Understand mean-reversion quant models — Hinglish notes

6.1.7 · Stock-Market › Algorithmic & Quant Trading

What Is Mean Reversion?

Teen pillars:

Stationary process: Mean aur variance time ke saath wildly drift nahi karte (ek trending stock ki tarah nahi jo continuously climb karta rahe).
Reverting force: Economic forces (profit-taking, value buyers) prices ko wapas kheenchte hain.
Quantifiable deviation: Hum "how far is too far" measure karte hain standard deviations (z-scores) ya Bollinger Bands se.

The Mathematical Foundation: Ornstein-Uhlenbeck Process

Mean reversion ko Ornstein-Uhlenbeck (OU) process se model kiya jaata hai, jo ek autoregressive process ka continuous-time version hai.

Derivation from First Principles

Step 1: Price change $d X_{t}$ ko do competing forces ke roop mein model karo:

Drift toward mean: Term $θ (μ - X_{t})$ kehta hai "agar $X_{t} > μ$ , toh neeche drift karo; agar $X_{t} < μ$ , toh upar drift karo." $μ$ se jitna door, pull utna hi zyada strong.
Random shock: $σ d W_{t}$ unpredictable noise hai (news, liquidity shocks).

Yeh step kyun? Hum systematic reversion (predictable) ko random walk (unpredictable) se alag karte hain.

Step 2: Expected value $E [X_{t} ∣ X_{0}]$ ke liye solve karo (price on average kahan hogi?).

Expectation ke liye stochastic term ko ignore karte hue, hamare paas hai: $\frac{d E [ X _{t} ]}{d t} = θ (μ - E [X_{t}])$

Yeh ek first-order linear ODE hai. Rearrange karo: $\frac{d E [ X _{t} ]}{μ - E [ X _{t} ]} = θ d t$

Dono sides integrate karo: $- ln ∣ μ - E [X_{t}] ∣ = θ t + C$

$t = 0$ par, $E [X_{0}] = X_{0}$ , isliye $C = - ln ∣ μ - X_{0} ∣$ .

Exponentiate karo: $μ - E [X_{t}] = (μ - X_{0}) e^{- θ t}$

Yeh step kyun? Hum solve kar rahe hain ki expected price kaise evolve hoti hai. Exponential decay ka matlab hai ki deviations time ke saath shrink hote hain.

Step 3: Forecast paane ke liye rearrange karo:

Yeh step kyun? Yeh formula hamara trading signal hai. Agar current price $μ$ se deviate karti hai, toh hum expected direction jaante hain aur apna bet size kar sakte hain.

Practical Implementation: Z-Score Strategy

OU process continuous-time hai. Practice mein, hum discrete returns ke saath kaam karte hain.

Trading Rules:

Entry: Agar $z_{t} < - 2$ (price mean se 2 SD neeche hai), toh BUY karo (upar reversion ki expect karo).
Entry: Agar $z_{t} > + 2$ (price mean se 2 SD upar hai), toh SELL/SHORT karo (neeche reversion ki expect karo).
Exit: Position close karo jab $z_{t}$ wapas 0 cross kare (price mean par wapas aaye).
Stop-loss: Agar $∣ z_{t} ∣ > 3$ (regime change? model breakdown?), losses limit karne ke liye exit karo.

$\pm 2$ SD kyun? Normal distribution assume karte hue, ~95% data 2 SD ke andar aata hai. Isse breach karna rare hai → reversion ki high probability. Lekin hume stationarity confirm karni hogi.

Figure — Understand mean-reversion quant models

Example 1: Single-Stock Mean Reversion

Setup: Stock XYZ mahino se $μ = 100$ ke aas-paas trade ho raha hai. Aaj, bad earnings → price $P_{t} = 85$ par drop karti hai. Historical volatility $σ = 5$ .

Step 1: Z-score compute karo. $z_{t} = \frac{85 - 100}{5} = \frac{- 15}{5} = - 3$

Yeh step kyun? Hum kitna extreme deviation hai yeh quantify karte hain. $z = - 3$ mean se 3 SD neeche hai, jo highly unusual hai.

Step 2: Decision — $z < - 2$ ek BUY signal trigger karta hai. Hum $μ = 100$ par reversion ki expect karte hain.

Yeh step kyun? Model kehta hai extreme deviations temporary hain. Hum bet laga rahe hain ki panic overdone hai.

Step 3: Revert hone ka time estimate karo. Agar hum ne $θ = 0.3$ per day (historical fitting se) estimate kiya, toh half-life: $t_{1/2} = \frac{l n 2}{0.3} \approx 2.3 days$

~2-3 dinon mein, hum expect karte hain ki half deviation close ho jaaye. Target exit: $z \approx 0$ , price around $97 - 103$ .

Yeh step kyun? Reversion speed jaanane se holding period aur profit expectations set karne mein madad milti hai.

Outcome: Teen din baad, stock $98$ par rebound karta hai jab earnings panic fade hoti hai. $z \approx - 0.4$ . $\approx 15%$ gain ke saath exit karo.

Example 2: Pairs Trading (Relative Mean Reversion)

Setup: Stocks A aur B ek hi sector mein hain (jaise Coca-Cola vs. Pepsi). Historically, unka ==price spread $S_{t} = P_{A} - P_{B}$ == $μ_{S} = 5$ ke aas-paas rehta hai. Aaj, $P_{A} = 50$ , $P_{B} = 38$ , isliye $S_{t} = 12$ . $σ_{S} = 2$ .

Step 1: Spread ka z-score compute karo. $z_{S} = \frac{12 - 5}{2} = 3.5$

Yeh step kyun? Spread normal se 3.5 SD upar hai. A ne B ko excessively outperform kiya hai.

Step 2: Convergence par bet lagane ke liye pair trade karo.

Short A (B ke relative overpriced).
Long B (A ke relative underpriced).
Yeh market-neutral hai: agar poora sector drop kare, toh hum B par lose karte hain lekin short A par gain karte hain. Hume sirf spread ka narrow hona matter karta hai.

Yeh step kyun? Ratio trade karke, hum mean-reversion ko isolate karte hain aur market risk hedge karte hain.

Step 3: Exit karo jab $z_{S} \approx 0$ (spread $μ_{S} = 5$ par wapas aaye). Maan lo $P_{A} = 48$ , $P_{B} = 43$ . Spread = 5.

Profit calculation:

Short A: 50 par becha, 48 par wapas kharida → gain $+ 2$ per share.
Long B: 38 par kharida, 43 par becha → gain $+ 5$ per share.
Net (equal position sizes assume karte hue): $+ 7$ per unit pair.

Yeh step kyun? Pairs trading relative mispricing se profit kamaata hai, absolute direction se nahi.

Testing for Mean Reversion: The ADF Test

Sabhi price series revert nahi hoti. Ek trending stock non-stationary hota hai. Hum Augmented Dickey-Fuller (ADF) test use karte hain.

Yeh step kyun? Mean-reversion ko ek trending stock par apply karne se paisa doobta hai (aap ek uptrend ko continuously short karte rehte ho). ADF isse rokta hai.

Common Mistakes

Active Recall Checkpoints

Recall Feynman Explain-to-a-12-Year-Old

Soch lo tumhare paas ek toy hai jo hamesha apne room ke middle mein rehna chahta hai. Agar tum use corner mein push karo, toh woh khud center mein wapas roll kar aata hai. Yahi mean reversion hai!

Stock prices kabhi kabhi us toy ki tarah hoti hain. Jab bad news price ko bahut neeche push karti hai, toh yeh aise hai jaise toy corner mein hai. Hamara model kehta hai, "Hey, yeh middle se bahut door hai—yeh probably wapas roll karega." Toh hum stock tab khareedte hain jab woh corner mein ho, phir bechte hain jab woh middle mein wapas roll kare aur paisa kamaate hain.

Lekin humein check karna hoga: Kya toy sach mein middle mein wapas jaane ke liye programmed hai, ya kisi ne middle ko move kar diya? Agar company actually tooti hui hai (jaise toy ki battery khatam ho gayi), toh woh wapas nahi aayegi, aur hum lose karte hain. Isliye hum ADF test se check karte hain ki "wapas rolling" ka pattern real hai ya nahi.

Connections

Statistical Arbitrage Strategies – mean reversion, stat-arb ke peeche ka engine hai
Bollinger Bands – mean ± 2 SD thresholds ka practical visualization
Cointegration Testing – pairs trading ke liye sirf correlation nahi, cointegration chahiye
Kalman Filter – advanced: time-varying $μ$ aur $θ$ dynamically estimate karo
Risk Management in Quant Trading – z-score magnitude se position sizing
Stationarity Tests (ADF, KPSS) – mean-reversion signals par trust karne se pehle prerequisites

#flashcards/stock-market

Mean-reversion models ki core assumption kya hai? :: Prices jo apne historical mean se significantly deviate karte hain woh temporary overreactions experience kar rahe hain aur time ke saath mean ki taraf wapas revert karenge.

Ornstein-Uhlenbeck SDE likhiye aur har term explain kijiye.

d X_{t} = θ (μ - X_{t}) d t + σ d W_{t}

θ

= reversion ki speed,

μ

= long-term mean,

X_{t}

= current price,

σ

= volatility,

d W_{t}

= random Brownian motion.

Mean-reversion trading mein z-score kya measure karta hai?

Current price moving average se kitne standard deviations door hai.

z_{t} = \frac{P _{t} - P ˉ _{n}}{σ _{n}}

Z-score mean-reversion strategy mein aap BUY kab karte ho?

Jab

z_{t} < - 2

(price mean se 2+ standard deviations neeche ho), upar reversion ki umeed mein.

Mean reversion ka half-life kya hai aur ise kaise calculate karte hain?

Woh time jisme price deviation ka aadha hissa mean ki taraf decay ho jaata hai.

t_{1/2} = \frac{l n 2}{θ}

, jahan

θ

speed parameter hai.

Mean-reversion apply karne se pehle ADF test kyun use karte hain?

Yeh test karne ke liye ki price series stationary (mean-reverting) hai ya nahi. Agar iska unit root hai (random walk), toh mean-reversion strategies fail ho jaayengi.

Pairs trading kya hai aur yeh mean reversion kaise use karta hai?

Do correlated stocks ko trade karna aur yeh bet lagana ki unka price spread historical mean par revert karega. Underperformer ko long karo, outperformer ko short karo, convergence se profit kamaao aur market risk hedge karo.

Mean-reversion models ka key risk kya hai? :: Regime shifts — jab fundamental mean permanently change ho jaata hai company deterioration, market structure change, ya crisis ki wajah se. Price purane mean par revert nahi karega.