CHAPTER 21

Statistical Arbitrage

Brian J. Jacobsen, Ph.D., J.D., CFA, CFP®

Chief Portfolio Strategist Wells Fargo Funds Management, LLC and Associate Professor Wisconsin Lutheran College

A mantra of investing is, “Buy low and sell high.” If one can simultaneously execute both sides of the transaction—the buying and the selling—without any commitment of capital, pure arbitrage exists (also known as riskless arbitrage). The activity of arbitrage tends to be self-exhausting: buying a lower priced good creates demand for it and drives up its price; while selling a higher priced good increases supply and drives down its price.

Why add “statistical” to arbitrage? Because statistical arbitrage relationships are much more prevalent than pure arbitrage opportunities. Statistical arbitrage strategies are based on the idea that we do not know with certainty what the future holds; so we can only make probabilistic statements about what may occur. Statistical arbitrage also refers to the use of statistics in identifying arbitrage opportunities.

Statistical arbitrage strategies are subject to myriad risks, but the most important one is model risk. Model risk refers to whether or not a model of the price process of a security is accurate (i.e., whether it conforms to reality). With an accurate model of security pricing, the identification and exploitation of statistical arbitrage opportunities becomes a relatively easy task. Making sure those models are indeed accurate is the golden key to investing.

Some additional, real-world problems with implementing a profitable statistical arbitrage strategy are in limitations on short selling, margin constraints, and timing issues. Not every security is available for short selling. For example, short selling a mutual fund is not possible—unless it is an exchange-traded fund. Also, whether a security can be sold short depends on the ability to obtain it from a security lender. Sometimes, the government will restrict the ability of investors to short sell and security lenders may change collateral requirements for borrowing a security.

An additional problem associated with shorting a security is that the investor might have to give back the security before he or she wants to. One cannot indefinitely short a security; and most of the time the security has to be returned on a moment's notice. This adds an additional dimension of risk—the risk of the early termination of the arbitrage positions.

Margin requirements can also limit one's ability to implement a statistical arbitrage strategy. Ideally, an equity would never fall below the maintenance margin level, but realistically, it might be subject to a margin call.

A great model can be constructed, but it can be grossly unprofitable with poor execution. As Robert Burns wrote in his 1785 poem “The Mouse,” “The best laid schemes of Mice and Men often go awry.” This is very true in investing since many things can come between a good idea and a profitable execution. For example, there could be a delay in getting the data to recognize an arbitrage opportunity, or a delay in acting on the opportunity (logging on to submit an order), or in how long it takes for the transaction to fill. Most arbitrage opportunities are only noticed after the opportunity has gone. Clearly, this is why institutional, and not individual, investors tend to dominate the arbitrage scene. Engaging in arbitrage is a full-time job that requires a large investment in information, processing, and execution technologies.

In this chapter, we first go through some rather simple models of statistical arbitrage—pairs trading and correlation trading. Then, we will move on to a more general method based on modeling the long-run relationships and short-run dynamics of the pricing processes. There are practical limitations to the use of statistical arbitrage that we do not incorporate in the models in this chapter: margin requirements, execution lags, and transaction costs, among others. All these things can give rise to the appearance of an arbitrage opportunity when, in fact, there was no opportunity because of the practical constraints.

Statistical arbitrage is an evolving field since the statistical tools at our disposal are constantly expanding. An additional source of dynamics in this area is in the feedback effects created by traders who use statistical arbitrage strategies: exploiting one opportunity forecloses that opportunity and creates new ones. All the models that are developed will become antiquated quickly, so the models themselves always need to be updated.