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## CLOSED FORM SOLUTION FOR FRACTIONAL STOCHASTIC VOLATILITY EXTENDING HESTON MODEL

For the model

$dS(t)=S(t) \sqrt{V(t)} dw^1(t)$

$D^\alpha_t V(t) = -\kappa(V(t)-\theta) dt + \nu\sqrt{V(t)} dw^2(t)$

$Corr(dw^1(t),dw^2(t)) = \rho dt$

The solution in the above screenshot from Heston’s paper remains valid with the change $\rho \rightarrow \rho S_{\alpha}(T-t)$ and $\nu \rightarrow \nu S_{\alpha}(T-t)$ where

$S_{\alpha}(s) = s^{\alpha-1}E_{\alpha,\alpha}(-\kappa s^\alpha)$

This follows from the computation of the Black-Scholes PDE computation in my last blog.

$C(t,\phi) = r\phi i t + \frac{a}{\sigma S_{\alpha}(T-t)^2}( (b_j - \rho \nu S_{\alpha}(T-t)^2}\phi i + d) t - 2 \log(\frac{1-g e^{dt}}{1-g}))$

$D(t,\phi) = \frac{b_j - S_{\alpha}(t)^2\rho\nu \phi i + d}{S_{\alpha}(t)^2\nu^2}( \frac{1-ge^{dt}}{1-g})$

$g = \frac{b_j + d - \rho\nu S_{\alpha}(t)^2 \phi i}{b_j - \rho\nu S_{\alpha}(t)^2 \phi i}$

$d = \sqrt{ (S_{\alpha}(t)^2 \rho\nu \phi i - b_j)^2 - \nu^2(2 u_f \phi i - \phi^2)}$

$f_j(x,v,t,\phi) = \exp( C(t,\phi) + D(t,\phi) v + i\phi)$

The probabilities that the option expires in-the-money are then exactly the same as the solution provided by Heston

$P_j(x,v,t,\phi) = \frac{1]{2} +\frac{1}{\pi} \int_0^\infty \frac{\exp(-i \phi\log(K) f_j(x,v,T,\phi)}{i\phi} d\phi$

The value of a call option is then:
$C(S,v,t) = SP_1 + e^{-r t}KP_2$