# Statistical arbitrage

Statistical arbitrage

In the world of finance and investments statistical arbitrage is used in two related but distinct ways:

* In academic literature, statistical arbitrage is opposed to (deterministic) arbitrage. In deterministic arbitrage a sure profit can be obtained from being long some securities and short others. In statistical arbitrage there is a statistical mispricing of one or more assets based on the expected value of these assets. (For a simple example, consider a game in which one flips a coin and collects \$1 on heads or pays \$0.50 on tails. In any single flip it is uncertain if one will win or lose money. However, in the statistical sense, there is an expected value of \$1&times;50% − \$0.50&times;50% = \$0.25 for each flip. According to the law of large numbers, the mean return on actual flips will approach this expected value as the number of flips increases. This is precisely the way in which a gambling casino makes a profit.) In other words, statistical arbitrage conjectures statistical mispricings or price relationships that are true "in expectation", in the long run when repeating a trading strategy.

* Among those who follow the hedge fund industry statistical arbitrage refers to a particular category of hedge funds (other categories include global macro, convertible arbitrage, and so on). In this narrower sense Statistical arbitrage is often abbreviated as "StatArb". According to Prof. Andrew Lo, StatArb "refers to highly technical short-term mean-reversion strategies involving large numbers of securities (hundreds to thousands, depending on the amount of risk capital), very short holding periods (measured in days to seconds), and substantial computational, trading, and IT infrastructure".

tatArb, the trading strategy

As a trading strategy, statistical arbitrage is a heavily quantitative and computational approach to equity trading. It involves data mining and statistical methods, as well as automated trading systems.

Historically StatArb evolved out of the simpler pairs trade strategy, in which stocks are put into pairs by fundamental or market-based similarities. When one stock in a pair outperforms the other, the poorer performing stock is bought long with the expectation that it will climb towards its outperforming partner, the other is sold short. This hedges risk from whole-market movements.

StatArb considers not pairs of stocks but a portfolio of a hundred or more stocks (some long, some short) that are carefully matched by sector and region to eliminate exposure to beta and other risk factors. Portfolio construction is automated and consists of two phases: in the first or 'scoring' phase each stock in the market is assigned a numeric score or rank that reflects its desirability; high scores indicate stocks that should be held long and low scores indicate stocks that are candidates for shorting. The details of the scoring formula vary and are highly proprietary, but generally (as in pairs trading) they involve a short term mean reversion principle so that, e.g., stocks that have done unusually well in the past week receive low scores and stocks that have underperformed receive high scores. In the second or 'risk reduction' phase the stocks are combined into a portfolio in carefully matched proportions so as to eliminate (or at least greatly reduce) market and factor risk. This phase often uses commercially available risk models like Barra/APT/Axioma/Northfield/ [http://www.finanalytica.com FinAnalytica] to constrain or eliminate various risk factors [For ex. Andrew Lo (op.cit.) states "the widespread use of standardized factor risk models such as those from MSCI/BARRA or Northfield Information Systems ... will almost certainly create common exposures among those managers to the risk factors contained in such platforms"] .

Broadly speaking, StatArb is actually any strategy that is bottom-up, beta-neutral in approach and uses statistical/econometric techniques in order to provide signals for execution. Signals are often generated through a contrarian mean-reversion principle, but can also be formed by lead/lag effects, extreme psychological barriers Fact|date=March 2007, corporate activity, as well as short-term momentum. This is usually referred to as a multi-factor approach to StatArb.

Because of the large number of stocks involved, the high portfolio turnover and the fairly small size of the effects one is trying to capture, the strategy is implemented in an automated fashion and great attention is placed on reducing trading costs.

Statistical arbitrage has become a major force at both hedge funds and investment banks. Many bank proprietary operations now center to varying degrees around statistical arbitrage trading.

Other forms of statistical arbitrage

Volatility arbitrage is a form of statistical arbitrage in which options, rather than equities, are the primary vehicle of the strategy.

Risks

In the general sense, statistical arbitrage only is demonstrably correct as the amount of trading time approaches infinity and the liquidity, or size of an allowable bet, approaches infinity. To put it another way, it does not take into consideration the same problems as the martingale betting system. Over any finite period of time, a series of low probability events may occur that impose heavy short term losses. If those short term losses are greater than the liquidity available to the trader, default may even occur, as in the case of Long-Term Capital Management.

Statistical arbitrage is also subject to model weakness as well as stock-specific risk.

The statistical relationship on which the model is based may be spurious, or may break down due to changes in the distribution of returns on the underlying assets. Factors which the model may not be aware of having exposure to could become the significant drivers of price action in the markets, and the inverse applies also. The existence of the investment based upon model itself may change the underlying relationship, particularly if enough entrants invest with similar principles. The exploitation of arbitrage opportunities themselves increases the efficiency of the market, thereby reducing the scope for arbitrage, so continual updating of models is necessary.

On a stock-specific level, there is risk of M&A activity or even default for an individual name. Such an event would immediately end any historical relationship assumed from empirical statistical analysis.

Events of Summer 2007

During July and August 2007 a number of StatArb (and other Quant type) Hedge funds experienced significant losses at the same time (which is difficult to explain unless there was a common risk factor). While the reasons are not yet fully understood, several published accounts blame the emergency liquidation of a fund that experienced customer withdrawals or margin calls. By closing out its positions quickly, the fund put pressure on the prices of the stocks it was long/short. Because other StatArb funds had similar positions (due to the similarity of their alpha models and risk-reduction models) the other funds experienced adverse returns. [ [http://web.mit.edu/alo/www/Papers/august07_2.pdf Andrew Lo: What Happened To The Quants In August 2007?] ]

In a sense, a stock "heavily involved in StatArb" is itself a risk factor, one that is new and thus was not taken into account by the StatArb models.

These events showed that StatArb has developed to a point where it is a significant factor in the marketplace, that existing funds have similar positions and are in effect competing for the same returns. Simulations of simple StatArb strategies by A. Lo show that the returns to such strategies have been reduced considerably from 1998 to 2007 (presumably because of competition).

ee also

*Volatility arbitrage
*Currency correlation

References

* Richard Bookstaber: A Demon of Our Own Design, Wiley (2006). Describes: the birth of Stat Arb at Morgan Stanley in the mid-1980’s, out of the pairs trading ideas of Gerry Bamberger. The eclipse of the concept after the departure of Bamberger for Newport/Princeton Partners and of D.E. Shaw to start his own StatArb firm. And finally the revival of StatArb at Morgan Stanley under Peter Muller in 1992. Includes this comment (p. 194): “Statistical arbitrage is now past its prime. In mid-2002 the performance of stat arb strategies began to wane, and the standard methods have not recovered.”

*Jegadeesh, N., 1990, 'Evidence of Predictable Behavior of Security Returns', Journal of Finance 45, p. 881-898. An important early article (along with Lehmann’s) about short term return predictability, the source of StatArb returns

*Lehmann, B., 1990, 'Fads, Martingales, and Market Efficiency', Quarterly Journal of Economics 105, p 1-28. First article in the open literature to document the short term return-reversal effect that early StatArb funds exploited.

* [http://media.wiley.com/product_data/excerpt/11/04700235/0470023511.pdf Ed Thorp: A Perspective on Quantitative Finance - Models for Beating the Market] Autobiographical piece describing Ed Thorp's stat arb work in the early and mid-1980's (see p. 5)

*Ed Thorp: Statistical Arbitrage, Wilmott Magazine, June 2008 ( [http://www.wilmott.com/pdfs/080617_thorp.pdf Part1] [http://www.wilmott.com/pdfs/080630_thorp.pdf Part 2] [http://www.wilmott.com/pdfs/080709_thorp.pdf Part3] ). More reminiscences from the early days of StatArb from one of its pioneers.

Footnotes

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