  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.
