Mean reversion vs momentum

The question

"The home side has steamed from 2.50 to 2.30 in half an hour. Will it keep shortening, or come back out?"

Traders tell both stories. Momentum says a move is information arriving, so follow it. Mean reversion says it's an overreaction, so fade it. Betfair price movements can show either, or neither, and the only way to know is to count.

The idea in one sentence

If moves tend to continue you have momentum, if they tend to reverse you have mean reversion, and if the next move is a coin toss you have a random walk, which is what an efficient market should look like.

The picture

Picture three prices over the hour before kick-off, all starting at 2.50 and all shortening to 2.30 in the first half-hour:

• Momentum. The move carries on to 2.20 by kick-off. Money arrived for a reason, perhaps team news, and more of it keeps coming.
• Mean reversion. The price drifts back to 2.40. A burst of money pushed it too far, and the market corrects.
• Random walk. From 2.30 it's equally likely to go either way. The first move told you nothing about the second.

Here's why the random walk is the starting point. If everyone could see that steamers keep steaming, they'd back them straight away, and the move would happen all at once. Predictable moves get traded away. So any pattern you find has to be tested against the random walk, not assumed.

Two sensible hypotheses, both worth testing, never assuming:

• Moves on real news (line-ups, injuries) are more likely to stick, because the fair price has genuinely changed.
• Moves on money alone, with no news, are more likely to come back part of the way.

Worked Betfair example

You test one rule on pre-match Match Odds: "If the home side's implied probability rises by 2 points or more between 60 and 30 minutes before kick-off, back it at 30 minutes and green up at kick-off." (Illustrative figures.)

1. Measure in probability, not ticks. A move from 2.50 to 2.30 is 1 ÷ 2.50 = 40.0% to 1 ÷ 2.30 = 43.5%, a rise of 3.5 points. Ticks are uneven (Lesson 8.2), so probability is the fair ruler.
2. Count what happened next. You find 800 such moves. In the next half-hour the price kept shortening in 452 (56.5%) and came back in 348.
3. Test against a coin toss. You'd expect 400 ± √(800 × 0.25) = ±14.1. You're (452 − 400) ÷ 14.1 = 3.7 standard errors above chance. The exact chance of 452 or more from 800 coin tosses is about 0.0001. That's momentum, statistically.
4. Price the trade. Say the typical case is a back of £100 at 2.30.
   • When the move continues, the price closes around 2.24. Lay £100 × 2.30 ÷ 2.24 = £102.68, locking +£2.68, or +£2.62 after 2% commission.
   • When it reverses, the price closes around 2.36. Lay £97.46, locking −£2.54.
5. Expected value. 0.565 × £2.625 − 0.435 × £2.542 = +£0.38 per £100 traded. Break-even hit rate is 2.542 ÷ (2.542 + 2.625) = 49.2%.
6. The honest checks. Was the 2-point trigger chosen after trying 1, 1.5, 2.5 and 3? Then the p-value is flattered (Lesson 7.4). Walk it forward season by season (Lesson 7.6), and use the prices on offer at 30 minutes and at kick-off, not the best seen (Lesson 7.5).

Verdict: a small, measurable momentum effect that pays about 38p per £100 before any hidden testing is accounted for. Worth confirming on unseen seasons; not worth staking on from one backtest.

The CLV link

Notice what this trade really is: a bet that your 2.30 will beat the closing price. When moves continue, backing at 30 minutes earns positive closing line value. Momentum and CLV are the same question asked two ways, which is why a genuine momentum edge shows up in CLV first.

The formula

Continuation rate test

$$z = \frac{k - n/2}{\sqrt{n/4}}$$

• k is the number of moves that continued.
• n is the number of moves in the sample.

In plain English: how far the continuation count sits from a coin toss. Well above zero is momentum; well below is mean reversion.

Lag-one autocorrelation

$$\rho_1 = \frac{\sum_{t} (x_t - \bar{x})(x_{t+1} - \bar{x})}{\sum_{t} (x_t - \bar{x})^2}$$

• x_t is the change in implied probability in period t (say, each 5 minutes).
• x̄ is the average change.

In plain English: does one period's move tend to be followed by a move the same way (positive ρ), the opposite way (negative ρ), or neither (ρ near zero)?

Variance ratio

$$\text{VR}(q) = \frac{\operatorname{Var}\big(x_t + x_{t+1} + \dots + x_{t+q-1}\big)}{q \times \operatorname{Var}(x_t)}$$

• q is the number of periods added together, e.g. 6 five-minute moves for a half-hour.
• Var is the variance.

In plain English: for a random walk, a half-hour move is exactly as variable as six five-minute moves added up, so VR = 1. Above 1, moves build on each other (momentum). Below 1, they cancel out (mean reversion).

Try it

Pen and paper: after 600 drifts of 2 points or more, the price kept drifting 270 times and came back 330 times. Is that momentum or mean reversion, and is it convincing?

Answer: 270 ÷ 600 = 45% continued, so it leans to mean reversion. z = (270 − 300) ÷ √150 = −2.45, a two-sided p of about 0.016. Interesting on one planned test, not convincing if it was one of many.

Common mistakes

• Measuring moves in ticks. Five ticks at 1.50 and five ticks at 5.0 are very different moves. Convert to implied probability.
• Remembering the big steamers. The moves that kept going are memorable; the ones that fizzled aren't. Count every case the rule would have caught.
• Using the closing price you already know. "It steamed and kept going" is only useful if you can spot it before it finishes. Trigger on what was known at the time.
• Confusing in-play time decay with momentum. In-play, a price drifting as minutes pass with no goal is predictable time decay, not momentum (Lesson 8.5).
• Assuming a pattern lasts. Once a pattern is known, it gets traded away. Monitor it like any strategy (Lesson 7.3).

Markets that learn are pitfall 3: why good models still lose money.