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.
The low volatility fluctuations regime of the exponential Ornstein-Uhlenbeck model / Bormetti, Giacomo; Cazzola, Valentina; Delpini, Danilo; Montagna, Guido; Nicrosini, Oreste. - In: JOURNAL OF PHYSICS. CONFERENCE SERIES. - ISSN 1742-6588. - 221:(2010), pp. 012014.012014-1-012014.012014-10. [10.1088/1742-6596/221/1/012014]