Multiplicative Langevin Process for Volatilities Produces Observed Q-Variance Regularities

Q-variance (so-called) posits a statistical relationship $\mathbf{E}({\sigma }^2\mathrm{\mid }z)={\sigma }_0^2+\frac{1}{2}{z}^2$ between an asset’s volatility ${\sigma }^2$, as observed in a time interval $T$, and its (suitably scaled) return $z$ in the same interval. We here show that this relationship is {\em exactly equivalent} to to positing an Inverse Gamma probability distribution for ${\sigma }^2$ itself. We then show that such a distribution is exactly generated by a multiplicative Langevin process with an arbitrary, settable coherence time ${\tau }_c$, so that very nearly the same Q-variance relationship will hold for all $T\ll {\tau }_c$.