Stochastic differential equations are commonly used to describe the dynamics of diffusive systems. Since their introduction, they have also been considered natural candidates for the description of the dynamics of financial returns. In this context we analyze the problem of the analytical characterization of the still unknown probability distribution of returns of the exponential Ornstein-Uhlenbeck model with stochastic volatility , where the price is driven by a Geometric Brownian motion, whose diffusion coefficient is expressed as an exponential function of an hidden variable Y governed by an Ornstein-Uhlenbeck 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 . Both analytical and numerical results we obtain can be exploited for useful applications in stock option pricing and risk management .
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 .
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Stochastic differential equations are commonly used to describe the dynamics of diffusive systems. Since their introduction, they have also been considered natural candidates for the description of the dynamics of financial returns. In this context we
We
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still unknown
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returns of
financial returns in
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, where the price is
. In this model the prices are
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as
through
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an Ornstein-Uhlenbeck
a mean-reverting
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. Both analytical and numerical results we obtain can be exploited for useful applications in stock option pricing and risk management
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