Pairs Trading

Pairs trading is a market-neutral strategy that takes a long position in one security and a short position in a closely related one when the usual relationship between their prices is disturbed, in the expectation that the relationship will be restored. The profit comes from the two prices converging, whatever the direction of the market as a whole. Typical pairs are two companies in the same industry, two share classes of one company, or an index fund and a near substitute.

The strategy is the simplest form of statistical arbitrage. It was developed in the mid-1980s by a quantitative group at Morgan Stanley led by Nunzio Tartaglia and has since become a standard technique at firms such as a hedge fund or proprietary trading desk. Despite the word arbitrage, it is not riskless: it is a bet on mean reversion that can fail.

The starting point is two securities that are driven by the same economic forces, so that news affecting one tends to affect the other. Statistical screening then confirms that their prices have in fact moved together. Three methods are common.

• Distance. Prices are rescaled to a common starting value, and pairs with the smallest accumulated squared difference over a formation period are chosen. This was the method tested in the best-known academic study of the strategy.
• Cointegration. A test such as the augmented Dickey-Fuller test or the Johansen test checks whether some weighted difference of the two prices is stationary. Cointegration is a stronger requirement than high correlation: two series can be highly correlated and still drift apart without limit.
• Clustering and factor methods. Shares are grouped by statistical similarity, and pairs are sought within groups.

Once a pair is chosen, the trader defines the spread between the two prices:

• $P_{A,t}$ and $P_{B,t}$ are the prices of the two securities at time $t$.
• $\beta$ is the hedge ratio, the number of units of B held against each unit of A, usually estimated by linear regression of one price on the other.

The spread is then standardised into a z-score using its historical mean and standard deviation:

Here $\mu$ and $\sigma$ are the mean and standard deviation of the spread over a look-back window. Suppose share A trades at 50, share B at 24, and the hedge ratio is 2. The spread is $50 - 2 \times 24 = 2$. If the spread has averaged 0.5 with a standard deviation of 0.5, the z-score is $(2 - 0.5) / 0.5 = 3$. A is expensive relative to B. The trader sells short one share of A and buys two shares of B. If the spread falls back to its mean of 0.5, the position gains 1.5 per unit, whether that happens because A falls, B rises or both.

A typical rule opens a trade when the z-score moves beyond 2 in either direction and closes it when the score returns near zero. Many traders add a stop, closing the trade if the score reaches 3 or 4, and a time limit after which an unconverged trade is abandoned.

Because the position is long one security and short a similar one, gains and losses from general market moves largely cancel. The portfolio's beta to the market is close to zero, and its return depends on the relative performance of the two legs. This is the appeal of the strategy: it adds a source of return that is not tied to the market's direction. Neutrality is approximate. The two legs may differ in their sensitivity to interest rates, size or other factors, and a pair that is neutral in normal times may not be in a crisis.

A study by Evan Gatev, William Goetzmann and Geert Rouwenhorst, published in 2006, tested a simple distance-based rule on US shares from the 1960s onward and found substantial excess returns that were not explained by standard risk factors. The same study and later work found that profits declined markedly over time, consistent with more capital pursuing the same opportunities and with falling trading costs allowing prices to be kept in line more tightly. Returns that remain are concentrated in less liquid shares and in periods of market stress, when capital to exploit divergences is scarce.

The central risk is that the spread does not come back. A relationship can break permanently when one company is taken over, loses a major customer, changes its business or faces a regulatory ruling. The statistical signal cannot tell a temporary divergence from a structural one, and a widening spread makes the trade look more attractive just as it becomes more dangerous.

The short leg adds risks of its own. Short selling requires borrowing shares, which costs a fee, and the lender can recall them. Losses on a short position are not capped. Because the expected profit on each pair is small, positions are often leveraged, and a sharp adverse move can force liquidation before convergence occurs. In August 2007 many quantitative equity funds holding similar long-short positions suffered sudden losses as they reduced exposure at the same time.

Research practice matters as much as market risk. Screening thousands of possible pairs and keeping those that pass a test guarantees some false positives, a form of overfitting. A sound backtest selects pairs using only earlier data, includes every transaction cost and borrowing fee, and allows for the delay between signal and execution.