The geometric phase of stock trading

Abstract

Geometric phases describe how in a continuous-Time dynamical system the displacement of a variable (called phase variable) can be related to other variables (shape variables) undergoing a cyclic motion, according to an area rule. The aim of this paper is to show that geometric phases can exist also for discrete-Time systems, and even when the cycles in shape space have zero area. A context in which this principle can be applied is stock trading. A zero-Area cycle in shape space represents the type of trading operations normally carried out by high-frequency traders (entering and exiting a position on a fast time-scale), while the phase variable represents the cash balance of a trader. Under the assumption that trading impacts stock prices, even zero-Area cyclic trading operations can induce geometric phases, i.e., profits or losses, without affecting the stock quote.