Rough paths in idealized financial markets

TL;DR

This paper shows that in an idealized financial market, typical price paths have a variation index at most 2, indicating they are not rougher than Brownian motion, without relying on stochastic assumptions.

Contribution

It establishes a non-stochastic framework demonstrating that typical price paths in idealized markets have bounded variation index, with explicit trading strategies illustrating this property.

Findings

Typical price paths have variation index at most 2

Trading strategies can exploit paths with higher variation index

Results do not rely on stochastic assumptions

Abstract

This paper considers possible price paths of a financial security in an idealized market. Its main result is that the variation index of typical price paths is at most 2, in this sense, typical price paths are not rougher than typical paths of Brownian motion. We do not make any stochastic assumptions and only assume that the price path is positive and right-continuous. The qualification "typical" means that there is a trading strategy (constructed explicitly in the proof) that risks only one monetary unit but brings infinite capital when the variation index of the realized price path exceeds 2. The paper also reviews some known results for continuous price paths and lists several open problems.