Abstract
Model predictive control (MPC) is a flexible yet tractable technique in control engineering that recently has gained much attention in the area of finance, particularly for its application to portfolio optimization. In this paper, we extend the MPC with linear feedback setting in Yamada and Primbs (in: Proceedings of the IEEE conference on decision and control, pp 5705–5710, 2012) by incorporating the following two important and practical issues: The first issue is gross exposure (GE), which is the total value of long and short positions invested in risky assets (or stocks) as a proportion of the wealth possessed by a hedge fund. This quantity measures the leverage of a hedge fund, and the fund manager may limit the amount of leverage by imposing an upper bound, i.e., a GE constraint. The second issue is related to transaction costs, where the MPC algorithm may require frequent trades of many stocks leading to large transaction costs in practice. Here we assume that the transaction cost is proportional to the change in the amount of money (i.e., the change of absolute values of long or short positions) invested in each stock. We formulate the MPC strategy based on a conditional meanvariance problem which we show reduces to a convex quadratic problem, even with gross exposure and proportional transaction cost constraints. Based on numerical experiments using Japanese stock data, we demonstrate that the incorporation of the transaction cost constraint improves the empirical performance of the wealth in terms of Sharpe ratio, which may be improved further by adding the GE constraint.
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Notes
An alternative approach may be to formulate a dynamic optimization problem for a specified (possibly sufficiently long) terminal time period \(T>0\) as provided in Mudchanatongsuk et al. (2008), in which the spreads are modeled by the continuous time Ornstein–Uhlenbeck (OU) processes (Uhlenbeck and Ornstein 1930) in log coordinates.
Note that the control technique demonstrated in this paper may be applied in the case where the spread process is given by a vector AR moving average (VARMA) model and can easily be extended to the case of vector cointegration in Johansen (1991), although we omit the details.
See Appendix Appendix A for the list of companies in the 27 pairs.
Note that all the computations in this paper are executed using MATLAB (R2017b). In particular, for solving the problems with transaction cost and/or GE constraints, i.e., (4.8) and (4.12), we have utilized the MATLAB function, quadprog.m, which is based on the interior point method for quadratic programming (see https://www.mathworks.com/help/optim/ug/quadprog.html).
See Dickey and Fuller (1979).
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This work is supported by GrantinAid for Scientific Research (A) 16H01833 from Japan Society for the Promotion of Science (JSPS).
Pairs Selection Procedure and the List of Selected Pairs
Pairs Selection Procedure and the List of Selected Pairs
We have selected 27 pairs using the procedure explained in Yamada and Primbs (2012) to perform empirical simulations in Sect. 5, which is briefly summarized as follows:

Apply a screening procedure based on the Dickey–Fuller (DF) statistic^{Footnote 6} and correlation coefficient for each pair in stock universe, and sort the pairs that passed the screening procedure by DF statistic from smallest to largest, where smaller DF statistic indicates more significance.

Select pairs from the top of the list down. Once a pair has been selected, remove all pairs further down in the list that contain either of the selected companies in the pair. Stop at a desired number of pairs or continue until the end of the list.
The selected pairs are listed in Table 1, which have DF statistics smaller than the 1% critical value and the absolute values of correlation coefficients greater than \(\sqrt{0.8}\). Note that an additional criterion that is sometimes used is to only select pairs where both stocks are in the same industry. We did not apply such a constraint, and as a result, we see that several pairs in the list are chosen from different industry categories.
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Yamada, Y., Primbs, J.A. Model Predictive Control for Optimal Pairs Trading Portfolio with Gross Exposure and Transaction Cost Constraints. AsiaPac Financ Markets 25, 1–21 (2018). https://doi.org/10.1007/s106900179236z
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DOI: https://doi.org/10.1007/s106900179236z