Tracking a financial benchmark using a few assets

We study the problem of tracking a financial benchmark – a continuously compounded growth rate or a stock market index – by dynamically managing a portfolio consisting of a small number of traded stocks in the market. In either case, we formulate the tracking problem as an instance of the stochastic linear quadratic control (SLQ), involving indefinite cost matrices. As the SLQ formulation involves a discounted objective over an infinite horizon, we first address the issue of stabilizability. We then use semidefinite programming (SDP) as a computational tool to generate the optimal feedback control. We present numerical examples involving stocks traded at the Hong Kong and New York Stock Exchanges to illustrate the various features of the model and its performance.