The low volatility fluctuations regime of the exponential Ornstein-Uhlenbeck model

In this work the analytical characterization of the probability density of financial
returns in the exponential Ornstein-Uhlenbeck model is addressed. Since prices are driven by
a Geometric Brownian motion, whose diffusion coefficient is expressed through an exponential
function of the mean-reverting process Y , an exact expression for the density is hard to be drawn.
However, we derive the closed-form formula for the characteristic function and for the cumulants
of the approximating linear model under the low volatility fluctuations regime. Theoretical
results are confirmed numerically by use of Monte Carlo simulations. The effectiveness of the
analytical predictions is tested on the German DAX30 Index, finding a good agreement between
the empirical data and the theoretical description.