Distributions of Historic Market Data -- Relaxation and Correlations

TL;DR

This paper analyzes the relaxation and correlation properties of mean-reverting stochastic variance models, deriving closed-form expressions and testing them against extensive historical financial data.

Contribution

It provides new analytical formulas for correlation functions and leverage in mean-reverting models, including the Heston and related models, with validation on real market data.

Findings

Derived closed-form correlation functions for various models

Identified multiple time scales in relaxation dynamics

Validated theoretical results with extensive market data

Abstract

We investigate relaxation and correlations in a class of mean-reverting models for stochastic variances. We derive closed-form expressions for the correlation functions and leverage for a general form of the stochastic term. We also discuss correlation functions and leverage for three specific models -- multiplicative, Heston (Cox-Ingersoll-Ross) and combined multiplicative-Heston -- whose steady-state probability density functions are Gamma, Inverse Gamma and Beta Prime respectively, the latter two exhibiting "fat" tails. For the Heston model, we apply the eigenvalue analysis of the Fokker-Planck equation to derive the correlation function -- in agreement with the general analysis -- and to identify a series of time scales, which are observable in relaxation of cumulants on approach to the steady state. We test our findings on a very large set of historic financial markets data.