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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. --

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Bibliographic Info

Paper provided by Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes in its series SFB 373 Discussion Papers with number 2003,25.

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Date of creation: 2003
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Handle: RePEc:zbw:sfb373:200325

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Keywords: implied volatility surface; smile; Black-Scholes formula; least squares kernel smoothing;

References

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  1. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  2. Schmalensee, Richard & Trippi, Robert R, 1978. "Common Stock Volatility Expectations Implied by Option Premia," Journal of Finance, American Finance Association, American Finance Association, vol. 33(1), pages 129-47, March.
  3. Matthias Fengler & Wolfgang Härdle & Christophe Villa, 2003. "The Dynamics of Implied Volatilities: A Common Principal Components Approach," Review of Derivatives Research, Springer, Springer, vol. 6(3), pages 179-202, October.
  4. Bernard Dumas & Jeff Fleming & Robert E. Whaley, 1998. "Implied Volatility Functions: Empirical Tests," Journal of Finance, American Finance Association, American Finance Association, vol. 53(6), pages 2059-2106, December.
  5. Latane, Henry A & Rendleman, Richard J, Jr, 1976. "Standard Deviations of Stock Price Ratios Implied in Option Prices," Journal of Finance, American Finance Association, American Finance Association, vol. 31(2), pages 369-81, May.
  6. George Skiadopoulos & Stewart Hodges & Les Clewlow, 2000. "The Dynamics of the S&P 500 Implied Volatility Surface," Review of Derivatives Research, Springer, Springer, vol. 3(3), pages 263-282, October.
  7. Beckers, Stan, 1981. "Standard deviations implied in option prices as predictors of future stock price variability," Journal of Banking & Finance, Elsevier, Elsevier, vol. 5(3), pages 363-381, September.
  8. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(1), pages 45-60.
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Cited by:
  1. 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.
  2. Fengler, Matthias R. & Härdle, Wolfgang K. & Villa, Christophe, 2001. "The dynamics of implied volatilities: A common principal components approach," SFB 373 Discussion Papers 2001,38, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  3. Matthias Fengler & Wolfgang Härdle & Enno Mammen, 2005. "A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics," SFB 649 Discussion Papers SFB649DP2005-020, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

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