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Mean-reversion and optimization

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Abstract

The purpose of these notes is to provide a systematic quantitative framework – in what is intended to be a ‘pedagogical’ fashion – for discussing mean-reversion and optimization. We start with pair trading and add complexity by following the sequence ‘mean-reversion via demeaning → regression → weighted regression → (constrained) optimization → factor models’. We discuss in detail how to do mean-reversion based on this approach, including common pitfalls encountered in practical applications, such as the difference between maximizing the Sharpe ratio and minimizing an objective function when trading costs are included. We also discuss explicit algorithms for optimization with linear costs, constraints and bounds. We also illustrate our discussion on an explicit intraday mean-reversion alpha.

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Notes

  1.  1 ‘Mean-reversion strategy’ is mostly trader lingo – which is what the author is accustomed to. Academic finance literature mostly uses ‘contrarian investment strategy’ instead. This article uses the term ‘mean-reversion (strategy)’ throughout.

  2.  2 ‘Specific risk’ is multi-factor risk model terminology. Some may prefer ‘idiosyncratic risk’.

  3.  3 For a partial list of references on mean-reversion, optimization and related topics, see, e.g., Adcock and Meade (1994); Allaj (2013); Atkinson et al (1997); Avellaneda and Lee (2010); Best and Hlouskova (2003); Black and Litterman (1991); Black and Litterman (1992); Cadenillas and Pliska (1999); Cheung (2010); Chu et al (2011); Cvitanić and Karatzas (1996); Daniel (2001); Da Silva et al (2009); Davis and Norman (1990); Drobetz (2001); Dumas and Luciano (1991); Fama and French (1993); Fama and MacBeth (1973); Gatev et al (2006); He and Litterman (1999); Hodges and Carverhill (1993); Idzorek (2007); Janeček and Shreve (2004); Jegadeesh and Titman (1993); Jegadeesh and Titman (1995); Knight and Satchell (2010); Liew and Roberts (2013); Lo and MacKinlay (1990); Magill and Constantinides (1976); Markowitz (1952); Merton (1969); Mitchell and Braun (2013); Mokkhavesa and Atkinson (2002); O’Tool (2013); Perold (1984); Poterba and Summers (1988); Rockafellar and Uryasev (2000); Satchell and Scowcroft (2000); Sharpe (1966) and Shreve and Soner (1994), and references therein.

  4.  4 Here and in the following refers to the cross-sectional mean return (not the time series mean return). Also, , and (see below) refer to the deviation from the mean return .

  5.  5 We assume no leverage and 0 margins. Nontrivial leverage simply rescales the investment level I. If margins are present, on top of I invested in stocks, we need an additional amount I′ to maintain margins, which simply reduces the strategy return due to the borrowing interest rate.

  6.  6 For example, we could have one group of stocks from the oil sector, the second group from technology, and the third group from, say, health care.

  7.  7 The R Package for Statistical Computing. Also, ‘~’ in (22) is R notation for a linear model.

  8.  8 We will discuss these subtleties below, when we discuss factor models and optimization.

  9.  9 As we discuss in the next section, this case is a certain limit of optimization.

  10. Be it owing to changes from one day to another, or due to computational uncertainties and so on.

  11. In this case, the actual portfolio consists of net long or short positions for individual stocks arising from long positions D i and short positions from the minimum variance portfolio.

  12. Above we considered homogeneous constraints (66). Technically, the same trick can be applied to inhomogeneous constraints of the form Σ i=1 N w i Y ia +y a =0. Everything goes through as above, except that now we have ρ a =−λy a . However, while this will give the correct solution to the minimization of the objective function, this is no longer necessarily the same as maximizing the Sharpe ratio with constraints: the latter explicitly break the invariance under the rescalings w i →ζw i (unless y a ≡0), which is what allowed us to rewrite the Sharpe ratio maximization problem in terms of the objective function minimization problem, whereby λ is fixed via (55). In the presence of inhomogeneous constraints this is no longer the case and some additional care is needed – see the section ‘Optimization with costs’. We will not need inhomogeneous constraints here, however.

  13. More precisely, usually one would use the unrotated factor loadings – recall that where is the Cholesky decomposition of the factor covariance matrix Φ. However, a rotation Y→YU by an arbitrary nonsingular m × m matrix U ab does not change the constraints (66).

  14. For ρ=±1, the Sharpe ratio goes to infinity if R A ≠±R B : the volatility vanishes for (recall that σ A =σ B ). If R A =±R B , then the two instruments A and – long for plus sign and short for minus sign – B are indistinguishable for optimization purposes.

  15. For the sake of simplicity, the transaction costs for buys and sells are assumed to be the same.

  16. Actually, w i derivatives are defined only for w i ≠0 and, for example, in the case of linear costs for w i ≠w* i – see the subsection ‘Optimization: general case’ for details.

  17. For any λ″>0, there is a unique optimum assuming C ij is positive-definite, and all L i ⩾0.

  18. We will discuss bounds below. Alternatively, one can squash returns to achieve the same.

  19. This argument also goes through for partially establishing and liquidating trades with ξ⪅1, that is, ξ≡Σ i=1 N|w* i | need not be equal 1. Uniformity of L i can also be relaxed (with some care).

  20. Note that stocks rarely jump (sub-)industries/sectors, and thus can be assumed to be static.

  21. More precisely, the intercept is already subsumed in Λ iA :Λ iA =1 if the stock labeled by i belongs to the cluster labeled by A=1, …, K; otherwise, Λ iA =0. Each stock belongs to one and only one cluster. This implies that Σ A=1 KΛ iA =1 for each i, thus a linear combination of the columns of Λ iA is the intercept.

  22. This is a so-called ‘delay-0’ alpha – P is O is used in the alpha, and as the establishing fill price.

  23. That is, to ensure that our results are not a mere consequence of the universe selection.

  24. Note that, as the alpha is purely intraday, this ‘rebalancing’ does not generate additional trades, it simply changes the universe that is traded for the next 21 days.

  25. The number of such tickers in our data is 3811. The number of BICS sectors is 10. The numbers of BICS industries is 48. The number of BICS sub-industries varies between 164 and 169 (due to small sub-industries, which are affected by the varying top-2000-by-ADDV universe).

  26. Plus we are assuming ‘delay-0’, meaning, we can place the trades ‘infinitely’ fast right after receiving the opening prints from the exchange(s) and get filled at the very same open prices. And, as mentioned above, we are ignoring trading costs and slippage, hence the rosy ROC, SR and CPS.

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    Zura Kakushadze

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Correspondence to Zura Kakushadze.

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DISCLAIMER: The correspondence address is used by the corresponding author for no purpose other than to indicate his professional affiliation as is customary in publications. In particular, the contents of this article are not intended as an investment, legal, tax or any other such advice, and in no way represent views of Quantigic® Solutions LLC, the website www.quantigic.com or any of their other affiliates.

1received his PhD in Theoretical Physics from Cornell University, was a Postdoctoral Fellow at Harvard University, and an assistant professor at the C.N. Yang Institute for Theoretical Physics at Stony Brook. Dr Kakushadze received the Alfred P. Sloan Fellowship in 2001. After expanding into quantitative finance, he was a Director at RBC Capital Markets, Managing Director at WorldQuant, Executive Vice President at Revere Data, and currently is the President and co-owner of Quantigic® Solutions, an Adjunct Professor at UConn, and a Full Professor at Free University of Tbilisi. He has 90+ publications in Theoretical Physics and Quantitative Finance.

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Kakushadze, Z. Mean-reversion and optimization. J Asset Manag 16, 14–40 (2015). https://doi.org/10.1057/jam.2014.37

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