13A.1 What statistical arbitrage is, and what it is not

Motivation

A pairs trade buys one asset and sells another because the gap between them looks too wide. Whether that is arbitrage, a statistical bet or a disguised directional position depends entirely on what ties the two prices together. This module gives the tools to tell the three apart and to size the bet when it is statistical.

The ladder of ties#

Rank relationships between two prices $P^1_t, P^2_t$ by what enforces them.

Contractual. A futures contract converges to its underlying at expiry (M07 · 7.3). After a merger is approved, the target's shares convert into the acquirer's at a fixed swap ratio. The spread is tied to zero by a contract. The only risks are timing, financing and the contract failing (deal break, M06A).
Economic, with costly arbitrage. MCX gold and COMEX gold converted at USDINR and grossed up for duty (M10 (10.15)). Anyone who can import can arbitrage, but the band is wide and the participants are few. The spread is mean-reverting because a physical trade caps it, not because of a statistical regularity.
Economic, without an arbitrage. Nifty Bank and Nifty 50. The banks are a large part of Nifty, so the indices share a common driver, but nothing forces their ratio back to any level.
Statistical. HDFC Bank and ICICI Bank. Same sector, same regulator, same macro exposures. Any tie is a regularity of the past sample.

"Statistical arbitrage" usually means level 3 or 4: a portfolio whose value is expected to revert, with no contract or physical trade forcing it. The word arbitrage is a misnomer. These are bets, and the tools below measure the odds.

Notation#

Prices are in logs unless stated: $p^i_t = \ln P^i_t$ . A spread is $s_t = p^1_t - \beta p^2_t - \mu$ , the residual of a linear combination; $\beta$ is the hedge ratio. In rupees, holding one rupee of asset 1 against $\beta$ rupees of asset 2 short gives P&L $\approx \Delta s_t$ per rupee of the long leg for small moves. $I(d)$ means integrated of order $d$ : $\Delta^d y_t$ is stationary with positive long-run variance. $W(r)$ is standard Brownian motion on $[0,1]$ ; " $\Rightarrow$ " is weak convergence. Equation numbers are (13A. $n$ ). The OU parameters are $(\kappa,\theta,\sigma)$ as in M10 (10.18).