Fractional Brownian Motion and the Inherent Exclusion of Arbitrage

Abstract

Over the last years, the usage of fractional Brownian motion for financial models was stuck. The favorable time-series properties of fractional Brownian motion exhibiting long-range dependence came along with an apparently insuperable shortcoming: the existence of arbitrage. In this article, we solve this dilemma: We provide a fractional analogue to the work of Sethi and Lehoczky (1981) thereby confirming that fractional Brownian motion and continuous tradability are incompatible. In the light of a market microstructure perspective to fractional Brownian motion, it becomes clear that the correct usage of fractional Brownian motion inherently implies dynamic market incompleteness.

Keywords: Fractional Brownian Motion, Wick-Itô Calculus, Fractional Stratonovich Calculus, Dynamic Incompleteness

JEL Classification: G12, G13