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TWO CORRELATED BROWNIAN MOTIONS FOR FRACTIONAL STOCHASTIC VOLATILITY

Let $w^2(t)$ be the Brownian motion such that volatility satisfies

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

and let $w^1(t)$ be the brownian motion driving the log-price

$dS(t)/S(t) = \sigma \sqrt{v(t)} dw^1(t)$

with correlation $\rho = Corr(w^1,w^2)$.  Then we can use the following to bring in $w^2(t)$ when computing option prices:

$dw^1 n= \rho dw^2 + \sqrt{1-\rho^2} dw^3$

in distribution.  Then we can plug in

$\sqrt{v(t)} dw^2 = \frac{1}{\nu S_\alpha(t-s)}(dv(s) - T_\alpha'(s) ds - \theta\kappa S_{\alpha}(t-s) ds)$

The correlation $\rho$ seems quite important for the smile issue, i.e. without the correlation, these stochastic volatility models do not seem to produce the smile.