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

Daniel Wei-Chung Miao (), Xenos Chang-Shuo Lin and Chang-Yao Lin
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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

Mathematical Methods of Operations Research, 2021, vol. 94, issue 3, No 6, 493-528

Abstract: 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.

Keywords: Option pricing; Black–Scholes formula; Implied volatility; Householder’s method (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s00186-021-00763-9

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