The Lab

Where it comes from

• The numbers. Leonardo of Pisa, known as Fibonacci, set a puzzle about breeding rabbits in his Liber Abaci (1202). Its answer is the sequence 1, 1, 2, 3, 5, 8, 13, ..., each number the sum of the two before. Indian scholars of poetry had found the same sequence centuries earlier, counting rhythms.
• The golden ratio. Divide each number by the next and the answer settles near 0.618. Its inverse, 1.618, is the "golden ratio", which Euclid had already described as a way of dividing a line. Johannes Kepler noticed the link to the sequence.
• The waves. Ralph Nelson Elliott, an American accountant, studied decades of stock charts in the 1930s. He described markets as moving in a repeating pattern of five waves with the trend and three against it, in The Wave Principle (1938). In Nature's Law (1946) he tied the waves' sizes to Fibonacci ratios.
• Who kept it going. Hamilton Bolton wrote about the waves through the 1950s and 60s. A. J. Frost and Robert Prechter's Elliott Wave Principle (1978) made them widely known, and is the standard text.
• Retracement levels. Traders draw lines at 23.6%, 38.2%, 50%, 61.8% and 78.6% of a move, as places a pullback might stop. We don't know who first used them this way, so we don't name anyone.

The maths, step by step

1. The sequence and its ratios. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

• 89 ÷ 144 = 0.618, 55 ÷ 144 = 0.382, 34 ÷ 144 = 0.236.
• As the numbers grow, these ratios settle on 1/φ, 1/φ² and 1/φ³, where φ = 1.618... is the golden ratio.
• 50% isn't a Fibonacci ratio at all. 78.6% is the square root of 0.618. Traders added them.

2. The levels. A price rises from 100 to 120, a move of 20. Pulled back by each share:

3. Why levels "work" on a chart. Give each level a band of ±2% of the move, since no price stops exactly on a line. Five bands cover 20% of all the places a pullback could stop between 0% and 100%.

• So a pullback that stops at random lands "on a Fibonacci level" one time in five.
• On any chart with many pullbacks, plenty will touch a level. That's what makes the levels look right. The test is whether they're touched more often than a band of the same width somewhere else.

What others found

This section is other people's work, not ours.

• Elliott waves have little rigorous testing. We found few careful academic tests of Elliott waves or Fibonacci levels, so we cite none for or against.
• Technical analysis more broadly:
  • William Brock, Josef Lakonishok and Blake LeBaron (1992) found simple moving-average rules seemed to work on a century of the Dow Jones.
  • Ryan Sullivan, Allan Timmermann and Halbert White (1999) allowed for how many rules had been tried. The best rules didn't keep working in the years after the original study.
  • Andrew Lo, Harry Mamaysky and Jiang Wang (2000) found some chart patterns carry a little information about what comes next. That's less than traders claim, but more than none.

What we tested and found

This section is ours.

• Fibonacci levels in trading: over two years of Bitcoin, Ether and Solana, trades placed at Fibonacci zones did no better than trades at plain fixed distances. Neither placement won steadily.
• Turning points: turns landed within ±2% of a Fibonacci ratio 25.5% of the time, and within the same-width bands just beside them 26.6% of the time. No clustering at the ratios.
• Elliott waves, counted as they happen over six years of hourly prices:
  • none of the Elliott zones held more turns than random zones of the same width, on any coin or size;
  • none of the seven wave trades beat random entries with the same stop and target;
  • wave 4s did end near 38.2% very often, but the hard rules force that: wave 4 can't fall into wave 1's range, which squeezes it into a narrow band. Inside that band, 38.2% was ordinary.
• More than one count often fits. At 16% to 28% of turns, the rules allowed more than one Elliott count. The labelling itself is often a choice.

The catch

• Hindsight makes them look perfect. On a finished chart, you can always find a swing that a Fibonacci level matches, and a count that fits the waves. The test is whether they help before the turn.
• Many levels, many chances. Five levels with a little leeway cover a fifth of all possible stops (step 3). Add more swings and more time frames, and almost every turn is "near" something.
• The rules shape the results. Some Elliott guidelines look true because the rules force them, as with wave 4.
• Popularity can't be ruled out as a cause. If many traders act at the same lines, prices might react there for a while. Our tests didn't find that, but they cover only a few coins and years.

Credits

• Leonardo of Pisa (Fibonacci), Liber Abaci, 1202.
• Euclid, Elements: the "extreme and mean ratio".
• Ralph Nelson Elliott, The Wave Principle, 1938, and Nature's Law, 1946.
• A. J. Frost and Robert R. Prechter Jr., Elliott Wave Principle, New Classics Library, 1978.
• William Brock, Josef Lakonishok and Blake LeBaron, "Simple Technical Trading Rules and the Stochastic Properties of Stock Returns", Journal of Finance, 1992.
• Ryan Sullivan, Allan Timmermann and Halbert White, "Data-Snooping, Technical Trading Rule Performance, and the Bootstrap", Journal of Finance, 1999.
• Andrew W. Lo, Harry Mamaysky and Jiang Wang, "Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation", Journal of Finance, 2000.
• Our tests: "Fibonacci confluence: does it time trades better than chance?", "Liquidation levels after a turn: targets, or just distance?" and "Elliott waves: do the zones and the wave trades hold up?".
• PerpGremlin and Claude, Fibonacci and Elliott waves: where the numbers come from, TerminalGremlin, 2026: the tests, the code and the words.