Understand statistical arbitrage basics — Hinglish notes

6.1.5 · Stock-Market › Algorithmic & Quant Trading

Statistical Arbitrage Kya Hai?

Teen Pillars

YEH kyun matter karte hain:

Pair Selection — galat pairs = koi mean reversion nahi = losses
Spread Calculation — entry/exit signals define karta hai
Position Sizing — har trade par risk control karta hai

Derivation: Statistical Arbitrage Spread

Step 1: Spread Define Karo

Do assets A aur B ke liye jinke prices $P_{A} (t)$ aur $P_{B} (t)$ hain, spread yeh hai:

$S (t) = P_{A} (t) - β \cdot P_{B} (t)$

β·P_B kyun subtract karein? Hum ek stationary series chahte hain. Raw price difference $P_{A} - P_{B}$ trend karta hai agar ek zyada tezi se badhta hai. Hedge ratio β us trend ko neutralize karta hai.

β kaise find karein? $P_{A}$ ka $P_{B}$ par linear regression:

$β = \frac{Cov ( P _{A} , P _{B} )}{Var ( P _{B} )} = \frac{\sum ( P _{A} - P ˉ _{A} ) ( P _{B} - P ˉ _{B} )}{\sum ( P _{B} - P ˉ _{B} ) ^{2}}$

Yeh step kyun? $S (t)$ ka variance minimize karta hai — spread ko constant mean ke around oscillate karata hai, drift nahi karne deta.

Step 2: Spread Standardize Karo (Z-Score)

Raw spread $S (t)$ ke arbitrary units hote hain. Z-score mein convert karo:

$z (t) = \frac{S ( t ) - μ _{S}}{σ _{S}}$

jahan:

$μ_{S} = \frac{1}{n} \sum_{i = 1}^{n} S (t_{i})$ (lookback window par mean spread)
$σ_{S} = \frac{1}{n} \sum_{i = 1}^{n} (S (t_{i}) - μ_{S})^{2}$ (standard deviation)

Z-score kyun? Dimensionless units se hum universal thresholds set kar sakte hain: "Enter when |z| > 2" kisi bhi pair ke liye kaam karta hai. Yeh yeh bhi measure karta hai ki current divergence kitni "extreme" hai.

2 kyun? Normal distribution assume karte hue, |z| > 2 approximately 5% time hota hai — mispricing signal karne ke liye rare enough, liquidity ke liye common enough.

Step 3: Trading Logic

Entry signals:

$z (t) > + 2$ : Spread bahut zyada high → Short A, Long B (bet ki spread giregi)
$z (t) < - 2$ : Spread bahut zyada low → Long A, Short B (bet ki spread badhegi)

Exit signals:

$z (t) \to 0$ : Position close karo (mean reversion complete)
$∣ z (t) ∣ > 3$ : Stop-loss (relationship break ho gayi hogi)

Yeh step kyun? High |z| temporary dislocation signal karta hai. z=0 par close karna mean reversion profit capture karta hai. Stop-loss structural breaks ke through hold karne se bachata hai.

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
β calculate karo: $β = 0.85 \cdot \frac{3}{6} = 0.425$

Yeh step kyun? β = ρ·(σ_A/σ_B) regression theory se. KO approximately 0.43 dollars move karta hai har dollar PEP move ke liye.

Day 251: Entry

KO = $62, P E P =$ 150
Spread: $S = 62 - 0.425 \times 150 = 62 - 63.75 = - 1.75$
Historical: $μ_{S} = - 5.75$ , $σ_{S} = 1.5$
Z-score: $z = \frac{- 1.75 - ( - 5.75 )}{1.5} = \frac{4.0}{1.5} = + 2.67$

Interpretation: Spread mean se 2.67 SD upar hai (KO, PEP ke relative overpriced hai).

Action: 100 shares KO short karo, 42.5 shares PEP long karo (β·100 = 42.5 dollar neutrality ke liye).

Yeh step kyun? Position sizes equal dollar exposure create karte hain: $100 \times 62 = 6200$ vs. $42.5 \times 150 = 6375$ (rounding ke saath close enough).

Day 260: Exit

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

Exit kyun? z 2.67 se 1.23 par aa gaya (substantial mean reversion). Profit lock in karo.

Profit calculation:

KO short: $62 p a r b ec ha,$ 59 par cover kiya → +$300
PEP long: $150 p a r k ha r i d a,$ 148 par becha → -$85
Net: $300 - 85 = * * +$ 215** (costs se pehle)

Profitable kyun? Spread narrow hua (mean reverted) jaise predict kiya tha. Market-neutral: dono stocks gir bhi sakte the — hum phir bhi profit karte agar spread converge hota.

Figure — Understand statistical arbitrage basics

Worked Example 2: Failed Trade (Learning Opportunity)

Setup

Gold miners: GDX vs. individual stock GOLD
Entry z = -2.5 (GOLD, GDX ke relative underpriced)
Action: GOLD long, GDX short

Outcome

Gold price industry-wide 15% crash karta hai
GOLD 18% girta hai, GDX 14% girta hai
Spread aur wide hota hai: z → -3.8
Stop-loss z = -3.0 par trigger hota hai
Loss: -$450

Yeh fail kyun hua?

Structural break: Cointegration ne stable relationship assume ki thi, lekin GOLD ne poor earnings announce ki
Beta instability: Crashes ke dauran, correlations 1 ki taraf spike karte hain (sab kuch saath girata hai)
Liquidity: Illiquid conditions mein widening spreads

Key lesson: StatArb stationarity assume karta hai. Cointegration check karo (Engle-Granger test) aur structural changes ke liye news monitor karo.

Key Formulas

Realistic Picture

Win Rate vs. Profit Factor

Typical StatArb performance:

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

Low win rate kyun kaam karta hai: Asymmetric payoff. Mean reversion chhote consistent wins deta hai; stop-loss bade losses cap karta hai. Yeh expectancy ke baare mein hai, accuracy ke baare mein nahi.

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

Upar ke liye: $(0.6 \times 150) - (0.4 \times 100) = 90 - 40 = +\$ 50$ per trade.

Risk: Mean Reversion Ka Half-Life

Half-life = spread ko mean ki taraf 50% decay hone mein lagaane wala time. AR(1) model se calculate karo:

$S (t) = ϕ \cdot S (t - 1) + ϵ, half-life = - \frac{l n ( 2 )}{l n ( ϕ )}$

Yeh kyun matter karta hai: Short half-life (minutes–hours) = fast profit/loss. Long half-life (weeks) = capital zyada der ke liye tied up, opportunity cost. Typical pairs trading: half-life 2–10 days.

Example: Agar $ϕ = 0.9$ (previous value ka 90% persist karta hai), half-life = $- ln (2) / ln (0.9) \approx 6.6$ periods.

Recall Feynman Explanation (ELI12)

Socho tumhare paas ek doosre ke saamne do lemonade stands hain. Usually, Stand A 50¢ charge karta hai aur Stand B 45¢ — 5¢ ka consistent fark hota hai kyunki Stand A ki location better hai.

Ek garmi wale din, Stand A panic ho jaata hai aur prices 70¢ tak badha deta hai jabki Stand B 45¢ par rehta hai. Ab fark 25¢ hai — aam se kaafi zyada! Tum jaante ho ki customers Stand B ki taraf jayenge jab tak Stand A prices nahi girata ya Stand B nahi badhaata.

StatArb aisa hai: "Main bet lagata hoon ki Stand A ki price neeche aayegi." Toh tum kisi ko Stand A lemonade (short) aaj ke high price par dene ka promise karte ho, saath mein Stand B lemonade (long) normal price par khareedte ho. Jab prices 5¢ fark par normalize ho jaati hain, tum change pocket kar lete ho.

"Statistical" part ka matlab hai tum pattern (5¢ fark) par bet laga rahe ho, guaranteed profit nahi hai. Kabhi kabhi pattern toot jaata hai (Stand A band ho jaata hai!), isliye tum "stop-loss" set karte ho — maximum loss jo tum accept karoge pattern change maan lene se pehle.

Connections

6.1.01-Introduction-to-algorithmic-trading — StatArb ek quantitative strategy hai jo execution ke liye algos chahti hai
6.1.04-Mean-reversion-strategies — StatArb, mean reversion ka pairs-trading implementation hai
6.2.03-Correlation-vs-cointegration — Pair selection ke liye distinguish karna critical hai
6.2.05-Backtesting-trading-strategies — Stationarity, costs, aur slippage ke liye backtest zaroori hai
5.3.02-Beta-and-systematic-risk — Yahan β hedge ratio hai; CAPM β se related hai lekin context alag hai
7.1.03-Value-at-Risk-VaR — StatArb position-level risk management ke liye VaR use karta hai

#flashcards/stock-market

Statistical arbitrage kya hai? :: Ek market-neutral strategy jo cointegrated asset pairs ke beech spreads ke mean reversion se profit karti hai, simultaneous long/short positions lekar jab spread >N standard deviations deviate kare.

Pairs trading mein hedge ratio β kya hai?

β = Cov(P_A, P_B) / Var(P_B), asset B ke shares ka ratio jo per share of A hold karna hai directional market risk eliminate karne ke liye (portfolio ko market-neutral banane ke liye).

Entry signals ke liye raw spread ki jagah z-score kyun use karein?

Z-score dimensionless aur normalized hota hai, jo kisi bhi pair ke liye universal thresholds allow karta hai (jaise |z| > 2) price scale se regardless, aur divergence ki statistical extremity measure karta hai.

Correlation aur cointegration mein key difference kya hai?

Correlation co-movement measure karta hai (permanently alag drift ho sakta hai); cointegration matlab hai prices ka ek linear combination stationary hai (mean-reverts), jo StatArb profitability ke liye required hai.

Dollar neutrality ke liye position sizes kaise calculate karte hain?

Shares of B = β × Shares of A, ensure karta hai ki A mein invested dollar value approximately B mein dollar value ke barabar ho market direction hedge karne ke liye.

Mean reversion mein half-life kya hai aur yeh kyun matter karta hai?

Half-life woh time hai jisme spread mean ki taraf 50% decay kare (AR(1) model se -ln(2)/ln(φ) se calculate hota hai). Short half-life = fast profit/loss; long half-life = capital zyada der ke liye tied up.

StatArb sirf 55-60% win rate ke saath profitable kyun ho sakta hai?

Expectancy = (P_win × Avg win) - (P_loss × Avg loss). Chhote consistent mean-reversion wins + stop-loss capped losses positive expectancy create karte hain low win rate ke baawajood bhi.

Typical entry aur exit z-score thresholds kya hain?

Entry: |z| > 2 (spread mean se 2 SD door); Exit: z → 0 (mean reversion) ya |z| > 3 (stop-loss, possible structural break).