2. Statistical Arbitrage

Stat arb = trading mathematical relationships between instruments, not directional bets on the market. Market-neutral, mean-reversion based, edge from many small wins.

The core idea — pairs trading

Two related stocks (e.g., HDFC Bank and ICICI Bank) tend to move together. When they diverge more than usual, bet that they’ll converge.

Spread = Price_A − β × Price_B

If spread is unusually high → short A, long β units of B
If spread is unusually low  → long A, short β units of B
Exit when spread returns to mean.

Profit doesn’t depend on market direction — only on the relationship reverting.

Cointegration vs correlation — the key distinction

Correlation: “Their returns move together day to day.” — Can be high but spread still drifts forever.
Cointegration: “Their prices are bound by an equilibrium.” — The spread is stationary and reverts.

Pairs trading needs cointegration, not just correlation.

The Engle-Granger test (in plain words)

Regress price of A on price of B → get hedge ratio β and residual (= spread).
Run an Augmented Dickey-Fuller (ADF) test on the residual.
If ADF p-value < 0.05 → spread is stationary → pair is cointegrated → tradeable.

Python:

from statsmodels.tsa.stattools import coint
score, pvalue, _ = coint(price_A, price_B)
if pvalue < 0.05:
    print("Cointegrated — candidate pair")

Workflow for pairs trading

Step 1: Universe & candidate pairs

Pick a homogeneous universe (e.g., NIFTY BANK constituents). Generate all $\binom{n}{2}$ pairs. With 12 banks → 66 pairs.

Step 2: Cointegration screen

For each pair, run cointegration test on rolling 1-year window. Keep pairs with p < 0.05.

Step 3: Compute spread Z-score

$z_t = \frac{\text{spread}_t - \mu_{spread}}{\sigma_{spread}}$

Using rolling 30-day or 60-day mean and std.

Step 4: Trade signals

$z > +2$ → spread too wide → short the spread (short A, long β·B).
$z < -2$ → spread too narrow → long the spread (long A, short β·B).
$|z| < 0.5$ → exit.

Step 5: Stop loss

$|z| > 4$ → relationship broke. Exit and re-evaluate cointegration.
Time stop: exit after N days regardless.

Sizing the legs

Position sizes are inversely proportional to prices and proportional to the hedge ratio:

If A = ₹1,500, B = ₹800, β = 1.6:

For ₹1,00,000 long A → buy 66 shares of A
Hedge: short 66 × 1.6 × (1500/800) = 198 shares of B (roughly equivalent rupee exposure)

Goal: dollar/rupee-neutral and beta-neutral leg sizes.

Real-world frictions

Cash-side shorting in India is restricted to intraday → most pairs trades use stock futures to short.
Different lot sizes between two F&O stocks → exact hedge ratios require rounding.
Dividend dates / corporate actions distort the spread temporarily — exclude or adjust.
Cointegration breaks — historically cointegrated pairs decouple (e.g., bank merger). Re-test monthly.
Crowded trade — popular pairs have thin edge; the Z-score reverts so quickly you can’t capture it.

Beyond pairs — multi-asset stat arb

Modern stat arb uses portfolios of N stocks, not just 2:

Run a regression / PCA on a basket → find the principal components.
Build a portfolio that’s “neutral” to the first few PCs (market, sector).
The residual portfolio is mean-reverting → trade it.

This is what funds like Renaissance, D.E. Shaw, Two Sigma do at scale (across thousands of stocks, with superior infrastructure and signal speed). Retail can play simple versions.

Beta-neutral & dollar-neutral strategies

Dollar-neutral: equal long and short capital ( $\sum w = 0$ ).
Beta-neutral: $\sum w_i \beta_i = 0$ (zero net market exposure).
Sector-neutral: equal long/short within each sector.

A market-neutral portfolio earns the alpha (selection skill) without taking beta (market direction). In return, you give up market upside but also crash protection.

Mean reversion at the single-stock level

Same idea, applied to one stock vs its own mean:

Bollinger reversion: stock 2 std away from 20 SMA → fade.
OU process fit: model log-price as Ornstein-Uhlenbeck mean-reverting → trade extremes.
Sigma reversion intraday: fade large moves in liquid F&O stocks at the open.

These work well in range-bound regimes and in liquid large-caps. They fail catastrophically in trending markets — always have a regime filter (see Chapter 5).

A working pairs-trade example

Pair: HDFC Bank (A) and ICICI Bank (B) Lookback: 1 year for cointegration, 60 days rolling for Z-score. Hedge ratio (β): 0.92 (computed via OLS on log prices).

Today’s signal:

Spread = log(HDFCBANK) − 0.92 × log(ICICIBANK)
Z-score = +2.3

Trade:

Short 100 shares HDFC Bank (futures, 1 lot ~550 shares — adjust)
Long ~92 equivalent ICICI Bank (futures)

Exit conditions:

Z-score returns to 0 (target).
Z-score breaches +4 (stop).
30 days elapsed (time stop).

Expected P&L per trade: small (50–80 bps) but high win rate (~70%). Run dozens of pairs simultaneously — the law of large numbers does the work.

When stat arb fails

Regime change: a structural event breaks historical relationships (e.g., a merger, regulatory change).
Crowding: retail/quant funds piling in arbitrage away the edge.
Volatility spikes: spreads widen to “infinity” in panic — your stop loss may be too late.
Black swan correlated to the spread: if both legs move on the same news, the hedge fails.

Always have a portfolio-level kill switch. If aggregate stat arb book drawdown > X%, deleverage. Don’t try to “average down” the spread — that’s the universal way stat arb shops blow up (LTCM, Amaranth lite-flavors).

Tools / data

Python: statsmodels for cointegration & ADF, pandas for data wrangling, vectorbt or custom for backtesting.
Data: minute-bar futures data is ideal; daily works for slower variants.
Execution: needs near-simultaneous fills in both legs → broker API + low latency.

Reading list

Statistical Arbitrage — Andrew Pole.
Algorithmic Trading: Winning Strategies and Their Rationale — Ernie Chan (chapters on pairs).
Pairs Trading: Quantitative Methods and Analysis — Ganapathy Vidyamurthy.