On market-neutral stock trading arbitrage via linear feedback

This paper develops a new version of the so-called Simultaneous Long-Short (SLS) linear feedback stock trading controller for a class of idealized markets characterized by continuously differentiable stock prices. The two salient features of our new controller are as follows: First, all trading gains are appropriately discounted to account for “opportunity costs” associated with an alternative investment in a risk-free bond. Second, the new feedback control strategy provides the investor with flexibility to obtain trading gains which are decoupled from the market behavior at large; i.e., a certain “market neutrality condition” is satisfied. The issue of arbitrage is the focal point of this paper. That is, even if interest rate discounting of trading gains is considered, whether the market goes up or down, the account value V(t) will exceed its initial value V(0) for all non-trivial stock price variations. In recognition of the fact that the idealized market only serves as a benchmark indicating the “realm of possibilities” for a real market, we also include numerical simulations indicating the performance of the controller using real-world historical prices.