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On the implied volatility of Inverse options under stochastic volatility models

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  • Elisa Al`os
  • Eulalia Nualart
  • Makar Pravosud

Abstract

In this paper we study short-time behavior of the at-the-money implied volatility for Inverse European options with fixed strike price. The asset price is assumed to follow a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating It^o's formula we first compute the level of the implied volatility of the option when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We also show that our results extend easily to Quanto-Inverse options. We apply our general results to the SABR and fractional Bergomi models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew. Finally, we provide an empirical application using Bitcoin options traded on Debirit to show how our theoretical formulas can be used to model real market data of such options.

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  • Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2023. "On the implied volatility of Inverse options under stochastic volatility models," Papers 2401.00539, arXiv.org, revised Sep 2024.
    • Handle: RePEc:arx:papers:2401.00539
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    References listed on IDEAS

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    1. Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2023. "On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model," Papers 2308.15341, arXiv.org, revised Sep 2024.
    2. Ai Jun Hou & Weining Wang & Cathy Y H Chen & Wolfgang Karl Härdle, 2020. "Pricing Cryptocurrency Options," Journal of Financial Econometrics, Oxford University Press, vol. 18(2), pages 250-279.
    3. Elisa Alòs & Jorge León & Josep Vives, 2007. "On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility," Finance and Stochastics, Springer, vol. 11(4), pages 571-589, October.
    4. Carol Alexander & Arben Imeraj, 2023. "Delta hedging bitcoin options with a smile," Quantitative Finance, Taylor & Francis Journals, vol. 23(5), pages 799-817, May.
    5. Gronwald, Marc, 2019. "Is Bitcoin a Commodity? On price jumps, demand shocks, and certainty of supply," Journal of International Money and Finance, Elsevier, vol. 97(C), pages 86-92.
    6. Garman, Mark B. & Kohlhagen, Steven W., 1983. "Foreign currency option values," Journal of International Money and Finance, Elsevier, vol. 2(3), pages 231-237, December.
    7. Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2022. "On the implied volatility of Asian options under stochastic volatility models," Papers 2208.01353, arXiv.org, revised Mar 2024.
    8. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107039124, October.
    9. Tak Kuen Siu & Robert J. Elliott, 2021. "Bitcoin option pricing with a SETAR-GARCH model," The European Journal of Finance, Taylor & Francis Journals, vol. 27(6), pages 564-595, April.
    10. Jovanka Lili Matic & Natalie Packham & Wolfgang Karl Härdle, 2023. "Hedging cryptocurrency options," Review of Derivatives Research, Springer, vol. 26(1), pages 91-133, April.
    11. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107611986, October.
    12. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    13. 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.
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