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Probability distribution of returns in the exponential Ornstein-Uhlenbeck model

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  • Giacomo Bormetti
  • Valentina Cazzola
  • Guido Montagna
  • Oreste Nicrosini

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.

Suggested Citation

  • Giacomo Bormetti & Valentina Cazzola & Guido Montagna & Oreste Nicrosini, 2008. "Probability distribution of returns in the exponential Ornstein-Uhlenbeck model," Papers 0805.0540, arXiv.org, revised Oct 2008.
    • Handle: RePEc:arx:papers:0805.0540
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    References listed on IDEAS

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    1. Josep Perello & Ronnie Sircar & Jaume Masoliver, 2008. "Option pricing under stochastic volatility: the exponential Ornstein-Uhlenbeck model," Papers 0804.2589, arXiv.org, revised May 2008.
    2. Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
    3. Roger Lord & Christian Kahl, 2006. "Why the Rotation Count Algorithm works," Tinbergen Institute Discussion Papers 06-065/2, Tinbergen Institute.
    4. E. Cisana & L. Fermi & G. Montagna & O. Nicrosini, 2007. "A Comparative Study of Stochastic Volatility Models," Papers 0709.0810, arXiv.org.
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    Cited by:

    1. Marcin Wk{a}torek & Jaros{l}aw Kwapie'n & Stanis{l}aw Dro.zd.z, 2021. "Financial Return Distributions: Past, Present, and COVID-19," Papers 2107.06659, arXiv.org.
    2. Giacomo Bormetti & Valentina Cazzola & Danilo Delpini, 2009. "Option pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model," Papers 0905.1882, arXiv.org, revised May 2010.

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