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Using Householder’s method to improve the accuracy of the closed-form formulas for implied volatility

Author

Listed:
  • Daniel Wei-Chung Miao

    (National Taiwan University of Science and Technology)

  • Xenos Chang-Shuo Lin

    (Aletheia University)

  • Chang-Yao Lin

    (National Taiwan University of Science and Technology)

Abstract

Existing closed-form formulas for implied volatilities perform differently for options with different moneyness and maturities. When the accuracy requirement is high, one usually resorts to Newton’s method to obtain accurate results. While this method works well, the procedure is no longer a closed-form expression and an unknown number of iterations are required. To achieve high accuracy over a wide range of moneyness and maturities without losing their closed-form nature, we propose to use Householder’s method to enhance the existing formulas. We derive the general form of the high order derivatives (with respect to volatility) of the Black–Scholes pricing function and its reciprocal function, which leads to the iterative formula of Householder’s method in closed-form. Our numerical analysis demonstrates the performance improvements when Householder’s method is applied to three best formulas in the literature and discusses how the required level of accuracy depends on moneyness and maturities.

Suggested Citation

  • Daniel Wei-Chung Miao & Xenos Chang-Shuo Lin & Chang-Yao Lin, 2021. "Using Householder’s method to improve the accuracy of the closed-form formulas for implied volatility," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 493-528, December.
    • Handle: RePEc:spr:mathme:v:94:y:2021:i:3:d:10.1007_s00186-021-00763-9
    DOI: 10.1007/s00186-021-00763-9
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    References listed on IDEAS

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    1. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
    2. Corrado, Charles J. & Miller, Thomas Jr., 1996. "A note on a simple, accurate formula to compute implied standard deviations," Journal of Banking & Finance, Elsevier, vol. 20(3), pages 595-603, April.
    3. Michele Mininni & Giuseppe Orlando & Giovanni Taglialatela, 2021. "Challenges in approximating the Black and Scholes call formula with hyperbolic tangents," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(1), pages 73-100, June.
    4. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    5. Curtis, Charles E., Jr. & Carriker, Gordon L., 1988. "Estimating Implied Volatility Directly from "Nearest-to-the-Money" Commodity Option Premiums," Working Papers 116875, Clemson University, Department of Agricultural and Applied Economics.
    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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