mean-reversion

Mean Reversion

Mean reversion is the statistical tendency for prices, spreads, or other financial variables to return toward a long-run average after deviating from it. A mean-reverting series overshoots its mean, then corrects back -- creating predictable oscillations that can be traded.

When Mean Reversion Works

• Ranging markets: Sideways price action with clear support/resistance
• Pairs spreads: Spread between cointegrated assets reverts to equilibrium
• Oversold/overbought extremes: RSI, Bollinger Band, or z-score extremes in stationary series
• Funding rate arbitrage: Perpetual funding rates revert to baseline
• Stablecoin depegs: Classic mean-reversion opportunity (peg = known mean)
• Post-dump recovery: Brief mean-reversion windows after initial PumpFun dumps

When Mean Reversion Fails

• Strong trending markets (most crypto most of the time)
• Regime changes: what was stationary becomes non-stationary
• Structural breaks: token migration, protocol upgrade, delistings
• Low liquidity: wide spreads consume mean-reversion profits

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Testing for Mean Reversion

Before trading mean reversion, you must statistically confirm the series is mean-reverting. Three complementary tests:

1. Augmented Dickey-Fuller (ADF) Test

Tests the null hypothesis that a series has a unit root (non-stationary).

from scipy import stats
import numpy as np

def adf_test(series: np.ndarray, max_lag: int = 0) -> dict:
    """Run ADF test. Reject null (p < 0.05) → stationary → mean-reverting."""
    # See references/statistical_tests.md for full implementation
    # Use statsmodels.tsa.stattools.adfuller for production
    pass

• p < 0.01: Strong evidence of stationarity
• p < 0.05: Evidence of stationarity
• p > 0.10: Cannot reject unit root -- likely non-stationary

2. Hurst Exponent

Measures the long-range dependence of a time series.

Hurst Value	Interpretation	Trading Implication
H < 0.5	Mean-reverting	Trade mean reversion
H = 0.5	Random walk	No edge
H > 0.5	Trending	Trade momentum

def hurst_exponent(series: np.ndarray) -> float:
    """Compute Hurst exponent via R/S method. H < 0.5 → mean-reverting."""
    # See references/statistical_tests.md for full R/S algorithm
    pass

3. Variance Ratio Test

Compares variance of multi-period returns to single-period variance.

• VR < 1: Negative autocorrelation (mean-reverting)
• VR = 1: Random walk
• VR > 1: Positive autocorrelation (trending)

def variance_ratio(series: np.ndarray, q: int = 5) -> float:
    """Compute variance ratio at horizon q. VR < 1 → mean-reverting."""
    returns = np.diff(np.log(series))
    var_1 = np.var(returns)
    returns_q = np.diff(np.log(series[::q]))
    var_q = np.var(returns_q)
    return var_q / (q * var_1)

See references/statistical_tests.md for complete implementations and interpretation guides.

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Half-Life Estimation

The half-life tells you how many periods it takes for a deviation to decay to half its size. This is the single most important parameter for mean-reversion trading.

AR(1) Regression Method

Fit the autoregressive model: delta_X_t = alpha + beta * X_{t-1} + epsilon

def half_life(series: np.ndarray) -> float:
    """Estimate mean-reversion half-life from AR(1) regression.

    Returns:
        Half-life in periods. Negative means non-mean-reverting.
    """
    y = np.diff(series)
    x = series[:-1]
    x = np.column_stack([np.ones(len(x)), x])
    beta = np.linalg.lstsq(x, y, rcond=None)[0][1]
    if beta >= 0:
        return -1.0  # Not mean-reverting
    return -np.log(2) / np.log(1 + beta)

Using Half-Life

Parameter	Rule of Thumb
Lookback window	2x half-life
Holding period	1x half-life
Maximum hold	3x half-life (stop)
Signal recalc	0.5x half-life

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Z-Score Signal Framework

The z-score normalizes the deviation from the mean, providing standardized entry/exit signals.

z = (price - rolling_mean) / rolling_std

Signal Rules

Condition	Signal	Action
z < -2.0	Buy	Enter long (price below mean)
z > +2.0	Sell	Enter short (price above mean)
z crosses 0	Exit	Close position (returned to mean)
abs(z) > 3.0	Stop	Close position (reversion failed)

Lookback Window

Set the rolling window to approximately 2x the half-life:

def z_score_signals(
    prices: np.ndarray,
    lookback: int,
    entry_z: float = 2.0,
    exit_z: float = 0.0,
    stop_z: float = 3.0,
) -> np.ndarray:
    """Generate z-score-based mean-reversion signals.

    Returns:
        Array of signals: 1 (long), -1 (short), 0 (flat).
    """
    rolling_mean = pd.Series(prices).rolling(lookback).mean().values
    rolling_std = pd.Series(prices).rolling(lookback).std().values
    z = (prices - rolling_mean) / rolling_std
    # See scripts/mean_reversion_test.py for full signal generation
    ...

Position Sizing with Z-Score

Scale position size with z-score magnitude for better risk-adjusted returns:

size = base_size * min(abs(z) / entry_threshold, max_scale)

See references/strategy_design.md for complete entry/exit framework and sizing.

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Ornstein-Uhlenbeck (OU) Process

The OU process is the continuous-time model of mean reversion:

dX = theta * (mu - X) * dt + sigma * dW

Parameter	Meaning	Estimation
theta	Speed of mean reversion	From AR(1) beta: theta = -ln(1+beta)/dt
mu	Long-run mean	From AR(1) intercept: mu = -alpha/beta
sigma	Volatility of innovations	Residual std from AR(1)

Parameter Estimation

def estimate_ou_params(series: np.ndarray, dt: float = 1.0) -> dict:
    """Estimate OU process parameters from observed series.

    Returns:
        Dict with keys: theta, mu, sigma, half_life.
    """
    y = np.diff(series)
    x = series[:-1]
    x_with_const = np.column_stack([np.ones(len(x)), x])
    params = np.linalg.lstsq(x_with_const, y, rcond=None)[0]
    alpha, beta = params[0], params[1]

    theta = -np.log(1 + beta) / dt
    mu = -alpha / beta if beta != 0 else np.mean(series)
    residuals = y - (alpha + beta * x)
    sigma = np.std(residuals) * np.sqrt(2 * theta / (1 - np.exp(-2 * theta * dt)))

    return {
        "theta": theta,
        "mu": mu,
        "sigma": sigma,
        "half_life": np.log(2) / theta if theta > 0 else -1,
    }

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Strategy Types

Single-Asset Mean Reversion

Apply z-score framework directly to a token's price series. Works best on:

• Stablecoins (USDC/USDT spread)
• Tokens in established ranges
• After confirming stationarity with ADF test

Pairs Trading

Trade the spread between two cointegrated assets:

1. Confirm cointegration (see cointegration-analysis skill)
2. Compute spread: S = Y - beta * X
3. Apply z-score framework to the spread
4. Go long spread (buy Y, sell X) when z < -2
5. Go short spread (sell Y, buy X) when z > +2

Statistical Arbitrage

Multi-asset extension of pairs trading:

• Eigenportfolios from PCA of correlated assets
• Trade the smallest eigenvalue portfolios (most mean-reverting)
• Requires larger universe (10+ assets)

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Crypto-Specific Considerations