Mean Reversion Explained | Generated by AI

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Question: What does “mean reversion” mean?

Answer:

Mean reversion is the statistical tendency of a series to pull back toward its long-run average after deviating from it. Formally: if x_t is far above μ, the expected next move E[x_{t+1} - x_t] is negative, and vice versa. The strength of the pull scales with the size of the deviation.

The canonical continuous-time model is the Ornstein–Uhlenbeck process:

dx_t = θ(μ - x_t) dt + σ dW_t

• μ — long-run mean the process reverts to
• θ — reversion speed (larger θ = faster snap-back); half-life of a deviation is ln(2)/θ
• σ — volatility
• dW_t — Brownian noise

Discretized, this is just an AR(1):

import numpy as np

def ou(n=1000, theta=0.1, mu=0.0, sigma=0.2, x0=2.0, dt=1.0):
    x = np.empty(n); x[0] = x0
    for t in range(1, n):
        x[t] = x[t-1] + theta*(mu - x[t-1])*dt + sigma*np.sqrt(dt)*np.random.randn()
    return x

Note x[t] = (1 - θ)·x[t-1] + θ·μ + noise — the coefficient (1-θ) < 1 is what guarantees reversion. If that coefficient is ≥ 1 the series is a random walk (no reversion) or explosive.

Where it shows up:

• Quant trading — pairs/stat-arb: trade the spread of two cointegrated assets, betting it returns to its mean. Test stationarity with ADF; estimate θ via OLS on the AR(1).
• Volatility / rates models — Vasicek, CIR for interest rates use OU dynamics.
• Performance attribution — “regression to the mean”: extreme outliers tend to be followed by less extreme values, a pure sampling artifact, not a force.

The mirror image is momentum/trending behavior, where deviations persist or compound rather than decay (coefficient ≥ 1).

Quick mean-reversion test — fit AR(1), check whether the slope is significantly below 1:

from statsmodels.regression.linear_model import OLS
import statsmodels.api as sm

x = ou()
X = sm.add_constant(x[:-1])
beta = OLS(x[1:], X).fit().params[1]   # < 1 ⇒ mean-reverting
half_life = -np.log(2) / np.log(beta)  # in steps

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