  In this paper an arbitrage strategy is constructed for the modified
Black-Scholes model driven by fractional Brownian motion or by a time changed
fractional Brownian motion, when the volatility is stochastic. This latter
property allows the heavy tailedness of the log returns of the stock prices to
be also accounted for in addition to the long range dependence introduced by
the fractional Brownian motion. Work has been done previously on this problem
for the case with constant `volatility' and without a time change; here these
results are extended to the case of stochastic volatility models when the
modulator is fractional Brownian motion or a time change of it. (Volatility in
fractional Black-Scholes models does not carry the same meaning as in the
classic Black-Scholes framework, which is made clear in the text.)
  Since fractional Brownian motion is not a semi-martingale, the Black-Scholes
differential equation is not well-defined sense for arbitrary predictable
volatility processes. However, it is shown here that any almost surely
continuous and adapted process having zero quadratic variation can act as an
integrator over functions of the integrator and over the family of continuous
adapted semi-martingales. Moreover it is shown that the integral also has zero
quadratic variation, and therefore that the integral itself can be an
integrator. This property of the integral is crucial in developing the
arbitrage strategy. Since fractional Brownian motion and a time change of
fractional Brownian motion have zero quadratic variation, these results are
applicable to these cases in particular. The appropriateness of fractional
Brownian motion as a means of modeling stock price returns is discussed as
well.
