We consider the European call option in a fractional Black-Scholes model with two instruments; a risk-less asset and a risky asset. A risky asset process X is governed by a standard Brownian motion W, whereas stochastic volatility is a function of fast mean-reverting fractional Ornstein-Uhlenbeck process Y which is driven by a fractional Brownian motion  with Hurst parameter  There are three parameters describing Y; (i) the effective volatility  which is obtained by the average with respect to the long-run distribution of Y, (ii) the rate of mean reversion a which is characterized in terms of  with a small parameter e, (iii) the variance of the long-run distribution  of Y which is dependent on Hurst parameter H.

In the case, where  Narita [18] obtains asymptotics of the price of a European call option as  Here we extend the precedent result by the author to the general case where  For this purpose, we take the stochastic integral for general integrands such that they are algebraically integrable in the sense of Hu [12]. Such an integral enables us to get a concrete Ito formula, and hence we can apply it to easy calculations of financial derivatives in a fractional Brownian environment with arbitrary Hurst parameter

, and obtain the corrected Black-Scholes price as sum of the classical Black-Scholes price with the effective constant volatility  and the corrected term. Further, we can obtain the explicit expression for the quantity appearing in the corrected term. Our theorem is automatically an extension of the results in Fouque et al. [9] and Kallianpur and Karandikar [16] to a fractional Black-Scholes model with uncorrelated W and