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Moment Explosions and Long-Term Behavior of Affine Stochastic Volatility Models

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Author Info
Martin Keller-Ressel
Abstract

We consider a class of asset pricing models, where the risk-neutral joint process of log-price and its stochastic variance is an affine process in the sense of Duffie, Filipovic and Schachermayer [2003]. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long-term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff-Nielsen-Shephard model.

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File URL: http://arxiv.org/abs/0802.1823
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File URL: http://arxiv.org/pdf/0802.1823
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Paper provided by arXiv.org in its series Quantitative Finance Papers with number 0802.1823.

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Date of creation: Feb 2008
Date of revision: Oct 2008
Handle: RePEc:arx:papers:0802.1823

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  1. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
  2. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January. [Downloadable!] (restricted)
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  3. Ole E. Barndorff-Nielsen, 1997. "Processes of normal inverse Gaussian type," Finance and Stochastics, Springer, vol. 2(1), pages 41-68. [Downloadable!] (restricted)
  4. Martin Keller-Ressel & Thomas Steiner, 2008. "Yield curve shapes and the asymptotic short rate distribution in affine one-factor models," Finance and Stochastics, Springer, vol. 12(2), pages 149-172, April. [Downloadable!] (restricted)
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