Microscopic Origin of Non-Gaussian Distributions of Financial Returns

TL;DR

This paper investigates the microscopic origins of heavy-tailed distributions in financial returns, demonstrating that certain stochastic volatility models naturally lead to Tsallis distributions, which fit real market data well.

Contribution

It extends stochastic volatility models to explain non-Gaussian distributions and links microscopic parameters to observable distribution features.

Findings

The scaling relation in zero-drift log-returns is verified for Dow Jones.

The Hull-White model yields Tsallis distributions for returns.

Empirical data fits well with the Tsallis distribution, confirming the model.

Abstract

In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumption, akin to the Born-Oppenheimer approximation, in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process. In this approximation, we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero-drift modified log-returns. We verify that the Dow-Jones Industrial Average index indeed follows this scaling. We then focus on two popular stochastic volatility models, the Heston and Hull-White models. In particular, we show that in the Hull-White model the resulting probability distribution of log-returns in this approximation corresponds to the Tsallis (t-Student) distribution. The Tsallis parameters are given in terms of the microscopic stochastic volatility model. Finally, we show that the log-returns for 30 years Dow Jones index data is well fitted by a Tsallis distribution, obtaining the relevant parameters.