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On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model

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  • Elisa Al`os
  • Eulalia Nualart
  • Makar Pravosud

Abstract

In this paper we study the short-time behavior of the at-the-money implied volatility for European and arithmetic Asian call options with fixed strike price. The asset price is assumed to follow the Bachelier model with 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 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 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.

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  • 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.
    • Handle: RePEc:arx:papers:2308.15341
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    References listed on IDEAS

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    1. Roza Galeeva & Ehud Ronn, 2022. "Oil futures volatility smiles in 2020: Why the bachelier smile is flatter," Review of Derivatives Research, Springer, vol. 25(2), pages 173-187, July.
    2. Walter Schachermayer & Josef Teichmann, 2008. "How Close Are The Option Pricing Formulas Of Bachelier And Black–Merton–Scholes?," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 155-170, January.
    3. Jean-Pierre Fouque & George Papanicolaou & K. Ronnie Sircar, 2001. "From The Implied Volatility Skew To A Robust Correction To Black-Scholes American Option Prices," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(04), pages 651-675.
    4. Masaaki Fukasawa, 2017. "Short-time at-the-money skew and rough fractional volatility," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 189-198, February.
    5. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107039124.
    6. Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107611986.
    7. 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.
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    Cited by:

    1. Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2023. "On the implied volatility of Inverse and Quanto Inverse options under stochastic volatility models," Papers 2401.00539, arXiv.org.

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