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## HESTON MODEL WITH FRACTIONAL BROWNIAN MOTION VOLATILITY I

The fractional Brownian motion $dw^H_t$ is not a semimartingale but there is an associated martingale $M_t = \int_0^t s^{H-1/2}(t-s)^{H-1/2} ds$ with $EM_t^2 = c_2^2 t^{2-2H}$ (see norros-valkeila-virtamo-fbm) so $dw_V(t) = t^{2H-1} dM_t$ has identical distribution to a Brownian motion.  We may want to consider then stochastic volatility models with

$dV(t) = \kappa (\theta - V(t))dt + \nu t^{1-2H} \sqrt{V(t)} dw_V(t)$

$dV(t) = \kappa(\theta - V(t))dt +\nu \sqrt{V(t)} dw_V(t)$

By the results of Norros-Valkeila-Virtamo, one can recover the fractional Brownian motion from the fundamental martingale $M_t$ by

$Y_t = 2 H \int_0^t (t-s)^{H-1/2} dM_s$ and $Z_t = \int_0^t s^{1/2-H} dY_s$

If you draw a log-log plot of the empirical $\int_0^t \phi(s) ds$ 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 $\phi(t) = t^a$ we see that $a+1 = 2.17$.  Now the Ito formula for the fractional Brownian motion in the special case of $f(x)=x^2$ is

$\int_0^t B_H(s) dB_H(s) = \frac{1}{2} B_H(t)^2 - \frac{1}{2} t^{2H}$

(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 $H\sim 0.585$ to model volatility.

Using Norros-Valkeila-Virtamo’s fundamental martingale $M^H_t$ associated with $B_H(t)$ with the same distribution as $t^{1-2H} dW_t$ where $W_t$ is a Brownian motion  one possibility is to use

$dV(t) = \kappa(\theta -V(t)) dt + \nu t^{1-2H} \sqrt{V(t)} dW_t$

with $1-2H = -0.17$ obviously.

In this case one would use as volatility in the log-price diffusion $\tilde{V}(t) = F(V(t))$ where $F$ is the transformation to recover fractional Brownian motion $Z_t$ from $M_t$.  Thus we would have the model

$dS_t = \mu S_t \sqrt{\tilde{V(t)}} dW^1(t)$

$\tilde{V}(t) = F(V(t))$

$dV(t) = \kappa(\theta -V(t)) dt + \nu t^{-0.17} \sqrt{V(t)} dW_t$