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## WAITING TIMES IN FINANCIAL SERIES

I have shown that the waiting times for volatility jumps have Mittag-Leffler distributions even with daily data.  Now Mittag-Leffler distributions were named Pillai-1990-MittagLefflerDistribution by R. N. Pillai who showed that they are attracted to stable law, i.e. that the extended central limit theorem holds for these.  So the appropriate modification for finance is subordination to these Mittag-Leffler distributed waiting times — so in a sense this extends the Peter Clark 1973 introduction of subordination into finance.  Luckily there is an treatment of Stanislavsky-BlackScholes-Subordination by Stanislavsky which features a call option pricing formula.  Mark Meerschaert’s approach to study of the exact time-change via the inverse stable subordinator is also directly applicable to all financial time series and provides possibly a complete explanation of long memory and non-Markovian features of financial time series (given my verification that Mittag-Leffler waiting times in fact are observed for daily stock returns).

The exact fractional differential equation (Piryatinska-Saichev-Woyczynski-2003) satisfied by an $\alpha$-stable process subordinated to Mittag-Leffler waiting times with index $\beta$ is:

$\partial_t^\beta f = \sigma^\alpha \partial_x^\alpha f + \tau^{-\beta}/\Gamma(1-\beta) \gamma(\tau) \delta(x)$

and the moments are given by:

$\langle |Y(t)|^\kappa \rangle = \frac{2 \sin(\pi\kappa/2) \Gamma(\kappa)}{\beta \sin(\pi\kappa/\alpha)\Gamma(\kappa\beta/\alpha)} \sigma^{\kappa} \tau^{\kappa\beta/\alpha}$