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Smile asymptotic for Bachelier Implied Volatility

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  • Roberto Baviera
  • Michele Domenico Massaria

Abstract

We investigate the asymptotic behaviour of the Implied Volatility in the Bachelier setting, extending the framework introduced by Benaim and Friz for the Black-Scholes setting. Exploiting the theory of regular variation, we derive explicit expressions for the Bachelier Implied Volatility in the wings of the smile, linking these to the tail behaviour of the underlying's returns' distribution. Furthermore, we establish a direct connection between the analyticity strip of the returns' characteristic function and the asymptotic formula for the Implied Volatility smile at extreme moneyness.

Suggested Citation

  • Roberto Baviera & Michele Domenico Massaria, 2025. "Smile asymptotic for Bachelier Implied Volatility," Papers 2506.08067, arXiv.org.
    • Handle: RePEc:arx:papers:2506.08067
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    File URL: http://arxiv.org/pdf/2506.08067
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    References listed on IDEAS

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    1. Michele Azzone & Roberto Baviera, 2022. "Additive normal tempered stable processes for equity derivatives and power-law scaling," Quantitative Finance, Taylor & Francis Journals, vol. 22(3), pages 501-518, March.
    2. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12, January.
    3. repec:dau:papers:123456789/1380 is not listed on IDEAS
    4. 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.
    5. Michele Azzone & Roberto Baviera, 2019. "Additive normal tempered stable processes for equity derivatives and power law scaling," Papers 1909.07139, arXiv.org, revised Jan 2022.
    6. Helyette Geman & C. Peter M. Dilip Y. Marc, 2007. "Self decomposability and option pricing," Post-Print halshs-00144193, HAL.
    7. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2007. "Self‐Decomposability And Option Pricing," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 31-57, January.
    8. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
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    Cited by:

    1. Roberto Baviera & Michele Domenico Massaria, 2025. "The Additive Bachelier model with an application to the oil option market in the Covid period," Papers 2506.09760, arXiv.org.

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