A note on the use of fractional Brownian motion for financial modeling

Abstract

In the second part of the past decade, 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. Within the last two years, several new models using fractional Brownian motion have been published. However, still the problem remains unsolved whether such models are reasonable choices from an economic perspective.In this article, we take on a straightforward mathematical argument in order to clarify when and why fractional Brownian motion is suited for economic modeling: We provide a fractional analog 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.Building a bridge to application, we show that one peculiar, but nevertheless popular result in the literature of fractional option pricing can be well explained by the fact that authors disobeyed this need for compatibility.