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Stochastic Models of Implied Volatility Surfaces

Author

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  • Rama Cont
  • Jose da Fonseca
  • Valdo Durrleman

Abstract

type="main" xml:lang="en"> We propose a market–based approach to the modelling of implied volatility, in which the implied volatility surface is directly used as the state variable to describe the joint evolution of market prices of options and their underlying asset. We model the evolution of an implied volatility surface by representing it as a randomly fluctuating surface driven by a finite number of orthogonal random factors. Our approach is based on a Karhunen–Loeve decomposition of the daily variations of implied volatilities obtained from market data on SP500 and DAX options. We illustrate how this approach extends and improves the accuracy of the well–known ‘sticky moneyness’ rule used by option traders for updating implied volatilities. Our approach gives a justification for the use of ‘Vegas’ for measuring volatility risk and provides a decomposition of volatility risk as a sum of independent contributions from empirically identifiable factors. (J.E.L.: G130, C14, C31).

Suggested Citation

  • Rama Cont & Jose da Fonseca & Valdo Durrleman, 2002. "Stochastic Models of Implied Volatility Surfaces," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 31(2), pages 361-377, July.
    • Handle: RePEc:bla:ecnote:v:31:y:2002:i:2:p:361-377
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    File URL: http://hdl.handle.net/10.1111/1468-0300.00090
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    Citations

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    Cited by:

    1. Mathias Barkhagen & Jörgen Blomvall, 2016. "Modeling and evaluation of the option book hedging problem using stochastic programming," Quantitative Finance, Taylor & Francis Journals, vol. 16(2), pages 259-273, February.
    2. Bas Peeters, 2012. "Risk premiums in a simple market model for implied volatility," Quantitative Finance, Taylor & Francis Journals, vol. 13(5), pages 739-748, January.
    3. Toby Daglish & John Hull & Wulin Suo, 2007. "Volatility surfaces: theory, rules of thumb, and empirical evidence," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 507-524.
    4. Marcel T. P. Van Dijk & Cornelis S. L. De Graaf & Cornelis W. Oosterlee, 2018. "Between ℙ and ℚ: The ℙ ℚ Measure for Pricing in Asset Liability Management," JRFM, MDPI, vol. 11(4), pages 1-23, October.
    5. Rene Carmona & Yi Ma & Sergey Nadtochiy, 2015. "Simulation of Implied Volatility Surfaces via Tangent Levy Models," Papers 1504.00334, arXiv.org.
    6. Samuel N. Cohen & Christoph Reisinger & Sheng Wang, 2021. "Arbitrage-free neural-SDE market models," Papers 2105.11053, arXiv.org, revised Aug 2021.
    7. Jan Kallsen & Paul Krühner, 2015. "On a Heath–Jarrow–Morton approach for stock options," Finance and Stochastics, Springer, vol. 19(3), pages 583-615, July.
    8. Aït-Sahalia, Yacine & Amengual, Dante & Manresa, Elena, 2015. "Market-based estimation of stochastic volatility models," Journal of Econometrics, Elsevier, vol. 187(2), pages 418-435.
    9. Vedant Choudhary & Sebastian Jaimungal & Maxime Bergeron, 2023. "FuNVol: A Multi-Asset Implied Volatility Market Simulator using Functional Principal Components and Neural SDEs," Papers 2303.00859, arXiv.org, revised Dec 2023.
    10. René Carmona & Sergey Nadtochiy, 2012. "Tangent Lévy market models," Finance and Stochastics, Springer, vol. 16(1), pages 63-104, January.
    11. Stephane Crepey, 2004. "Delta-hedging vega risk?," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 559-579.
    12. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    13. Juho Kanniainen & Martin Magris, 2018. "Option market (in)efficiency and implied volatility dynamics after return jumps," Papers 1810.12200, arXiv.org.
    14. Truc Le, 2014. "Intrinsic Prices Of Risk," Papers 1403.0333, arXiv.org, revised Aug 2014.
    15. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    16. Xixuan Han & Boyu Wei & Hailiang Yang, 2018. "Index Options And Volatility Derivatives In A Gaussian Random Field Risk-Neutral Density Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(04), pages 1-41, June.
    17. Mehdi El Amrani & Antoine Jacquier & Claude Martini, 2019. "Dynamics of symmetric SSVI smiles and implied volatility bubbles," Papers 1909.10272, arXiv.org, revised Feb 2021.

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