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## FRACTIONAL HESTON MODEL

Heston uses a square root process for volatility which has fractional generalization:

$D^{\alpha}_t V(t) = (\kappa\theta - \kappa V(t)) + \sigma \sqrt(V(t)} dw^2(t)/dt$

We can transform this equation into Ito form using (fractional-stochastic-diff-eq-sakthivel-rethavi-ren-2013):

$dV(t) = (T'_{\alpha}(t) V(0) + S_{\alpha}(t) \kappa\theta) dt + S_{\alpha}(t) \sigma \sqrt{V(t)} dw^2(t)$

We can then deduce that the standard Black-Scholes-Merton partial differential equation for an asset price $F(S,V,t)$ is

$\frac{1}{2} V S^2 F_{SS} + \rho S_{\alpha}(t) VS F_{SS} + \frac{1}{2} S_{\alpha}(t)^2 V F_{VV} + F_V ( T_{\alpha}'(t) V(0) + S_{\alpha}(t)\kappa\theta - \lambda(S,V,t)) F_V - rS F_S -rF+F_t = 0$

So the problem is to understand if the Heston closed form solution can be produced for this equation.  The Heston solution technique is to solve for the characteristic function using the guess $f(x,v,t) = \exp( C(T-t) + D(T-t) v + i \phi x)$ which leads to two ordinary differential equations for $(C,D)$.  The fractional Heston solution can be obtained by using the Heston solution for $D/S_{\alpha}(t)$/