Multiplicative Langevin Process for Volatilities Produces Observed Q-Variance Regularities

2026-05-30

AI Summary

This study addresses the empirically observed Q-variance relationship between asset volatility and contemporaneous returns. To explain this phenomenon, the authors construct a multiplicative Langevin process with tunable correlation time and rigorously demonstrate, for the first time, that the Q-variance relation holds approximately at any timescale shorter than the correlation time if and only if the volatility follows an inverse gamma distribution. By combining stochastic process modeling with methods from statistical physics, the theoretical analysis shows that the proposed model accurately reproduces the empirical Q-variance scaling when the observation window is much smaller than the correlation time. This provides a microscopic mechanism and a generative model underlying the observed statistical regularity in financial time series.

Abstract

Q-variance (so-called) posits a statistical relationship $\mathbf{E}(σ^2 | z) = σ_0^2 + \tfrac{1}{2}z^2$ between an asset's volatility $σ^2$, 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 $σ^2$ itself. We then show that such a distribution is exactly generated by a multiplicative Langevin process with an arbitrary, settable coherence time $τ_c$, so that very nearly the same Q-variance relationship will hold for all $T \ll τ_c$.

Problem

• Q-variance
• volatility
• Langevin process
• Inverse Gamma distribution
• financial returns

Innovation

• multiplicative Langevin process
• Q-variance
• inverse Gamma distribution
• volatility modeling
• coherence time