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## THE BLACK-SCHOLES PDE FOR FRACTIONAL STOCHASTIC VOLATILITY MODEL

Let $T_\alpha(t) = E_\alpha(-\kappa t^\alpha)$ and $S_\alpha(t) = t^{\alpha-1} E_{\alpha,\alpha}(-\kappa t^\alpha)$.  The fractional stochastic volatility model that takes into account empirical long memory features of volatility is

$dS_t = S_t \sqrt{V_t} dw^1_t$

$D^\alpha_tV_t = -\kappa(V_t - \theta) + \nu \sqrt{V_t} dw^2_t/dt$

where $\langle dw^1_t, dw^2_t \rangle = \rho dt$.  If $F(S,V,t)$ is the price of some derivative, under this model,

$dF = F_t dt +F_S dS + \frac{1}{2} F_{SS} d\langle S\rangle + F_{SV} dV dS + F_V dV + \frac{1}{2} F_{VV} d\langle V \rangle$ (1)Now we construct the portfolio $\Pi = -F + \Delta_1 S + \Delta_2 V$ assuming the market price of risk is zero so  $d\Pi = r\Pi dt$ as in the derivations of the standard Black-Scholes PDE, implying in (1) the $F_S dS$ and $F_V dV$ terms are subtracted.  We have

$r\Pi dt = (rF - r\Delta_1 S - r\Delta_2 V)dt$

$d \Pi = dF -F_S dS - F_V dV$

$= F_t dt + \frac{1}{2} F_{SS} S^2 V dt$

$latex +\rho S_{\alpha} (T-t) \sqrt{V} F_{SV} dt$

$+\frac{1}{2} F_{VV} \nu^2 S_\alpha (T-t)^2 V$

The partial differential equation that must be satisfied is thus

$F_t + \frac{1}{2} F_{SS} S^2 V + \frac{1}{2} F_{VV} S_{\alpha}(T-t)^2 \nu^2 V + \rho \nu S_{\alpha}(T-t)F_{SV} S V - rF =0$

In fact an inspection of Steven Heston’s 1993 (Heston-original) closed form solution for his short memory model shows that with the substitution $\rho \rightarrow \rho S_\alpha(T-t)$ and $\nu \rightarrow \nu S_\alpha(T-t)$ the analysis of this fractional stochastic volatility model and a closed form solution for it is identical to the Heston model.