Volatility Calibration

An hour is not an hour

Every projection of how far price can still travel assumes the day spends its volatility at a steady rate. It does not. Halfway through the session between 59% and 68% of the movement is already behind you, and the number is different in every market.

Atty Labs · 19 September 2026 · 11 min read

7,351 sessions across four futures markets, 1-minute bars

68% of gold’s daily movement is done at the halfway mark on the clock

13% of gold’s movement is left for its entire final quarter

1.38× how much too wide a clock-based projection draws gold at 3pm London

Where this came from

We needed an honest answer to a small question while building an expected-move tool: at 11:40 in the morning, how much of today's range is still ahead? The textbook answer is the fraction of the clock still ahead, because that is what a square-root-of-time projection assumes. We measured it instead, on one market, and the assumption was wrong enough to matter. Then we measured it on three more and found that it is wrong by a different amount and in a different shape in each of them, which is the more interesting result and the reason this is an article rather than a constant in a script.

The assumption everything rests on

Price uncertainty is normally projected forward with the square root of time. Two hours of risk is √2 times one hour of risk. It is the arithmetic behind an expected-move cone, behind scaling a daily volatility number to an intraday horizon, and behind every "the market has X left in it today" estimate.

That arithmetic is exactly right if variance arrives at a constant rate through the session. It is wrong in proportion to how badly that fails. So we measured the rate directly: for every session, the squared one-minute returns, bucketed into 26 equal slices of the day, each session normalised to its own total so that a single violent day cannot dominate the average.

Cumulative share of the session’s variance against elapsed time. The dashed diagonal is the flat assumption. Every market bulges above it, and no two bulge the same way.

Every market front-loads, and none of them agree on the rest

The first quarter of the session holds 38.9% of ES's variance, 45.5% of NQ's, 34.7% of crude's and 35.8% of gold's, against the 25% a flat day would give it. At the halfway point on the clock, the share of the day's movement already spent is 58.6% on ES, 64.9% on NQ, 65.1% on crude and 68.3% on gold.

The end of the session is where the markets part company. Gold spends 13.0% of its variance in its entire final quarter and crude 17.0%, while NQ keeps 19.0% and ES keeps 23.5% — which is very nearly the flat 25%. The tidy story that markets are loud in the morning and quiet in the afternoon is true of gold, roughly true of crude, and not true of the S&P at all. ES has a closing auction at the end of its day, and the auction is a real event that puts the variance back.

Gold's front-loading has an equally concrete cause. The COMEX day session opens at 08:20 New York, ten minutes before the US data releases at 08:30. Its first quarter is not busy because mornings are busy; it is busy because it contains the calendar.

First quarter against last, per market, with the flat 25% marked. The spread in the last quarter — 13% on gold against 23.5% on ES — is the part that breaks a shared assumption.

What that does to a projection

Turn the profile into the number a projection actually needs. A wall-clock projection says the width still to come is proportional to √(1 − elapsed). The measured answer is proportional to √(1 − variance spent). Divide one by the other and you have how wrong the clock is, at every moment of the day.

At the midpoint the wall clock is 10% too wide on ES, about 20% too wide on NQ and crude, and 26% too wide on gold. Three-quarters of the way through, gold is the extreme: 1.38× — a cone drawn on the clock is nearly 40% wider than the market's own history says it should be. ES runs the other way late in the day and finishes 0.93×, slightly too narrow into the close, for the same auction reason.

The wall-clock projection as a multiple of the measured one. One number does not fix this for four markets: the curves cross, and one of them ends on the wrong side of the line.

This is an old idea with a specific modern price tag

That price moves on a clock of its own is not new. Clark proposed in 1973 that prices follow a process subordinated to a trading clock rather than to time¹, and Ané and Geman showed in 2000 that returns measured in transaction time are far closer to normal than returns measured in calendar time.² The intraday U-shape has been documented since the mid-1980s.³⁴ What this study adds is the size of the error, per market, in the one place a retail trader meets it: the width of the cone on the screen.

Does replacing the clock actually help?

A calibration is worth something only if it holds up on data it never saw. So each market's profile is built on an early period and then scored, unchanged, on a later one: 997 fitted and 247 held back on ES and NQ, 1,031 and 254 on crude, 3,413 and 165 on gold.

The test is whether the normalised move stays the same size all day. If a projection is calibrated, then dividing the actual move-to-the-close by the projected width should give a number with the same average at 10am as at 3pm. The dispersion of that number across the day is the score, and lower is better.

The variance clock wins in every market in the fitting period — ES 0.064 to 0.029, NQ 0.063 to 0.031, crude 0.100 to 0.045, gold 0.143 to 0.033 — and it wins in every market in the held-back period too, by a narrower margin. Gold gains most because the wall clock suits it worst.

Gold is also where the honest warning lives. Its held-back dispersion is 0.107 against 0.033 in fitting, a much larger degradation than the other three. Sixteen years of gold is not sixteen years of one behaviour, and its profile deserves more suspicion than ES's does.

Each row is one market: the pair of dots on top is the fitting period, the faded pair below it the held-back period. Every arrow points the same way.

What the dispersion number is measuring. The wall clock runs too wide in the middle of the day and too narrow at the end; on gold the effect is severe.

The timing is market-specific. The shape is not.

One more question has to be answered before any of this is usable: once the clock is right, where do the outcomes actually fall? A Gaussian assumption says 68% of them land inside one sigma and 95% inside two.

Measured, the 68% boundary sits at 0.88 sigma on ES, 0.87 on NQ, 0.89 on crude and 0.81 on gold. The 95% boundary sits at 2.31, 2.21, 2.18 and 2.21. Every market is tighter than the bell curve in the middle and wider in the tail, and the four are close enough to each other to share one pair of multipliers — which is striking, because nothing else in this study is shared between them.

That is the finding worth carrying away: when the variance arrives is specific to each market; what the distribution looks like once it does is not. A single distributional shape with four different clocks describes all four markets better than one clock and one shape describes any of them.

Measured containment against the Gaussian 1.00 and 2.00. Four unrelated markets, nearly the same answer.

The same fact seen directly: more quiet days than a bell curve allows, and a longer tail on both sides. Excess kurtosis runs from 3.6 on NQ to 6.2 on ES.

What to do with this

Stop reading the clock as a progress bar. At lunchtime the session is half over and roughly two-thirds done. A quiet afternoon is not the market waiting to move; on gold and crude it is the market having already moved.
Don’t carry one market’s profile to another. Running the S&P's shape on gold overstates gold's late-day risk by nearly 40% at the three-quarter mark. If a tool draws a cone, it needs a profile per instrument, or it is guessing in three of four cases.
Size the late session differently from the early one. The same stop distance means something different at 9:45 and at 2:30, because the amount of movement left to hit it is not the same. That is a risk statement, not a signal.
Use measured containment, not sigma multiples. One sigma contains 68% of outcomes only if outcomes are Gaussian, and these are not. The honest band is drawn at the quantile you actually measured.

What we’d test next

Whether a profile should be conditioned rather than fixed. These are averages over years. A day with a scheduled release at 2pm has a different shape from a day without one, and an event-aware profile is the obvious next version.
Whether the profile is stable enough to be worth re-fitting. Gold's degradation out of sample says the answer is different per market; a rolling re-fit against a frozen one is a cheap test.
Whether the containment multipliers hold at horizons shorter than the rest of the session. Everything here is measured to the close.

Method

Data: 1-minute futures bars, day session only. ES and NQ 1,244 sessions each and CL 1,285, all 2021-08-30 to 2026-08-27; GC 3,578 sessions, 2010-06-07 to 2026-09-09. Sessions are the conventional day sessions, which are not the same window: 09:30–16:00 ET for the index contracts, 09:00–14:30 for crude, 08:20–13:30 for gold. A session needs at least 75% of its expected bars to be included.
The variance profile: Squared one-minute log returns, measured from the session open, bucketed into 26 equal slices of the session. Each session is normalised to its own total before being added to the average, so the profile describes shape rather than level and a single violent day cannot dominate it. Profiles are built on the fitting period only.
Fitting and holdout: ES, NQ and CL are fitted up to 2025-09-01 and scored on everything after. GC is fitted up to 2024-01-01 and scored after, on a smaller held-back sample of 165 sessions, which is the weakest holdout in the study. No parameter is re-fitted on held-back data.
Scoring a clock: At each of nineteen checkpoints through the session, the actual move from that point to the close is divided by what each clock projects is left: the day's ex-ante volatility (the standard deviation of the previous 20 sessions' close-to-open moves) times √(1 − elapsed) for the wall clock, or √(1 − variance spent) for the variance clock. A calibrated clock produces the same average absolute value at every checkpoint; the score is the coefficient of variation of that average across the day.
Containment: The 68.3rd and 95.4th percentiles of the absolute normalised move, pooled across all checkpoints in the fitting period. The distribution chart uses one observation per session instead — close against open, divided by the previous 20 sessions' volatility — so that it contains no overlapping windows.
Limits: No costs, no trades and no strategy anywhere in this study; it is a measurement of when markets move, not a claim that knowing it makes money. Profiles are unconditional averages and will misdescribe any individual day with a scheduled event in it. Gold's holdout is small and its out-of-sample degradation is reported rather than smoothed over.

References

Clark, Peter K. “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices.” Econometrica 41, no. 1 (1973): 135–155. Link
Ané, Thierry, and Hélyette Geman. “Order Flow, Transaction Clock, and Normality of Asset Returns.” The Journal of Finance 55, no. 5 (2000): 2259–2284. Link
Wood, Robert A., Thomas H. McInish, and J. Keith Ord. “An Investigation of Transactions Data for NYSE Stocks.” The Journal of Finance 40, no. 3 (1985): 723–739. Link
Harris, Lawrence. “A Transaction Data Study of Weekly and Intradaily Patterns in Stock Returns.” Journal of Financial Economics 16, no. 1 (1986): 99–117. Link
Andersen, Torben G., and Tim Bollerslev. “Intraday Periodicity and Volatility Persistence in Financial Markets.” Journal of Empirical Finance 4, nos. 2–3 (1997): 115–158. Link
Mandelbrot, Benoit. “The Variation of Certain Speculative Prices.” The Journal of Business 36, no. 4 (1963): 394–419. Link

Historical behaviour of futures contracts, not a strategy or a recommendation. Full disclaimer.