The fractional Brownian motion is not a semimartingale but there is an associated martingale with (see norros-valkeila-virtamo-fbm) so has identical distribution to a Brownian motion. We may want to consider then stochastic volatility models with
instead of the Heston model
By the results of Norros-Valkeila-Virtamo, one can recover the fractional Brownian motion from the fundamental martingale by
If you draw a log-log plot of the empirical over 1900 stocks what you will find is a slope of 2.17 +/- 0.626. If these volatility processes were a diffusion process, the slope would be 2.0. Here is a sampling of figures. Assuming we see that . Now the Ito formula for the fractional Brownian motion in the special case of is
(given explicitly as Example 3.13 in Hu-Oksendal’s hu-oksendal-fractional-whitenoise-calculus-2000). So these examples suggest using a fractional Brownian motion with to model volatility.
Using Norros-Valkeila-Virtamo’s fundamental martingale associated with with the same distribution as where is a Brownian motion one possibility is to use
In this case one would use as volatility in the log-price diffusion where is the transformation to recover fractional Brownian motion from . Thus we would have the model