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## PRODUCT RULE FOR CAPUTO FRACTIONAL DERIVATIVES IS THE KEY TO MITTAG-LEFFLER AUTOCORRELATIONS IN VOLATILITY

The Caputo fractional derivative is the convolution of the normal derivative with a fractional kernel:

$D^\mu f(t) = 1/\Gamma(1-\mu) \int_0^t (t-s)^{-\mu} f'(t) ds$

The fact that the derivative is in the integrand allows us to apply the usual product rule to

$D^\mu (x_t x_{t+h}) = 1/\Gamma(1-\mu) \int_0^t (t-s)^{-\mu} x_{s+h}Dx_s + x_s Dx_{s+h} ds$

Now assume $D^\mu x_t = -\lambda x_t + \sigma dw_t/dt$ with a white noise $dw_t/dt$.  Since $E(dw_t/dt) = 0$ we have

$D^\mu E(x_t x_{t_h}) = E( x_{t+h} D^\mu x_t + x_t D^\mu x_{t+h}) + ER$ where the white noise terms all have zero expectation.  Then $D^\mu E(x_t x_{t+h}) = -2\lambda E(x_t x_{t+h}) + ER$.  Assuming the remainder term is small, this produces for the autocorrelation the fractional equation $D^\mu f(t) = -2\lambda f(t)$ approximately therefore the autocorrelation will be a Mittag-Leffler function.  This simple analysis is important because the empirical volatility autocorrelations are well-matched by $(a,b,c) \rightarrow E_{a,b}(-ct^a)$ and the analysis provides us with the steps toward modeling volatility with closer match to empirical data.

Now let us take a look at some of the fits to empirical data to get a sense for why these sorts of models are useful.