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Arbitrage-Free Smoothing of the Implied Volatility Surface

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  • Matthias R. Fengler

Abstract

The pricing accuracy and pricing performance of local volatility models crucially depends on absence of arbitrage in the implied volatility surface: an input implied volatility surface that is not arbitrage-free invariably results in negative transition probabilities and/ or negative local volatilities, and ultimately, into mispricings. The common smoothing algorithms of the implied volatility surface cannot guarantee the absence arbitrage. Here, we propose an approach for smoothing the implied volatility smile in an arbitrage-free way. Our methodology is simple to implement, computationally cheap and builds on the well-founded theory of natural smoothing splines under suitable shape constraints. Unlike other methods, our approach also works when input data are scarce and not arbitrage-free. Thus, it can easily be integrated into standard local volatility pricers.

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Bibliographic Info

Paper provided by Sonderforschungsbereich 649, Humboldt University, Berlin, Germany in its series SFB 649 Discussion Papers with number SFB649DP2005-019.

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Length: 37 pages
Date of creation: Mar 2005
Date of revision:
Handle: RePEc:hum:wpaper:sfb649dp2005-019

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Keywords: Arbitrage-Free Smoothing; Volatility; Implied Volatility Surface;

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References

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  1. Härdle, Wolfgang & Hlávka, Zdenek, 2009. "Dynamics of state price densities," Journal of Econometrics, Elsevier, vol. 150(1), pages 1-15, May.
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  6. Enno MAMMEN & C. THOMAS-AGNAN, 1996. "Smoothing Splines And Shape Restrictions," SFB 373 Discussion Papers 1996,87, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  7. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
  8. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
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  12. 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.
  13. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
  14. E. Mammen, 1999. "Smoothing Splines and Shape Restrictions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 26(2), pages 239-252.
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Citations

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Cited by:
  1. Fengler, Matthias & Hin, Lin-Yee, 2011. "Semi-nonparametric estimation of the call price surface under strike and time-to-expiry no-arbitrage constraints," Economics Working Paper Series 1136, University of St. Gallen, School of Economics and Political Science, revised May 2013.
  2. Bernd Engelmann & Matthias Fengler & Morten Nalholm & Peter Schwendner, 2006. "Static versus dynamic hedges: an empirical comparison for barrier options," Review of Derivatives Research, Springer, vol. 9(3), pages 239-264, November.
  3. Kim, Namhyoung & Lee, Jaewook, 2013. "No-arbitrage implied volatility functions: Empirical evidence from KOSPI 200 index options," Journal of Empirical Finance, Elsevier, vol. 21(C), pages 36-53.
  4. Stefano Galluccio & Yann Le Cam, 2005. "Implied Calibration of Stochastic Volatility Jump Diffusion Models," Finance 0510028, EconWPA.
  5. Michal Benko & Wolfgang Härdle & Alois Kneip, 2006. "Common Functional Principal Components," SFB 649 Discussion Papers SFB649DP2006-010, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  6. Jim Gatheral & Antoine Jacquier, 2012. "Arbitrage-free SVI volatility surfaces," Papers 1204.0646, arXiv.org, revised Mar 2013.
  7. Gaoyue Guo & Antoine Jacquier & Claude Martini & Leo Neufcourt, 2012. "Generalised arbitrage-free SVI volatility surfaces," Papers 1210.7111, arXiv.org, revised Oct 2013.
  8. Wolfgang Härdle & Zdenek Hlavka, 2005. "Dynamics of State Price Densities," SFB 649 Discussion Papers SFB649DP2005-021, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  9. Andrey Itkin, 2014. "One more no-arbitrage parametric fit of volatility smile," Papers 1407.0256, arXiv.org, revised Jul 2014.
  10. Laurini, Márcio P., 2007. "Imposing No-Arbitrage Conditions In Implied Volatility Surfaces Using Constrained Smoothing Splines," Insper Working Papers wpe_89, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
  11. Bo Zhao & Stewart Hodges, 2013. "Parametric modeling of implied smile functions: a generalized SVI model," Review of Derivatives Research, Springer, vol. 16(1), pages 53-77, April.
  12. Sylvain Corlay, 2013. "B-spline techniques for volatility modeling," Papers 1306.0995, arXiv.org, revised May 2014.
  13. Carol Alexander & Johannes Rauch, 2014. "Discretisation-Invariant Swaps," Papers 1404.1351, arXiv.org.
  14. Gabriel Drimus & Walter Farkas, 2013. "Local volatility of volatility for the VIX market," Review of Derivatives Research, Springer, vol. 16(3), pages 267-293, October.

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