The Lab

Where it comes from

• The word "regression" comes from Francis Galton (1886). He saw that very tall parents tend to have children who are tall, but less so: an extreme tends to be followed by something closer to the average.
• The standard model is George Uhlenbeck and Leonard Ornstein's (1930), from physics: a particle knocked about at random but pulled back towards a centre. In finance it became the model for interest rates (Oldřich Vašíček, 1977) and for spreads between related prices.
• Prices themselves were long thought to wander at random. Maurice Kendall (1953) found stock prices moved with little memory. Paul Samuelson (1965) and Eugene Fama (1965 and 1970) gave the reasons: in a market full of traders, a predictable move gets traded away.
• Then came evidence of some coming back:
  • over three to five years, stock losers recovered (Werner De Bondt and Richard Thaler, 1985);
  • stock prices showed some reversion over years (James Poterba and Lawrence Summers, 1988; Fama and Kenneth French, 1988);
  • weekly and monthly losers bounced (Narasimhan Jegadeesh, 1990; Bruce Lehmann, 1990);
  • the gap between two linked stocks tended to close (Evan Gatev, William Goetzmann and Geert Rouwenhorst, 2006, testing "pairs trading").

The maths, step by step

1. The half-life. If a fixed share of a gap closes each day, the gap shrinks like interest in reverse. The half-life is how long half of it takes to close.

• Example: 0.83% a day closes. Half has closed after ln 0.5 ÷ ln(1 − 0.0083) = 83 days.
• A tenth an hour would halve it in 6.6 hours.

2. The z-score: how unusual is today? Take a window of readings, their average and their usual spread (σ).

• Example: readings 0, 0, 0, 0 and 0.1. The average is 0.02 and σ is 0.04. The last reading is (0.1 − 0.02) ÷ 0.04 = 2.0 spreads out.
• A reversion trader would sell at 2 and buy at −2.

3. The trap. The gap jumps to 0.1 and simply stays there. As the window fills with the new level, the z-score falls:

2.00 → 1.22 → 0.82 → 0.50

The gap never closed. The average moved up to meet it. A rule that exits "when the z-score is back near zero" closes the trade and books the loss as if it had come back.

4. A test for coming back. For a price that wanders at random, the spread over 4 days is √4 = 2 times a day's.

• At 3% a day, a random walk's 4-day spread is 6%.
• If the real 4-day spread is only 5%, the "variance ratio" is (5 ÷ 6)² = 0.69. Below 1, moves partly undo each other. That's reversion.
• Above 1 would mean moves tend to continue: momentum.

What we tested and found

This section is ours.

• Pairs of coins: six pairs, four years, 576 settings. No pair's gap reliably came back: gaps took from 20 to over 1,000 days to halve, and every pair lost money. More than half of trades won, but the losers were larger. Most "came back" only in the sense of step 3: 204 of 208 Ether/Bitcoin trades on a 7-day window closed "back at 0".
• Buying sharp drops: on fresh 2020–21 prices, the bounce after a sharp one-hour fall was no bigger than buying at that hour on a nearby day.
• Grid bots: on 14 coins, real prices paid a grid more than at least 15 of 20 shuffled copies on only 2 to 4 of the 14. With no edge, a coin does that by luck 6 times in 21, so luck alone gives about 14 × 6 ÷ 21 = 4 coins. Over those 30 months, prices didn't return to their range more than a random path would.
• A crowded market's bounce was the whole market bouncing, not the coin.

So in our crypto tests, the coming back that the theory describes for gaps and short spans didn't show up strongly enough to trade after costs.

Credits

• Francis Galton, "Regression towards Mediocrity in Hereditary Stature", Journal of the Anthropological Institute, 1886.
• G. E. Uhlenbeck and L. S. Ornstein, "On the Theory of the Brownian Motion", Physical Review, 1930.
• Oldřich Vašíček, "An Equilibrium Characterization of the Term Structure", Journal of Financial Economics, 1977.
• Maurice G. Kendall, "The Analysis of Economic Time-Series, Part I: Prices", Journal of the Royal Statistical Society, 1953.
• Paul A. Samuelson, "Proof That Properly Anticipated Prices Fluctuate Randomly", Industrial Management Review, 1965.
• Eugene F. Fama, "The Behavior of Stock-Market Prices", Journal of Business, 1965; "Efficient Capital Markets: A Review of Theory and Empirical Work", Journal of Finance, 1970.
• Werner F. M. De Bondt and Richard Thaler, "Does the Stock Market Overreact?", Journal of Finance, 1985.
• James M. Poterba and Lawrence H. Summers, "Mean Reversion in Stock Prices: Evidence and Implications", Journal of Financial Economics, 1988.
• Eugene F. Fama and Kenneth R. French, "Permanent and Temporary Components of Stock Prices", Journal of Political Economy, 1988.
• Narasimhan Jegadeesh, "Evidence of Predictable Behavior of Security Returns", Journal of Finance, 1990; Bruce N. Lehmann, "Fads, Martingales, and Market Efficiency", Quarterly Journal of Economics, 1990.
• Andrew W. Lo and A. Craig MacKinlay, "Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test", Review of Financial Studies, 1988.
• Evan Gatev, William N. Goetzmann and K. Geert Rouwenhorst, "Pairs Trading: Performance of a Relative-Value Arbitrage Rule", Review of Financial Studies, 2006.
• Our tests: "Pairs trading: does the gap between two coins come back?", "Buying a sharp drop: does the market bounce the next day?", "Grid bots on 14 coins: what 30 months of prices say", and "Forced selling and crowded trades: open interest, funding, long/short ratios".
• PerpGremlin and Claude, Mean reversion: does what drifts away come back?, TerminalGremlin, 2026: the tests, the code and the words.