Statistical Arbitrage

Statistical arbitrage, often shortened to stat arb, is a family of trading strategies that use statistical models to find securities that appear mispriced relative to one another. The trader buys those judged cheap, sells short those judged expensive, and expects the gap to narrow. Positions are spread across many securities and balanced so that the portfolio has little exposure to the direction of the overall market.

Despite the name, the approach is not arbitrage in the strict sense of a riskless profit. Each trade is a bet with an uncertain outcome, and the expected profit arises only on average over a large number of bets. Statistical arbitrage is practised mainly by hedge funds and proprietary trading firms and is a core part of quantitative trading.

The approach grew out of pairs trading developed at Morgan Stanley in the mid-1980s. Gerry Bamberger, a programmer at the firm, is credited with the original idea, and a group led by Nunzio Tartaglia expanded it into a large automated operation. David Shaw, who had worked in that group, founded the firm D. E. Shaw in 1988 and became one of the best-known practitioners.

Over the following decades the method moved from individual pairs to portfolios of hundreds or thousands of shares, and from holding periods of weeks to days and then to minutes. The fastest forms now overlap with high-frequency trading.

• Pairs and baskets. Two shares, or a share and a basket of related shares, are found whose prices have moved together. Statistical tests such as cointegration are used to judge whether the relationship is stable.
• Factor-based residuals. Each share's return is split by a factor model into a part explained by common influences, such as the market and its industry, and a share-specific remainder. The strategy trades the remainder, on the expectation that it reverts toward zero.
• Cross-sectional signals. Shares are ranked on a predictive measure, such as their recent relative return in short-term reversal, and the strategy holds the top group against the bottom group.

In each case the portfolio is constructed to be neutral to the market, and usually to industries and other common factors, so that results depend on relative prices alone. Most strategies rely on mean reversion, though some use momentum or other patterns.

A common way to decide when to trade is to standardize the spread between two instruments as a z-score:

• $s$ is the current value of the spread.
• $\mu$ is the average of the spread over a look-back period.
• $\sigma$ is the standard deviation of the spread over that period.

Suppose the spread between two bank shares has averaged $1.50 with a standard deviation of $0.45, and it now stands at $2.40. The z-score is $(2.40 - 1.50) / 0.45 = 2.0$. A rule that trades when the score passes 2 would sell the spread, shorting the relatively expensive share and buying the cheaper one, and would close the position when the score returns near zero. If the spread instead widens further, the position loses money, and most rules include a level at which the trade is abandoned.

Two explanations are offered. The first is that the strategies supply liquidity. When an investor sells a share in a hurry and pushes its price down relative to its peers, the statistical arbitrageur buys it and is paid for absorbing the temporary pressure. The second is that prices sometimes overreact or underreact to information, and the models detect the error. Both imply that returns shrink as more capital pursues the same signals, and the simple versions of these strategies have become much less profitable since the 1990s.

• Divergence. A statistical relationship can break permanently, for example after a takeover bid or a change in a company's business. A test that showed stationarity in the past does not bind the future.
• Leverage. Expected gains per trade are small, so funds use leverage, which turns a moderate adverse move into a large loss.
• Crowding. Firms using similar data and models hold similar positions. In August 2007 many equity market-neutral funds lost heavily within a few days as some reduced positions and pushed prices against the others.
• Costs and financing. High turnover makes every transaction cost significant, and short selling depends on being able to borrow shares at a reasonable fee.
• Model error. A model tuned too closely to history suffers from overfitting and fails on new data.

The near-collapse of Long-Term Capital Management in 1998, though centred on bond markets, is the standard illustration of how leveraged convergence trades can fail when spreads widen instead of narrowing and capital runs out before the relationship recovers.