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## FRACTIONAL POISSON WAITING TIMES FOR EQUITY VOLATILITY BUT IS IT REALLY A CONTINUOUS TIME RANDOM WALK?

In the early 2000s Gorenflo, Mainardi, Scalas had promoted the CTRW model for tick data in finance. We know now that even for daily data not the price changes but volatility shocks have fractional Poisson waiting times. The evidence is pretty universal as the figures of fits of empirical waiting times show.  But is volatility really a continuous-time random walk, i.e. subordinated to these waiting times, are the jumps a random walk?  Well, no.  In fact, volatility shocks have long memory subordinated to these waiting times.  A simple coarse check on this is to examine the fractional differencing parameter of a fitted ARFIMA model for values of volatility at jump times.

In fact, if we consider the volatility jump sizes at the event that the lag-1 jump exceeds a threshold we find that the function

$\mu t^{\nu-1} E_{\nu,\nu}(-\mu t^\nu)$ (*)

fits the AR(8) coefficients of the subseries.  The fits are not as clear as the waiting time distribution as the coefficients are noisier.  We take the negative of the AR(8) coefficient fits and add a small constant 0.1 to make the entire graph positive and fit the function (*) and fit reasonably good fits.  Note that (*) happens to be the solution of a time-fractional differential equation of type $D^\nu f(t) = -\mu f(t)$.