Understand statistical arbitrage basics

What is Statistical Arbitrage?

The Three Pillars

WHY these matter:

1. Pair Selection — wrong pairs = no mean reversion = losses
2. Spread Calculation — defines entry/exit signals
3. Position Sizing — controls risk per trade

Derivation: The Statistical Arbitrage Spread

Step 1: Define the Spread

For two assets A and B with prices $P_A(t)$ and $P_B(t)$, the spread is:

$S(t) = P_A(t) - \beta \cdot P_B(t)$

WHY subtract β·P_B? We want a stationary series. Raw price difference $P_A - P_B$ trends if one grows faster. The hedge ratio β neutralizes that trend.

HOW to find β? Linear regression of $P_A$ on $P_B$:

$\beta = \frac{\text{Cov}(P_A, P_B)}{\text{Var}(P_B)} = \frac{\sum (P_A - \bar{P}_A)(P_B - \bar{P}_B)}{\sum (P_B - \bar{P}_B)^2}$

Why this step? Minimizes variance of $S(t)$ — makes spread oscillate around constant mean instead of drifting.

Step 2: Standardize the Spread (Z-Score)

Raw spread $S(t)$ has arbitrary units. Convert to z-score:

$z(t) = \frac{S(t) - \mu_S}{\sigma_S}$

where:

• $\mu_S = \frac{1}{n}\sum_{i=1}^{n} S(t_i)$ (mean spread over lookback window)
• $\sigma_S = \sqrt{\frac{1}{n}\sum_{i=1}^{n} (S(t_i) - \mu_S)^2}$ (standard deviation)

WHY z-score? Dimensionless units let us set universal thresholds: "Enter when |z| > 2" works for any pair. Also measures how "extreme" current divergence is.

Why 2? Assuming normal distribution, |z| > 2 happens ~5% of time — rare enough to signal mispricing, common enough for liquidity.

Step 3: Trading Logic

Entry signals:

• $z(t) > +2$: Spread too high → Short A, Long B (bet spread falls)
• $z(t) < -2$: Spread too low → Long A, Short B (bet spread rises)

Exit signals:

• $z(t) \to 0$: Close position (mean reversion complete)
• $|z(t)| > 3$: Stop-loss (relationship may have broken)

Why this step? High |z| signals temporary dislocation. Closing at z=0 captures mean reversion profit. Stop-loss prevents holding through structural breaks.

Worked Example 1: Coca-Cola vs. Pepsi

Setup

• Historical data: 250 trading days
• KO prices: mean $58, σ =$3
• PEP prices: mean $145, σ =$6
• Correlation: 0.85
• Calculate β: $\beta = 0.85 \cdot \frac{3}{6} = 0.425$

Why this step? β = ρ·(σ_A/σ_B) from regression theory. KO moves ~0.43 dollars per dollar PEP moves.

Day 251: Entry

• KO = $62, PEP =$150
• Spread: $S =62 - 0.425 \times 150 = 62 - 63.75 = -1.75$
• Historical: $\mu_S = -5.75$, $\sigma_S = 1.5$
• Z-score: $z = \frac{-1.75 - (-5.75)}{1.5} = \frac{4.0}{1.5} = +2.67$

Interpretation: Spread is 2.67 SD above mean (KO overpriced relative to PEP).

Action: Short 100 shares KO, Long 42.5 shares PEP (β·100 = 42.5 for dollar neutrality).

Why this step? Position sizes create equal dollar exposure: $100 \times 62 = 6200$ vs. $42.5 \times 150 = 6375$ (close enough with rounding).

Day 260: Exit

• KO = $59, PEP =$148
• New spread: $S = 59 - 0.425 \times 148 = 59 - 62.9 = -3.9$
• New z-score: $z = \frac{-3.9 - (-5.75)}{1.5} = \frac{1.85}{1.5} = +1.23$

Why exit? z dropped from 2.67 → 1.23 (substantial mean reversion). Lock in profit.

Profit calculation:

• KO short: Sold $62, covered$59 → +$300
• PEP long: Bought $150, sold$148 → -$85
• Net: $300 - 85 = **+$215** (before costs)

Why profitable? Spread narrowed (mean reverted) as predicted. Market-neutral: both stocks could've fallen — we'd still profit if spread converged.

Worked Example 2: Failed Trade (Learning Opportunity)

Setup

• Gold miners: GDX vs. individual stock GOLD
• Entry z = -2.5 (GOLD underpriced vs. GDX)
• Action: Long GOLD, Short GDX

Outcome

• Gold price crashes 15% industry-wide
• GOLD falls 18%, GDX falls 14%
• Spread widens further: z → -3.8
• Stop-loss triggered at z = -3.0
• Loss: -$450

Why did this fail?

1. Structural break: Cointegration assumed stable relationship, but GOLD announced poor earnings
2. Beta instability: During crashes, correlations spike toward 1 (everything falls together)
3. Liquidity: Widening spreads in illiquid conditions

Key lesson: StatArb assumes stationarity. Check for cointegration (Engle-Granger test) and monitor news for structural changes.

Key Formulas

The Realistic Picture

Win Rate vs. Profit Factor

Typical StatArb performance:

• Win rate: 55–65% (barely better than coin flip!)
• Avg win: $150
• Avg loss: $100
• Profit factor: (0.60 × 150) / (0.40 × 100) = 2.25

Why low win rate works: Asymetric payoff. Mean reversion gives small consistent wins; stop-loss caps large losses. It's about expectancy, not accuracy.

$\text{Expectancy} = (P_{\text{win}} \times \text{Avg win}) - (P_{\text{loss}} \times \text{Avg loss})$

For above: (0.6 \times 150) - (0.4 \times 100) = 90 - 40 = +\50 per trade.

Risk: The Half-Life of Mean Reversion

Half-life = time for spread to decay50% toward mean. Calculate from AR(1) model:

$S(t) = \phi \cdot S(t-1) + \epsilon, \quad \text{half-life} = -\frac{\ln(2)}{\ln(\phi)}$

Why it matters: Short half-life (minutes–hours) = fast profit/loss. Long half-life (weeks) = capital tied up, opportunity cost. Typical pairs trading: half-life 2–10 days.

Example: If $\phi = 0.9$ (90% of previous value persists), half-life = $-\ln(2)/\ln(0.9) \approx 6.6$ periods.

Connections

• 6.1.01-Introduction-to-algorithmic-trading — StatArb is a quantitative strategy requiring algos for execution
• 6.1.04-Mean-reversion-strategies — StatArb is the pairs-trading implementation of mean reversion
• 6.2.03-Correlation-vs-cointegration — Critical to distinguish for pair selection
• 6.2.05-Backtesting-trading-strategies — Must backtest for stationarity, costs, and slippage
• 5.3.02-Beta-and-systematic-risk — β here is hedge ratio; related to CAPM β but different context
• 7.1.03-Value-at-Risk-VaR — StatArb uses VaR for position-level risk management

#flashcards/stock-market

What is statistical arbitrage? :: A market-neutral strategy that profits from mean reversion of spreads between cointegrated asset pairs by taking simultaneous long/short positions when spread deviates >N standard deviations.

Concept Map

Hinglish (regional understanding)