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Volatility Processes

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Analytically Tractable Stochastic Stock Price Models

Part of the book series: Springer Finance ((FINANCE))

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Abstract

The book begins with an overview of the properties of several stochastic processes that play an important role in modeling of the volatility of the stock. The family of volatility processes includes Brownian motion, geometric Brownian motion, Ornstein-Uhlenbeck processes, squared Bessel processes, and Cox-Ingersoll-Ross processes (CIR-processes). In the first chapter, means and variances of these processes are computed and marginal distributions associated with the volatility processes are found. For a geometric Brownian motion, marginal distributions are log-normal, for an Ornstein-Uhlenbeck process, they are Gaussian, while for a CIR-process, they coincide with noncentral chi-square distributions. The chapter also includes the proof of the Pittman-Yor theorem. This theorem concerns certain exponential functionals of squared Bessel processes.

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Authors and Affiliations

  1. Department of Mathematics, Ohio University, Athens, OH, USA

    Archil Gulisashvili

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  1. Archil Gulisashvili
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Gulisashvili, A. (2012). Volatility Processes. In: Analytically Tractable Stochastic Stock Price Models. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31214-4_1

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