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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

Using the option delta systematically, we derive tighter lower and upper bounds of the Black-Scholes implied volatility than those in Tehranchi [SIAM J. Financ. Math. 7 (2016), 893-916]. As an application, we propose a Newton-Raphson algorithm on the log price that converges rapidly for all price ranges when using a new lower bound as an initial guess. Our new algorithm is a better alternative to the widely used naive Newton-Raphson algorithm, whose convergence is slow for extreme option prices.

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  • 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, revised Oct 2024.
    • 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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