Solving the index tracking problem: a continuous optimization approach

Abstract

Investing vast amounts of money with the goal of fostering medium to long-term growth in returns is a challenging task in financial optimization. A method might be mirroring the market index as closely as possible by choosing from the stocks that make up the index. This approach is known as index tracking and the objective of this paper is to address this problem in order to solve it by means of mathematical programming techniques. In particular, we are interested in investigating the index tracking problem (ITP) as a mixed integer linear program in presence of some real-world constraints known as cardinality constraints as well as transaction costs. These ITP models are NP-hard, and consequently, difficult to solve by classical exact methods even for medium-sized instances. In order to overcome this issue, we propose a method based on nonconvex programming techniques. More precisely, we reformulate the problem as a difference of convex functions (DC) program and solve it by means of an approach known as DC algorithm. In order to evaluate the performance of the proposed algorithm, we conducted numerical experiments using benchmark instances. The results of the algorithm are compared with those provided by the state-of-the-art MILP solver Gurobi. The numerical results confirm the efficiency of the method in solving the ITP.