Journal of Scientific Computing

, Volume 75, Issue 2, pp 1156–1186 | Cite as

Three \(l_1\) Based Nonconvex Methods in Constructing Sparse Mean Reverting Portfolios

  • Xiaolong Long
  • Knut Solna
  • Jack Xin


We study the problem of constructing sparse and fast mean reverting portfolios. The problem is motivated by convergence trading and formulated as a generalized eigenvalue problem with a cardinality constraint (d’Aspremont in Quant Finance 11(3):351–364, 2011). We use a proxy of mean reversion coefficient, the direct Ornstein–Uhlenbeck estimator, which can be applied to both stationary and nonstationary data. In addition, we introduce three different methods to enforce the sparsity of the solutions. One method uses the ratio of \(l_1\) and \(l_2\) norms and the other two use \(l_1\) norm. We analyze various formulations of the resulting non-convex optimization problems and develop efficient algorithms to solve them for portfolio sizes as large as hundreds. By adopting a simple convergence trading strategy, we test the performance of our sparse mean reverting portfolios on both synthetic and historical real market data. In particular, the \(l_1\) regularization method, in combination with quadratic program formulation as well as difference of convex functions and least angle regression treatment, gives fast and robust performance on large out-of-sample data set.


Mean reversion Sparse estimation Convergence trading \(l_1\) and \(l_2\) norms 

Mathematics Subject Classification

62P05 90C20 90C26 



We would like to thank Dr. Wuan Luo for bringing reference [6] to our attention and for helpful communication.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Playa Del ReyUSA
  2. 2.Department of MathematicsUniversity of California at IrvineIrvineUSA

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