A contribution to the systematics of stochastic volatility models

TL;DR

This paper systematically compares various stochastic volatility models, revealing their different long-term and short-term return distribution behaviors, and analyzing autocorrelation decay in volatility.

Contribution

It provides a comprehensive comparison of stochastic volatility models, identifying their distributional characteristics and autocorrelation properties.

Findings

Long-time return distribution is Gaussian or power-law tail.

Short-time return distribution is stretched-exponential or algebraic.

Autocorrelation decay of volatility is characterized.

Abstract

We compare systematically several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can assume also an algebraic decay, in the family of models which we call ``GARCH''-type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We calculate also the decay of the autocorrelation function of volatility.