Mean reversion and pairs trading

Simulate a spread that snaps back to its average and trade it, until the relationship breaks.

Pull toward the average0.050

θ: the share of the gap to the average closed each day.

Long-run average0.0

Daily noise1.0

Entry threshold±1.5 sd

Bet on a snap back once the spread is this many standard deviations from its average.

Half-life13.5 days

Trades7

Win rate86%

Final profit+24.9In spread points, one unit

The first 100 days are used to measure the average and the bands. Teal shading: long the spread. Coral: short.

The spread closes half its gap to the average in about 13.5 days. Fading moves beyond ±1.5 standard deviations, the strategy won 86% of 7 trades. High win rates are typical here. The danger is the one trade that never comes back.

Profit over time, same shocks

Across 300 simulated markets

Profitable, mean-reverting spread100%

Profitable, random-walk spread50%

Average profit, mean-reverting34.3

One good chart proves little. The edge is real only if it holds across many markets, and only while the relationship itself holds.

Try this

Set the pull to zero. The bands still trigger trades, but across 300 markets the strategy is a coin flip.
Raise the pull toward 0.3. The half-life shrinks to a couple of days and trades come quickly.
Switch on the relationship break and watch one open trade wipe out months of small wins.

The math

x_{t+1} = x_t + \theta\,(\mu - x_t) + \sigma\,\varepsilon_t, \quad t_{1/2} = \frac{\ln 2}{-\ln(1-\theta)}

$x_t$: the spread on day t, for example the price gap between two related stocks
$\theta$: speed of mean reversion
$\mu$: long-run average
$\sigma$: size of daily noise
$\varepsilon_t$: a standard normal shock
$t_{1/2}$: half-life: days to close half the gap

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