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Tighter 'Uniform Bounds for Black-Scholes Implied Volatility' and the applications to root-finding

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  • Jaehyuk Choi
  • Jeonggyu Huh
  • Nan Su

Abstract

This note improves the lower and upper bounds of the Black-Scholes implied volatility (IV) in Tehranchi (SIAM J. Financial Math., 7 (2016), p. 893). The proposed tighter bounds are systematically based on the bounds of the option delta. While Tehranchi used the bounds to prove IV asymptotics, we apply the result to the accurate numerical root-finding of IV. We alternatively formulate the Newton-Raphson method on the log price and demonstrate that the iteration always converges rapidly for all price ranges if the new lower bound found in this study is used as an initial guess.

Suggested Citation

  • Jaehyuk Choi & Jeonggyu Huh & Nan Su, 2023. "Tighter 'Uniform Bounds for Black-Scholes Implied Volatility' and the applications to root-finding," Papers 2302.08758, arXiv.org.
    • Handle: RePEc:arx:papers:2302.08758
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    References listed on IDEAS

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    1. Chance, Don M, 1996. "A Generalized Simple Formula to Compute the Implied Volatility," The Financial Review, Eastern Finance Association, vol. 31(4), pages 859-867, November.
    2. Minqiang Li & Kyuseok Lee, 2011. "An adaptive successive over-relaxation method for computing the Black-Scholes implied volatility," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1245-1269.
    3. Chambers, Donald R & Nawalkha, Sanjay K, 2001. "An Improved Approach to Computing Implied Volatility," The Financial Review, Eastern Finance Association, vol. 36(3), pages 89-99, August.
    4. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2017. "Shapes of implied volatility with positive mass at zero," Working Papers 2017-77, Center for Research in Economics and Statistics.
    5. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2013. "Shapes of implied volatility with positive mass at zero," Papers 1310.1020, arXiv.org, revised May 2017.
    6. Li, Minqiang, 2008. "Approximate inversion of the Black-Scholes formula using rational functions," European Journal of Operational Research, Elsevier, vol. 185(2), pages 743-759, March.
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