On an MIT course on discrete stochastic processes, Prof. Gallagher talks about the problem of theory of probability theory being that the problems that are solved are well-posed while the hard problem is to create the right models for the ‘real world’. This exercise has simply never been done satisfactorily for volatility despite the voluminous literature. The features that I have discovered that are not fully absorbed in the literature are:
(a) the autocorrelations are fit by Mittag-Leffler functions (this part is a refinement of Ding-Granger-Engle 1993 and is partially handled by ARFIMA-type long memory processes)
(b) the shocks to (equity) volatility are fractional Poisson
(c) the Fisher Black 1976 ‘leverage effect’ is the only correlation between price process and volatility so all these continuous-time stochastic volatility models are off by a lag — future volatility is correlated to past returns at lag 1 (with no discernible decay for individual stocks beyond noise).
These are the features on which we have to build the right model for stochastic volatility. We want something like: volatility follows a fractional Poisson process that is correlated with shocks correlated with the Brownian motion driving the price process at lag 1.