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A Risk-Neutral Stochastic Volatility Model

Author

Listed:
  • Yingzi Zhu

    (Citibank, N.A., 909 Third Avenue, 29th Floor, Zone 1, New York, NY 10043, USA)

  • Marco Avellaneda

    (Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA)

Abstract

We construct a risk-neutral stochastic volatility model using no-arbitrage pricing principles. We then study the behavior of the implied volatility of options that are deep in and out of the money according to this model. The motivation of this study is to show the difference in the asymptotic behavior of the distribution tails between the usual Black–Scholes log-normal distribution and the risk-neutral stochastic volatility distribution.In the second part of the paper, we further explore this risk-neutral stochastic volatility model by a Monte-Carlo study on the implied volatility curve (implied volatility as a function of the option strikes) for near-the-money options. We study the behavior of this "smile" curve under different choices of parameter for the model, and determine how the shape and skewness of the "smile" curve is affected by the volatility of volatility ("V-vol") and the correlation between the underlying asset and its volatility.

Suggested Citation

  • Yingzi Zhu & Marco Avellaneda, 1998. "A Risk-Neutral Stochastic Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(02), pages 289-310.
    • Handle: RePEc:wsi:ijtafx:v:01:y:1998:i:02:n:s0219024998000163
    DOI: 10.1142/S0219024998000163
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    Cited by:

    1. Cristian Homescu, 2011. "Implied Volatility Surface: Construction Methodologies and Characteristics," Papers 1107.1834, arXiv.org.
    2. Max O. Souza & Jorge P. Zubelli, 2007. "On The Asymptotics Of Fast Mean-Reversion Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(05), pages 817-835.
    3. Tak Siu, 2006. "Option Pricing Under Autoregressive Random Variance Models," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 62-75.
    4. Carr, Peter & Wu, Liuren, 2016. "Analyzing volatility risk and risk premium in option contracts: A new theory," Journal of Financial Economics, Elsevier, vol. 120(1), pages 1-20.
    5. Archil Gulisashvili & Elias M. Stein, 2009. "Implied Volatility In The Hull–White Model," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 303-327, April.

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