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In this paper, we develop an It\^o type integration theory for fractionalBrownian motions with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2})$ via roughpath theory. The It\^o type integrals are path-wise defined and have zeroexpectations. We establish the fundamental tools associated with this It\^otype integration such as It\^o formula, chain rule and etc. As an applicationwe apply this path-wise It\^o integration theory to the study of a fractionalBlack-Scholes model with Hurst parameter less than half, and prove that thecorresponding fractional Black-Scholes market has no arbitrage under a class oftrading strategies.

It\^o integrals for fractional Brownian motion and applications to option pricing