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Valuation of volatility derivatives as an inverse problem

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  • Peter Friz
  • Jim Gatheral

Abstract

Ground-breaking recent work by Carr and Lee extends well-known results for variance swaps to arbitrary functions of realized variance, provided a zero-correlation assumption is made. We give a detailed mathematical analysis of some of their computations and work out the cases of volatility swaps and calls on variance. The latter leads to an ill-posed problem that we solve using regularization techniques. The sum is divergent, that means we can do something Heaviside†

Suggested Citation

  • Peter Friz & Jim Gatheral, 2005. "Valuation of volatility derivatives as an inverse problem," Quantitative Finance, Taylor & Francis Journals, vol. 5(6), pages 531-542.
    • Handle: RePEc:taf:quantf:v:5:y:2005:i:6:p:531-542
    DOI: 10.1080/14697680500362452
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    References listed on IDEAS

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    1. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    2. 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.
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    Cited by:

    1. Andrew Papanicolaou, 2021. "Extreme-Strike Comparisons and Structural Bounds for SPX and VIX Options," Papers 2101.00299, arXiv.org, revised Mar 2021.
    2. Akihiko Takahashi & Yukihiro Tsuzuki & Akira Yamazaki, 2009. "Hedging European Derivatives with the Polynomial Variance Swap under Uncertain Volatility Environments," CARF F-Series CARF-F-161, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Leonidas S. Rompolis & Elias Tzavalis, 2017. "Pricing and hedging contingent claims using variance and higher order moment swaps," Quantitative Finance, Taylor & Francis Journals, vol. 17(4), pages 531-550, April.
    4. Claudio Albanese & Harry Lo & Aleksandar Mijatovic, 2009. "Spectral methods for volatility derivatives," Quantitative Finance, Taylor & Francis Journals, vol. 9(6), pages 663-692.
    5. Carlos Fuertes & Andrew Papanicolaou, 2012. "Implied Filtering Densities on Volatility's Hidden State," Papers 1203.6631, arXiv.org, revised Mar 2017.
    6. Akihiko Takahashi & Yukihiro Tsuzuki & Akira Yamazaki, 2010. "Hedging European Derivatives with the Polynomial Variance Swap under Uncertain Volatility Environments," CARF F-Series CARF-F-238, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    7. Stamatis Leontsinis & Carol Alexander, 2017. "Arithmetic variance swaps," Quantitative Finance, Taylor & Francis Journals, vol. 17(4), pages 551-569, April.
    8. Elyas Elyasiani & Silvia Muzzioli & Alessio Ruggieri, 2016. "Forecasting and pricing powers of option-implied tree models: Tranquil and volatile market conditions," Department of Economics 0099, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".
    9. Elisa Alos & Kenichiro Shiraya, 2017. "Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach," CARF F-Series CARF-F-407, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Nov 2018.
    10. Akihiko Takahashi & Yukihiro Tsuzuki & Akira Yamazaki, 2011. "Hedging European Derivatives With The Polynomial Variance Swap Under Uncertain Volatility Environments," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(04), pages 485-505.
    11. Robert J. Elliott & Katsumasa Nishide & Carlton‐James U. Osakwe, 2016. "Heston‐Type Stochastic Volatility with a Markov Switching Regime," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 36(9), pages 902-919, September.
    12. Gabriel G. Drimus, 2012. "Options on realized variance by transform methods: a non-affine stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 12(11), pages 1679-1694, November.
    13. A. Papanicolaou, 2016. "Analysis of VIX Markets with a Time-Spread Portfolio," Applied Mathematical Finance, Taylor & Francis Journals, vol. 23(5), pages 374-408, September.
    14. Akihiko Takahashi & Yukihiro Tsuzuki & Akira Yamazaki, 2009. "Hedging European Derivatives with the Polynomial Variance Swap under Uncertain Volatility Environments," CIRJE F-Series CIRJE-F-653, CIRJE, Faculty of Economics, University of Tokyo.
    15. Elisa Alòs & Kenichiro Shiraya, 2019. "Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach," Finance and Stochastics, Springer, vol. 23(2), pages 423-447, April.
    16. Elisa Alos & Frido Rolloos & Kenichiro Shiraya, 2019. "On the difference between the volatility swap strike and the zero vanna implied volatility," Papers 1912.05383, arXiv.org, revised Dec 2020.

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