The geometric phase of stock trading

TL;DR

This paper extends the concept of geometric phases from continuous to discrete systems, demonstrating how high-frequency trading cycles can generate profits or losses without changing stock prices.

Contribution

It introduces the idea that geometric phases can occur in discrete-time systems and applies this to model profit generation in high-frequency trading.

Findings

Zero-area cycles in shape space can induce profit or loss.

Geometric phases can exist even when cycles have zero area.

High-frequency trading operations can generate profits without price impact.

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.