Computation of the Ex-Post Optimal Strategy for the Trading of a Single Financial Asset

In this paper we explain how to compute the maximum amount of money one investor can earn in trading a single financial asset under a set of trading constraints. The obtained algorithm allows to identify the ex-post optimal strategy

S

* over a set of (known) prices, which is unconventional in Finance. We deliberately adopt such a simplification to show that

even

if one posits a complete knowledge of the “future”, the determination of

S

* is far from triviality, especially in a framework with transaction costs. We review some solutions that are exponential and propose a new polynomial algorithm. Among others, our results shed light on a not so documented aspect of financial markets complexity, propose an absolute boundary for the profits one can realize in a specific time window and against which any investment strategy can be gauged.