Probability distribution of returns in the exponential Ornstein-Uhlenbeck model

TL;DR

This paper derives analytical expressions for the probability distribution of returns in an exponential Ornstein-Uhlenbeck model with stochastic volatility, validating results with numerical simulations and real financial data.

Contribution

It provides closed-form solutions for the return distribution in specific limit cases of the model, enhancing understanding of stochastic volatility effects.

Findings

Analytical formulas match Monte Carlo simulations.

Good agreement with empirical data from DAX30 and Dow Jones.

Model captures key features of financial return distributions.

Abstract

We analyze the problem of the analytical characterization of the probability distribution of financial returns in the exponential Ornstein-Uhlenbeck model with stochastic volatility. In this model the prices are driven by a Geometric Brownian motion, whose diffusion coefficient is expressed through an exponential function of an hidden variable Y governed by a mean-reverting process. We derive closed-form expressions for the probability distribution and its characteristic function in two limit cases. In the first one the fluctuations of Y are larger than the volatility normal level, while the second one corresponds to the assumption of a small stationary value for the variance of Y. Theoretical results are tested numerically by intensive use of Monte Carlo simulations. The effectiveness of the analytical predictions is checked via a careful analysis of the parameters involved in the numerical implementation of the Euler-Maruyama scheme and is tested on a data set of financial indexes. In particular, we discuss results for the German DAX30 and Dow Jones Euro Stoxx 50, finding a good agreement between the empirical data and the theoretical description.