Z-Score Mean Reversion

Z-score is the mathematical concept behind Bollinger Bands. BB shows price ± 2 standard deviations as bands on a price chart. Z-score shows the same information as an oscillator (-2 to +2). Z-score of 2 = price at upper Bollinger Band.

20 periods is standard for daily charts. Shorter periods (10-15) are more responsive but noisier. Longer periods (30-60) are smoother but lag more. Start with 20 and backtest alternatives for your instrument.

Yes! Unlike bounded indicators (RSI 0-100), z-score is unbounded. During extreme events, z-score can reach ±4, ±5, or beyond. These are very rare (0.006% for ±4) and often indicate regime change or major events.

Z-score assumes the mean is stable. During trends or regime changes, the mean itself shifts, invalidating the z-score signal. Also, markets aren't perfectly normal - fat tails mean extremes occur more often than theory suggests.

Primary target is z-score = 0 (price at mean). This represents complete mean reversion. Extended target is the opposite threshold (z = +2 for a long from -2). Exit 50% at z = 0, trail the rest.

Half-life is the expected time for z-score to halve (e.g., from -2 to -1). Shorter half-life means faster reversion and more tradeable. Instruments with half-life > 20 days are close to random walks and shouldn't be traded with z-score.

Use the Engle-Granger or Johansen cointegration test. In Python: `from statsmodels.tsa.stattools import coint; result = coint(price_a, price_b)`. P-value < 0.05 indicates cointegration. Only trade pairs that pass this test.

Hurst exponent (H) measures mean reversion tendency. H < 0.5 = mean-reverting (good for z-score). H = 0.5 = random walk (no edge). H > 0.5 = trending (avoid z-score). Calculate using R/S analysis or DFA.

Use linear regression: Price_A = alpha + beta × Price_B. The beta is the hedge ratio. For every share of A, trade beta shares of B. This creates a market-neutral position where only the spread matters.

Dynamic thresholds (based on historical percentiles) often work better because different instruments have different z-score distributions. Calculate 5th/95th percentile of historical z-scores for instrument-specific thresholds.

OU process models mean reversion: dX = θ(μ-X)dt + σdW. θ = mean reversion speed, μ = long-term mean, σ = volatility. Half-life = ln(2)/θ. Use regression to estimate parameters: ΔX = a + bX + ε, where θ = -b.

Use rolling Hurst exponent, rolling half-life, or Hidden Markov Models. If half-life exceeds 2× historical average or Hurst crosses above 0.5, pause z-score trading. Regime detection prevents trading during trends.

Diversify across uncorrelated instruments and pairs. Use correlation matrix to limit exposure to correlated positions. Risk budget based on signal strength (z-score extremity + half-life). Maximum 60% total exposure.

Kelly fraction: f* = (p×b - q) / b, where p = win rate, q = 1-p, b = avg win / avg loss. For z-score: p ≈ 0.68, b ≈ 1.5, so f* ≈ 30%. Use fractional Kelly (25-50% of f*) for safety due to estimation uncertainty.

Z-score extremes guide direction: Z < -2 suggests calls or bull spreads. Use half-life for DTE selection (DTE ≈ 1.5× half-life). IV percentile guides strategy: high IV = sell premium at extremes, low IV = buy premium.