Optimal trading with linear costs

ABSTRACT We address the problem of finding the optimal trading strategy where linear costs apply, with a strict cap on the position allowed in the market. Using Bellman's backward recursion method, we show that the optimal strategy is to switch between the maximum allowed long position and the maximum allowed short position whenever the predictor exceeds a threshold value, for which we establish an exact equation. This equation can be solved explicitly in the case of a discrete Ornstein-Uhlenbeck predictor.We discuss in detail the dependence of this threshold value on transaction costs. Finally, we establish a strong connection between our problem and the case of a quadratic risk penalty, where our threshold becomes the size of the optimal nontrading band.