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Dynamics of symmetric SSVI smiles and implied volatility bubbles

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  • Mehdi El Amrani
  • Antoine Jacquier
  • Claude Martini

Abstract

We develop a dynamic version of the SSVI parameterisation for the total implied variance, ensuring that European vanilla option prices are martingales, hence preventing the occurrence of arbitrage, both static and dynamic. Insisting on the constraint that the total implied variance needs to be null at the maturity of the option, we show that no model--in our setting--allows for such behaviour. This naturally gives rise to the concept of implied volatility bubbles, whereby trading in an arbitrage-free way is only possible during part of the life of the contract, but not all the way until expiry.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:1909.10272
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    References listed on IDEAS

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    1. Mark Davis & Jan Obloj, 2007. "Market completion using options," Papers 0710.2792, arXiv.org, revised Oct 2008.
    2. Frédéric Abergel & Riadh Zaatour, 2012. "What drives option prices ?," Post-Print hal-00687675, HAL.
    3. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    4. Marc Romano & Nizar Touzi, 1997. "Contingent Claims and Market Completeness in a Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 399-412, October.
    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    6. 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.
    7. 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.
    8. Jacopo Corbetta & Pierre Cohort & Ismail Laachir & Claude Martini, 2019. "Robust calibration and arbitrage-free interpolation of SSVI slices," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 665-677, December.
    9. Gaoyue Guo & Antoine Jacquier & Claude Martini & Leo Neufcourt, 2012. "Generalised arbitrage-free SVI volatility surfaces," Papers 1210.7111, arXiv.org, revised May 2016.
    10. Martin Schweizer & Johannes Wissel, 2008. "Term Structures Of Implied Volatilities: Absence Of Arbitrage And Existence Results," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 77-114, January.
    11. Antoine Jacquier & Claude Martini & Aitor Muguruza, 2018. "On VIX futures in the rough Bergomi model," Quantitative Finance, Taylor & Francis Journals, vol. 18(1), pages 45-61, January.
    12. Jim Gatheral & Antoine Jacquier, 2011. "Convergence of Heston to SVI," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1129-1132.
    13. Jean Jacod & Philip Protter, 2010. "Risk-neutral compatibility with option prices," Finance and Stochastics, Springer, vol. 14(2), pages 285-315, April.
    14. Martin Schweizer & Johannes Wissel, 2008. "Arbitrage-free market models for option prices: the multi-strike case," Finance and Stochastics, Springer, vol. 12(4), pages 469-505, October.
    15. repec:hal:wpaper:hal-00687675 is not listed on IDEAS
    16. Pierre Cohort & Jacopo Corbetta & Claude Martini & Ismail Laachir, 2018. "Robust calibration and arbitrage-free interpolation of SSVI slices," Papers 1804.04924, arXiv.org, revised Mar 2019.
    17. 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.
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    Cited by:

    1. Claude Martini & Iacopo Raffaelli, 2021. "Revisiting the Implied Remaining Variance framework of Carr and Sun (2014): Locally consistent dynamics and sandwiched martingales," Papers 2105.06390, arXiv.org.

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