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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.

Suggested Citation

  • Matthias R. Fengler, 2005. "Arbitrage-Free Smoothing of the Implied Volatility Surface," SFB 649 Discussion Papers SFB649DP2005-019, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2005-019
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    Cited by:

    1. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    2. Härdle, Wolfgang & Hlávka, Zdenek, 2009. "Dynamics of state price densities," Journal of Econometrics, Elsevier, vol. 150(1), pages 1-15, May.
    3. Stefano Galluccio & Yann Le Cam, 2005. "Implied Calibration of Stochastic Volatility Jump Diffusion Models," Finance 0510028, EconWPA.
    4. Yanlin Qu & Randall R. Rojas, 2017. "Closed-form Solutions of Relativistic Black-Scholes Equations," Papers 1711.04219, arXiv.org.
    5. Carol Alexander & Johannes Rauch, 2014. "Model-Free Discretisation-Invariant Swaps and S&P 500 Higher-Moment Risk Premia," Papers 1404.1351, arXiv.org, revised Feb 2016.
    6. 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.
    7. repec:eee:apmaco:v:258:y:2015:i:c:p:372-387 is not listed on IDEAS
    8. 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.
    9. Pierre M. Blacque-Florentin & Badr Missaoui, 2015. "Nonparametric and arbitrage-free construction of call surfaces using l1-recovery," Papers 1506.06997, arXiv.org, revised Aug 2016.
    10. Курочкин С.В., 2016. "Выпуклость Множества Цен Опционов Как Необходимое И Достаточное Условие Отсутствия Арбитража," Журнал Экономика и математические методы (ЭММ), Центральный Экономико-Математический Институт (ЦЭМИ), vol. 52(2), pages 103-111, апрель.
    11. Michal Benko & Wolfgang Härdle & Alois Kneip, 2006. "Common Functional Principal Components," SFB 649 Discussion Papers SFB649DP2006-010, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    12. Itkin, Andrey, 2015. "To sigmoid-based functional description of the volatility smile," The North American Journal of Economics and Finance, Elsevier, vol. 31(C), pages 264-291.
    13. 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.
    14. 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.
    15. Johannes Rauch & Carol Alexander, 2016. "Tail Risk Premia for Long-Term Equity Investors," Papers 1602.00865, arXiv.org.
    16. 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.
    17. Bernales, Alejandro & Guidolin, Massimo, 2015. "Learning to smile: Can rational learning explain predictable dynamics in the implied volatility surface?," Journal of Financial Markets, Elsevier, vol. 26(C), pages 1-37.
    18. Carol Alexander & Alexander Rubinov & Markus Kalepky & Stamatis Leontsinis, 2012. "Regime‐dependent smile‐adjusted delta hedging," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 32(3), pages 203-229, March.
    19. repec:gam:jijfss:v:6:y:2018:i:1:p:30-:d:135806 is not listed on IDEAS
    20. repec:spr:comgts:v:14:y:2017:i:4:d:10.1007_s10287-017-0283-8 is not listed on IDEAS
    21. Sylvain Corlay, 2013. "B-spline techniques for volatility modeling," Papers 1306.0995, arXiv.org, revised Jun 2015.
    22. Gaoyue Guo & Antoine Jacquier & Claude Martini & Leo Neufcourt, 2012. "Generalised arbitrage-free SVI volatility surfaces," Papers 1210.7111, arXiv.org, revised May 2016.
    23. Fengler, Matthias R. & Hin, Lin-Yee, 2015. "Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints," Journal of Econometrics, Elsevier, vol. 184(2), pages 242-261.
    24. 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.

    More about this item

    Keywords

    Arbitrage-Free Smoothing; Volatility; Implied Volatility Surface;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C81 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Methodology for Collecting, Estimating, and Organizing Microeconomic Data; Data Access
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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