Forex Maths & Physics
Mean Reversion — The Elastic Band of the Markets
If the random walk is a drunk wandering aimlessly, mean reversion is a drunk on a leash. Stretch too far from the post and something pulls you back. Many spreads, rate differentials and relative-value pairs behave this way — and the maths is surprisingly clean.
The model: Ornstein–Uhlenbeck
The standard description is the Ornstein–Uhlenbeck process:
Read it in plain English: the change in X has two parts. The first, θ(μ − X), is the pull back toward the long-run mean μ — the further you are, the harder the tug, scaled by the speed θ. The second, σ dW, is the usual random noise trying to push you away.
The half-life — the number that matters
The single most useful quantity is the half-life: how long it takes, on average, for a deviation to shrink by half.
- A short half-life (hours, days) means a fast, tradeable snap-back.
- A long half-life (months) means you might be right but tie up capital forever — and the "mean" may drift before you get paid.
The trap: when the band snaps
Mean reversion is seductive because it feels like buying low and selling high. The danger is regime change. The "fair value" μ is not fixed by nature — it can shift when a central bank changes policy or a structural break occurs. A pair that reverted reliably for years can suddenly trend away and never come back. That is the elastic band snapping rather than pulling.
How to use it sensibly
- Estimate the half-life before trading, not after. It sets your holding period and your stop.
- Test that the mean is stable. Cointegration (a future topic) is the proper tool for confirming a real long-run relationship rather than a coincidence.
- Cap the loss. Pure mean-reversion strategies make many small gains and occasionally one large loss when the relationship breaks — exactly the fat-tail shape to respect.
Mean reversion is real, common, and profitable — right up until the leash breaks. The maths tells you how hard the pull is; your risk management decides whether you survive the day it stops pulling.
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