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Rough Paths In Idealized Financial Markets

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Executive Summary

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. The authors 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.

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