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Fitting the Smile Revisited: A Least Squares Kernel Estimator for the Implied Volatility Surface

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  • Fengler, Matthias R.
  • Wang, Qihua

Abstract

Nonparametric methods for estimating the implied volatility surface or the implied volatility smile are very popular, since they do not impose a specific functional form on the estimate. Traditionally, these methods are two-step estimators. The first step requires to extract implied volatility data from observed option prices, in the second step the actual fitting algorithm is applied. These two-step estimators may be seriously biased when option prices are observed with measurement errors. Moreover, after the nonlinear transformation of the option prices the error distribution will be complicated and less tractable. In this study, we propose a one-step estimator for the implied volatility surface based on a least squares kernel smoother of the Black-Scholes formula. Consistency and the asymptotic distribution of the estimate are provided. We demonstrate the estimator using German DAX index option data to recover the smile and the implied volatility surface.

Suggested Citation

  • Fengler, Matthias R. & Wang, Qihua, 2003. "Fitting the Smile Revisited: A Least Squares Kernel Estimator for the Implied Volatility Surface," SFB 373 Discussion Papers 2003,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    • Handle: RePEc:zbw:sfb373:200325
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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. Matthias Fengler & Wolfgang Härdle & Christophe Villa, 2003. "The Dynamics of Implied Volatilities: A Common Principal Components Approach," Review of Derivatives Research, Springer, vol. 6(3), pages 179-202, October.
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    11. Beckers, Stan, 1981. "Standard deviations implied in option prices as predictors of future stock price variability," Journal of Banking & Finance, Elsevier, vol. 5(3), pages 363-381, September.
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    Cited by:

    1. Matthias Fengler & Wolfgang Härdle & Christophe Villa, 2003. "The Dynamics of Implied Volatilities: A Common Principal Components Approach," Review of Derivatives Research, Springer, vol. 6(3), pages 179-202, October.
    2. Vedant Choudhary & Sebastian Jaimungal & Maxime Bergeron, 2023. "FuNVol: A Multi-Asset Implied Volatility Market Simulator using Functional Principal Components and Neural SDEs," Papers 2303.00859, arXiv.org, revised Dec 2023.
    3. repec:hum:wpaper:sfb649dp2005-020 is not listed on IDEAS
    4. Fengler, Matthias R. & Härdle, Wolfgang & Mammen, Enno, 2003. "Implied volatility string dynamics," SFB 373 Discussion Papers 2003,54, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    5. Fengler, Matthias R. & Härdle, Wolfgang Karl & Mammen, Enno, 2005. "A dynamic semiparametric factor model for implied volatility string dynamics," SFB 649 Discussion Papers 2005-020, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.

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