Motivated by the industry practice of pairs trading and long/short equity strategies, we study an approach that combines statistical learning and optimization to construct portfolios with mean-reverting price dynamics.

Our main objectives are:

• Design a portfolio with mean-reverting price dynamics, with parameters estimated by maximum likelihood;
• Select portfolios with desirable characteristics, such as high mean reversion;
• Build a parsimonious portfolio, i.e. find a small subset from a larger collection of assets for long/short positions.

In this article, we present the full problem formulation and discuss a specialized algorithm that exploits the problem structure. Using historical price data, we illustrate the method in a series of numerical examples .

Problem Formulation

Given historical data for m assets observed over _T _time-steps. Our main goal is to find the vector w, the linear combination of assets that comprise our portfolio, such that the corresponding portfolio price process best follows an OU process. The likelihood of an OU process observed over _T _time-steps is given by

A major feature of our joint optimization approach is that we simultaneously solve for the optimal portfolio and corresponding parameters for maximum likelihood.

Minimizing the negative log-likelihood results in the optimization problem

Sparsity and Speed

Given a set of candidate assets, we want to select a small parsimonious subset to build a portfolio. This feature is useful in practice since it reduces transaction costs, execution risks, and the burden of monitoring many stock prices.

To add this feature to the model, we want to impose a sparsity penalty on the portfolio vector w. While the 1-norm is frequently used, in our case we have already imposed the 1-norm equality constraint ||w||₁ = 1 . To obtain sparse solutions (i.e. limiting non-zero weights to a small number), we use the 0-norm and apply a cardinality constraint ||w||₀ ≤η to the optimization problem. This constraint limits the maximum number of assets in the portfolio, and is nonconvex.

In addition to sparsifying the solution, we may also want to promote other features of the portfolio. The penalized likelihood framework is flexible enough to allow these enhancements. An important feature is encapsulated by the mean-reverting coefficient μ; a higher may be desirable. We can seek a higher by promoting a lower c , e.g. with a linear penalty.

#optimization #algorithms #portfolio #trading