Learn pairs trading and cointegration — Hinglish notes

6.1.6 · Stock-Market › Algorithmic & Quant Trading

The Statistical Foundation

What Is Cointegration?

Yeh definition kyun matter karta hai:

Individual stock prices random walks hote hain (non-stationary): $P_{t} = P_{t - 1} + ϵ_{t}$ . Koi mean reversion nahi.
Lekin agar $P_{A}$ aur $P_{B}$ cointegrated hain, toh unka spread $S_{t} = P_{A} - β P_{B}$ zaroor mean-revert karta hai. Hum usse trade kar sakte hain.

$β$ kaise dhundhein? ==Ordinary Least Squares (OLS) regression== run karo:

Y_{t} = α + β X_{t} + ϵ_{t}

Slope $β$ hi ==hedge ratio== hai — kitne units of $X$ trade karne hain per unit of $Y$ taaki market-neutral position bane.

Deriving the Spread from First Principles

Do stock prices se shuru karo:

Stock A: $P_{A} (t)$
Stock B: $P_{B} (t)$

Step 1: $P_{A}$ ko $P_{B}$ pe regress karo

P_{A} (t) = α + β P_{B} (t) + ϵ (t)

Kyun? Hum linear relationship dhundh rahe hain. $β$ batata hai "B mein har $1 m o v e k e l i y e, A a v er a g e p e$ \beta$ dollars move karta hai."

Step 2: Spread construct karo

S (t) = P_{A} (t) - β P_{B} (t) - α

$α$ kyun subtract karein? Spread ko zero ke around center karne ke liye. Ab $S (t)$ long-run relationship se deviation represent karta hai.

Step 3: Test karo ki $S (t)$ stationary hai ya nahi
==Augmented Dickey-Fuller (ADF) test== use karo. Null hypothesis: "S mein unit root hai (non-stationary)." Agar hum reject kar dein (p-value < 0.05), toh $S$ stationary hai → pair cointegrated hai.

Figure — Learn pairs trading and cointegration

Z-score kyun? Yeh unit-agnostic hai (chahe stocks $10 k e h o n y a$ 1000 ke) aur probabilistic hai: $∣ Z ∣ > 2$ sirf ~5% time hota hai agar $S$ Gaussian hai.

Step-by-Step: Building a Pairs Trade

Example 1: Coca-Cola (KO) vs. PepsiCo (PEP)

Context: Dono beverage giants hain. Similar business models, consumer spending ke saath correlated.

Step 1: 2 saal ke liye daily prices collect karo
Maano:

KO aaj $60 p ec l ose k a r t ahai, P E P$ 180 pe.

Step 2: KO ko PEP pe regress karo
OLS deta hai $β = 0.32$ (har $1 P E P m o v e k e l i y e, K O$ 0.32 move karta hai).

Step 3: Spread calculate karo

S (t) = KO (t) - 0.32 \times PEP (t)

Aaj: $S = 60 - 0.32 \times 180 = 60 - 57.6 = 2.4$ .

Step 4: Historical $S$ pe ADF test run karo
p-value = 0.01 mila → Unit root reject → Cointegrated ✓

Step 5: Z-score compute karo
Historical $μ_{S} = 1.0$ , $σ_{S} = 0.8$ .

Z = \frac{2.4 - 1.0}{0.8} = 1.75

Interpretation: Spread mean se 1.75 standard deviations upar hai. Hamare +2 threshold tak nahi pahuncha, toh hum wait karte hain.

Agla din: KO $62 t ak j u m p k a r t ahai, P E P$ 180 pe rehta hai.

S = 62 - 0.32 \times 180 = 4.4, Z = \frac{4.4 - 1.0}{0.8} = 4.25

Action: 100 shares KO short karo, 32 shares PEP long karo ( $β = 0.32$ ratio match karne ke liye). Hamara exposure: market moves ke liye neutral, sirf spread converge hone se profit hoga.

Yeh step kyun? Jab spread abnormally wide hota hai, KO overpriced hota hai PEP ke relative mein. Hum mean reversion expect karte hain.

Exit: Jab $Z$ 0.5 se neeche aa jaaye, dono legs close karo. Agar aisa kabhi nahi hua (spread hamesha ke liye diverge ho gaya), toh loss uthana padega — isliye stop-losses zaroori hain.

Example 2: Gold Miners (GDX) vs. Gold (GLD)

Context: GDX gold mining stocks hold karta hai. GLD physical gold hold karta hai. Miners gold prices pe leveraged bets hote hain.

Regression: $GDX (t) = α + 1.8 \times GLD (t) + ϵ (t)$ → $β = 1.8$ .

Scenario: Gold rally karta hai, lekin miners peeche reh jaate hain.

GLD = $200, G D X =$ 350.
Expected GDX = $1.8 \times 200 + α = 365$ (maano $α = 5$ ).
Spread $S = 350 - 365 = - 15$ → Miners gold ke relative mein underpriced hain.

Z-score = -2.3 (maano $μ_{S} = 0$ , $σ_{S} = 6.5$ ).

Trade: GDX long karo, GLD short karo. Catch-up ka wait karo.

Yeh step kyun? Negative spread ka matlab hai miners ne abhi gold ki move follow nahi ki. Historically, woh karte hain.

Common Mistakes (Steel-man Your Errors)

Common mistake Mistake 3: Using the Wrong Hedge Ratio

Galat idea: "Bas har stock mein equal dollar amounts trade karo."

Kyun sahi lagta hai: Simplicity. Agar tum A ka $10 k s h or t k a r o a u r B k a$ 10k long karo, toh tum market-neutral ho.

Fix: Sahi hedge ratio $β$ (OLS regression slope, volatility measure nahi) ke bina, tumhari "neutral" position directional exposure carry karti hai. Hedge ratio tumhe batata hai A ki price ki dollar sensitivity B ki price ke against. Agar $β = 2$ hai, toh matlab A historically B ke har $1 move ke liye \$ 2 move karta hai — isliye tumhein A mein aadhe dollars chahiye (ya B ke twice as many shares) legs balance karne ke liye, equal dollars nahi.

Example: Maano A ko B pe regress karne pe $β = 2$ hai. Tum A ka $10k short karo aur B ka $10k long karo (naive equal-dollar approach). Ab B upar move karta hai aisa ki A ki fair value ko B ke har $1 pe $2 badhni chahiye. Kyunki tum equal dollars hold karte ho, A leg ka dollar move B leg se roughly double hoga. Tumhare legs cancel nahi hote: tumhare paas residual directional exposure hai B ke moves ka aur tum market-neutral nahi rehe. Fix yeh hai ki hedge ratio ke hisaab se size karo — A ka $10k short karo lekin B ka $20k long karo (ya shares match karo $n_{B} = β n_{A}$ ke anusaar) — taaki dono legs ki dollar sensitivities offset ho jaayein aur sirf spread tumhara P&L drive kare.

Key distinction: Hedge ratio $β$ (regression slope) aur relative volatility alag concepts hain. $β$ cointegrating relationship mein average dollar-for-dollar co-movement capture karta hai; volatility random fluctuations ki size measure karta hai. Legs size karne ke liye $β$ use karo — volatility ratio nahi.

The Math Behind Cointegration Tests

Augmented Dickey-Fuller (ADF) Test

Hum test karte hain ki spread $S_{t}$ follow karta hai:

Δ S_{t} = α + γ S_{t - 1} + i = 1 \sum p β_{i} Δ S_{t - i} + ϵ_{t}

Null hypothesis: $γ = 0$ (unit root → non-stationary).
Alternative: $γ < 0$ (mean-reverting).

Yeh form kyun? $Δ S_{t} = S_{t} - S_{t - 1}$ first difference hai. Agar $γ < 0$ hai, toh $S$ ki past values future changes ko negatively affect karti hain → mean ki taraf pull back karta hai.

Decision rule: Agar ADF statistic < critical value (ya p-value < 0.05), toh null reject karo → cointegrated.

Practical note: 1-year se 2-year daily data use karo. Bahut short → false positives. Bahut long → regime changes miss ho jaate hain.

The Johansen Test (Multiple Pairs)

$n > 2$ stocks ke basket ke liye, ==Johansen test== use karo. Yeh error-correction matrix ke eigenvalue decomposition ke through saari cointegrating relationships simultaneously dhundh leta hai.

Kyun? 3+ stocks ke saath, multiple cointegrating vectors ho sakte hain (jaise A-B cointegrated hai, B-C cointegrated hai → A-C transitivity se cointegrated hai). Johansen sab pakad leta hai.

Building a Robust Pairs Trading System

Pair Selection:
Screen karo:
- Same sector (energy, financials, etc.).
- Similar market cap (mega-cap ko small-cap ke saath pair karne se bachho).
- High correlation + ADF test pass karna.
In-Sample Testing:
$β$ , $μ_{S}$ , $σ_{S}$ estimate karne ke liye pehle 70% data use karo.
Out-of-Sample Validation:
Baaki 30% pe test karo. Kya historical Z-scores ne actual reversions predict kiye?
Risk Management:
- Stop-loss agar $∣ Z ∣ > 4$ (spread diverge ho raha hai, revert nahi ho raha).
- Max hold period (jaise 30 days).
- Position sizing: kabhi bhi per pair > 2% capital risk mat karo.
Execution:
- Simultaneously enter karo (market orders → slippage risk).
- Earnings, dividends, corporate actions ke liye monitor karo (cointegration break kar sakte hain).

Recall Explain Like I'm 12

Socho do best friends hain, Alice aur Bob, jo hamesha apni Halloween candy 50-50 share karte hain. Ek din, Alice ke paas 80 pieces hain, Bob ke paas 40. Yeh toh strange hai! Tum bet lagate ho ki agले hafte woh zyada evenly share karenge — shayad Alice kuch Bob ko de de, ya Bob aur trade kar le. Yahi hai pairs trading: jab do cheezein jo aadat se saath rehti hain bahut alag ho jaati hain, tum bet lagate ho ki woh wapas aayengi. Cointegration math ka woh tarika hai kehne ka ki "yeh do dost hamesha balance ho jaate hain, chahe thodi der ke liye drift kar jaayein."

Connections

Mean Reversion Strategies – Pairs trading ek subset hai; Ornstein-Uhlenbeck processes samajhna tumhari understanding gehri karega.
Statistical Arbitrage – Pairs trading sabse simple stat-arb strategy hai; baskets tak scale hoti hai.
Market Neutral Strategies – Hedge ratio zero market beta ensure karta hai.
Time Series Analysis – Stationarity, autocorrelation, aur ARMA models spread modeling ko underpin karte hain.
Risk Management in Algo Trading – Stop-losses aur position sizing critical hain jab spreads diverge karein.
Backtesting and Walk-Forward Analysis – Out-of-sample validate karke overfitting se bacho.

#flashcards/stock-market

What is cointegration in pairs trading? :: Do non-stationary price series cointegrated hoti hain agar unka linear combination (spread) stationary (mean-reverting) ho, jisse hum spread trade kar sakein.

How do you calculate the hedge ratio β in pairs trading?

OLS regression Y = α + βX + ε run karo. Slope β batata hai kitne units of X trade karne hain per unit of Y taaki market-neutral position bane.

What does a Z-score > +2 signal in pairs trading?

Spread abnormally wide hai (mean se 2 std devs upar) → outperformer ko short karo, underperformer ko long karo, mean reversion expect karte hue.

Why can't you rely on correlation alone for pairs trading?

High correlation matlab co-movement hai lekin yeh guarantee nahi karta ki spread mean-reverting hai. Do stocks correlate kar sakte hain phir bhi permanently diverge ho sakte hain. Cointegration (ADF se test ki gayi) zaroori hai.

What is the ADF test and why do we use it?

Augmented Dickey-Fuller test check karta hai ki spread mein unit root hai ya nahi (non-stationary). Null reject karna (p < 0.05) confirm karta hai ki spread stationary hai → pair cointegrated hai.

How do you construct the spread in pairs trading?

S(t) = P_A(t) - β P_B(t) - α, jahaan β regression se hedge ratio hai aur α spread ko zero ke around center karta hai.

When should you exit a pairs trade?

Jab |Z| < 0.5 ho (spread mean pe wapas aa gaya) ya stop-loss hit ho (|Z| > 4, jo divergence indicate karta hai reversion ki jagah).

What is the half-life of mean reversion and why does it matter?

Spread ko apne mean se aadha wapas decay hone mein kitna time lagta hai. Agar bahut zyada lamba ho (>30 days), toh capital inefficiently tie up hoti hai. AR(1) se calculate karo: half-life = ln(0.5)/ln(φ).

Why must pairs be from the same sector?

Woh common economic drivers share karte hain (energy ke liye oil prices, banks ke liye interest rates). Alag sectors chance se correlate kar sakte hain lekin fundamental linkage nahi hoti → cointegration toot jaati hai.

Why is the hedge ratio β different from relative volatility?

β OLS regression slope hai jo cointegrating relationship mein average dollar-for-dollar co-movement capture karta hai; volatility random fluctuations ki size measure karta hai. Legs ko β se size karo, volatility ratio se nahi.