step by step

1. Why uneven trading makes fat tails. Say half of all hours are quiet, with a spread of 1, and half are busy, with a spread of 2 (a variance of 4). Each hour on its own is a bell curve.

• Mixed together, the kurtosis is 3 × (1² + 4²) ÷ 2 ÷ 2.5² − 3 = 1.08, above a bell curve's 0.
• Count time in trading instead, and each bar holds the same amount of activity, so the mixing goes away. That is Clark's idea.

2. A volume bar. For Bitcoin, a bar closed each time 4,234 BTC had traded, which gave as many bars as there were hours. A busy hour makes several bars; a quiet weekend night may make one bar over several hours.

3. The touch map on a volume clock. On Bitcoin, the busiest hour of the week traded 2.7 times the average hour. A one-hour line starting just before it is drawn about √2.72 = 1.65 times as wide. Over a whole week, the busy and quiet hours even out, and the lines stay as they are.

4. The score. Bitcoin, a day ahead: the Brier score fell from 0.1400 to 0.1353, 3.4% lower. The band of the gain ran from +0.0041 to +0.0052: clear of luck.

What we found

1. Volume bars are closer to a bell curve, on every coin and in every year.

• Lower kurtosis and a higher Hill index in all 15 cases: three coins, over the four years and in each of the four years.
• Still not a bell curve. Jarque–Bera rejected it for every clock on every coin, over the four years and in each year. The closest was Bitcoin's last year on volume bars: a kurtosis of 0.4, yet still rejected (a score of 40, against 6 for 5% odds).

2. Dollar bars: 12 of 15. They failed on Solana over the four years and on Solana and Bitcoin in October 2023 to October 2024. A fixed dollar size drifts as the price moves: in Solana's first year, at low prices, there were only 2,137 dollar bars against 8,759 hours.

3. The touch map: no, by the bar set in advance.

• The volume clock scored better an hour ahead (by 1.9% to 4.9%), four hours ahead (1.9% to 5.2%) and a day ahead (1.0% to 3.5%), on all three coins in both looks, clear of luck every time.
• A week ahead, nothing (−0.1% to +0.1%, all within luck). A week holds the whole weekly cycle of busy and quiet hours, so a clock built from that cycle has little to add.
• The bar asked for both the day and the week, so the answer is no.

Credits

• Benoit Mandelbrot, "The Variation of Certain Speculative Prices", Journal of Business, 1963: fat tails.
• Benoit Mandelbrot and Howard M. Taylor, "On the Distribution of Stock Price Differences", Operations Research, 1967: business time.
• Peter K. Clark, "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices", Econometrica, 1973: the volume clock, and step 1's mixing.
• Thierry Ané and Hélyette Geman, "Order Flow, Transaction Clock, and Normality of Asset Returns", Journal of Finance, 2000.
• Marcos López de Prado, Advances in Financial Machine Learning, Wiley, 2018: volume and dollar bars.
• Carlos M. Jarque and Anil K. Bera, "Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals", Economics Letters, 1980.
• Bruce M. Hill, "A Simple General Approach to Inference About the Tail of a Distribution", Annals of Statistics, 1975.
• Torben G. Andersen and Tim Bollerslev, "Intraday Periodicity and Volatility Persistence in Financial Markets", Journal of Empirical Finance, 1997: the daily pattern of activity.
• J.P. Morgan and Reuters, RiskMetrics — Technical Document, 1996: the volatility of the moment.
• Glenn W. Brier, "Verification of Forecasts Expressed in Terms of Probability", Monthly Weather Review, 1950.
• Bradley Efron, "Bootstrap Methods: Another Look at the Jackknife", Annals of Statistics, 1979, and Edward Carlstein (1986), Annals of Statistics: re-drawing blocks of a series.
• The touch map, the expected-trading clock for it, and these tests are our own.
• Data: Bybit's public candles.
• PerpGremlin and Claude, The volume clock: counting time in trading instead of hours, TerminalGremlin, 2026: the tests, the code and the words.