The Lab

The maths, step by step

The example: BTC is at 60,000. Will it touch 63,000 (3,000 higher) within one day? Say BTC typically moves 3% in a day these days.

1. The distance, in percent: 3,000 is 5% of 60,000. The formula measures it as a growth rate, which makes it a little smaller: 4.88%.
2. The volatility, called σ (sigma): how much the price typically moves in a set time, up or down. Here it is 3% a day.
   • In the tests it was measured from the last few days of hourly moves, with the newest hours counting most (an exponentially weighted average, as in J.P. Morgan's RiskMetrics). It looks back about four days.
3. The volatility over the wait grows with the square root of the time, written √time.
   • Over one day: 3% × √1 = 3%.
   • Over four days: 3% × √4 = 3% × 2 = 6%, not 12%. Up days and down days partly cancel, so the spread grows more slowly than the time.
4. z = distance ÷ volatility over the wait. For one day: 4.88 ÷ 3 = 1.63. The level is 1.63 typical moves away.
5. The bell curve, called N (the normal distribution), describes where the price ends up under this model. Most endings are near the start, and fewer are far out. The share of the curve beyond 1.63 typical moves is 5.2%. That is the chance BTC ends the day above 63,000.
6. Double it for a touch: 2 × 5.2% = 10.4%, written 2·N(−z).
   • Why double: a price that touches 63,000 and falls back has a mirror image that went on up and ended above it. The two are equally likely.
   • So touches are twice the "ends above" share.
7. Drift is any steady lean up or down. We assumed none. The exact formula still carries a tiny lean from the maths of percentages. It gives 10.1% in place of 10.4%; over a day the difference is small.

The same level, waited four days: z = 4.88 ÷ 6 = 0.81, and the chance rises to 42% (41% with the tiny lean).

What "fat tails" means: big moves happen more often in crypto than the bell curve allows. So the curve's thin far end underprices far levels. That is what the test below found.

The repair, the market's own curve: take each coin's first year. Sort every test by its z, and count how often levels at each z were really touched. For example, of all the tests with z between 2 and 3, what share touched? Then use that table in place of the bell curve.

The score: each prediction is scored by the square of its miss.

• A 30% prediction that touched scores (1 − 0.3)² = 0.49.
• If it didn't touch: (0 − 0.3)² = 0.09.
• The average over all predictions is the score. Lower is better. (Its name is the Brier score, after Glenn Brier, 1950.)

The test

How: BTC, ETH and SOL hourly candles from Bybit, October 2022 to October 2026. We started a test every two hours, with waits of 4 hours, a day, 3 days and a week, and levels from 0.5% to 12% above and below the price. That made about 2.9 million predictions. First, on a fake market built to follow the formula's own assumptions, its predictions matched within a point or two.

What the real market did:

• Rare touches came far more often than the formula says.
  • Where it said 0.6–2.4% (z between 2 and 3), 4–7% happened.
  • Beyond z = 3 it says "almost never". In fact 2.3% happened between z = 3 and 4, and 0.5% beyond 4.
  • Levels it gave under 2% were touched 10.5% of the time within a week.
  • These are crypto's fat tails and its sudden changes of mood.
• Likely-looking touches came a little less often, by 4–5 points in the 30–70% band. Big hours push the volatility estimate up, then the market calms before reaching the level.
• The longer the wait, the worse the tails. Volatility itself changes over a week, and the formula assumes it doesn't.
• A slower volatility memory worked better. About four days beat 16 hours and a day and a half, on every coin.

The repair that worked on unseen years: the market's own curve. From each coin's first year, learn how often levels were really touched at each z. Use that in place of the formula. On the next three years, which it never saw, its odds were right to within 2.5 points in every band. It matched or beat the formula on every coin in every year.

How much does today's volatility add? We compared the own curve with simply knowing each level's usual touch rate on that coin. Knowing today's volatility lowered the score (the average squared miss) by only 0.4–7.6%, with no pattern from year to year:

What it means:

• For liquidation prices and far stops, don't trust the plain formula. It underprices far levels several times over, exactly where the damage is.
• The market's own curve gives fair odds. That is honest information to plan with, not a trading edge.
• Cautions:
  • The tests overlap (a start every two hours, waits up to a week), so the counts overstate the evidence.
  • Candles show the last traded price, but liquidations fire on the mark price, so a wick can overstate a liquidation touch.
    • A wick is the thin line above or below a candle's body: how far the price went during the candle before coming back. For example, a candle that opens at 100, dips to 95 and closes at 99 has a wick down to 95.
  • Levels below the price were touched a little less than levels above at the same odds. These were four mostly rising years, so read that as the period, not a law.
  • Three coins, and the candles are one exchange's (Bybit).

Update, 27 October 2026: where the idea comes from

Added after this page was published. The text above is left as it was.

• "Will the price touch this level?" is the first passage problem: when does a path that wanders at random first reach a line? Louis Bachelier modelled prices as such a path in 1900.
• Its best-known use in finance is the barrier option: an option that switches off, or on, when the price touches a level. Robert Merton priced one, the "down-and-out" call, in 1973. The doubling rule in step 6 above is at its heart.
• The maths is worked through above, in "The maths, step by step".
• The catch, in two parts:
  • Big moves come more often than the bell curve allows (Benoit Mandelbrot, 1963). That is why far levels were touched several times more often than the formula said.
  • Volatility changes over time. Robert Engle's ARCH model (1982) was the first to describe calm and wild spells as they come and go. The formula takes one volatility for the whole wait. For example: fed a quiet 2% a day, it gives a level 6% away about a 0.4% chance of a touch in a day. If the next day turns out to move 4%, the same formula gives about 14%.
• Theirs and ours. The model, the formula, the RiskMetrics volatility and the Brier score are theirs, credited below. The four-year test and the market's own touch curve are ours.

Credits

• Louis Bachelier, "Théorie de la spéculation", Annales scientifiques de l'École Normale Supérieure, 1900.
• Paul A. Samuelson, "Rational Theory of Warrant Pricing", Industrial Management Review, 1965: geometric Brownian motion.
• The doubling rule in step 6 is the "reflection principle", a standard result for Brownian motion. We aren't sure who first stated it, so we don't name anyone.
• J.P. Morgan and Reuters, RiskMetrics — Technical Document, 1996: the exponentially weighted volatility.
• Glenn W. Brier, "Verification of Forecasts Expressed in Terms of Probability", Monthly Weather Review, 1950.
• Benoit Mandelbrot, "The Variation of Certain Speculative Prices", Journal of Business, 1963: fat tails and volatility clustering.
• Robert C. Merton, "Theory of Rational Option Pricing", Bell Journal of Economics and Management Science, 1973: the down-and-out option.
• Robert F. Engle, "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation", Econometrica, 1982: volatility that changes over time.
• Our finding that the formula underprices far levels repeats, on crypto, what Mandelbrot's fat tails predict. The market's own touch curve is ours.
• Data: Bybit's public candles.
• PerpGremlin and Claude, First passage: the chance a price touches a level, TerminalGremlin, 2026: the tests, the code and the words.