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Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models

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  • Archil Gulisashvili
  • Josep Vives

Abstract

In this paper, we obtain sharp asymptotic formulas with error estimates for the Mellin convolution of functions, and use these formulas to characterize the asymptotic behavior of marginal distribution densities of stock price processes in mixed stochastic models. Special examples of mixed models are jump-diffusion models and stochastic volatility models with jumps. We apply our general results to the Heston model with double exponential jumps, and make a detailed analysis of the asymptotic behavior of the stock price density, the call option pricing function, and the implied volatility in this model. We also obtain similar results for the Heston model with jumps distributed according to the NIG law.

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  • Archil Gulisashvili & Josep Vives, 2014. "Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models," Papers 1403.5302, arXiv.org.
    • Handle: RePEc:arx:papers:1403.5302
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    References listed on IDEAS

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    1. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    2. Peter Friz & Stefan Gerhold & Archil Gulisashvili & Stephan Sturm, 2011. "On refined volatility smile expansion in the Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1151-1164.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    5. del Baño Rollin, Sebastian & Ferreiro-Castilla, Albert & Utzet, Frederic, 2010. "On the density of log-spot in the Heston volatility model," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2037-2063, September.
    6. Leif Andersen & Alexander Lipton, 2013. "Asymptotics For Exponential Lévy Processes And Their Volatility Smile: Survey And New Results," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(01), pages 1-98.
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    Cited by:

    1. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Extreme-Strike Asymptotics for General Gaussian Stochastic Volatility Models," Papers 1502.05442, arXiv.org, revised Feb 2017.

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