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Arbitrage-free smoothing of the implied volatility surface

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  • Matthias Fengler

Abstract

The pricing accuracy and pricing performance of local volatility models depends on the absence of arbitrage in the implied volatility surface. An input implied volatility surface that is not arbitrage-free can result in negative transition probabilities and consequently mispricings and false greeks. We propose an approach for smoothing the implied volatility smile in an arbitrage-free way. The method is simple to implement, computationally cheap and builds on the well-founded theory of natural smoothing splines under suitable shape constraints.

Suggested Citation

  • Matthias Fengler, 2009. "Arbitrage-free smoothing of the implied volatility surface," Quantitative Finance, Taylor & Francis Journals, vol. 9(4), pages 417-428.
  • Handle: RePEc:taf:quantf:v:9:y:2009:i:4:p:417-428
    DOI: 10.1080/14697680802595585
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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.
    2. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    3. 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.
    4. Yatchew, Adonis & Hardle, Wolfgang, 2006. "Nonparametric state price density estimation using constrained least squares and the bootstrap," Journal of Econometrics, Elsevier, vol. 133(2), pages 579-599, August.
    5. Hardle, Wolfgang & Linton, Oliver, 1986. "Applied nonparametric methods," Handbook of Econometrics,in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 38, pages 2295-2339 Elsevier.
    6. 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.
    7. J. Michael Harrison & Stanley R. Pliska, 1981. "Martingales and Stochastic Integrals in the Theory of Continous Trading," Discussion Papers 454, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    8. 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.
    9. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    10. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    11. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    12. M. A. H. Dempster & D. G. Richards, 2000. "Pricing American Options Fitting the Smile," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 157-177.
    13. Hentschel, Ludger, 2003. "Errors in Implied Volatility Estimation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 38(04), pages 779-810, December.
    14. Hardle, Wolfgang & Linton, Oliver, 1986. "Applied nonparametric methods," Handbook of Econometrics,in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 38, pages 2295-2339 Elsevier.
    15. Harvey, Campbell R & Whaley, Robert E, 1991. " S&P 100 Index Option Volatility," Journal of Finance, American Finance Association, vol. 46(4), pages 1251-1261, September.
    16. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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    Cited by:

    1. 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.
    2. Michal Benko & Wolfgang Härdle & Alois Kneip, 2006. "Common Functional Principal Components," SFB 649 Discussion Papers SFB649DP2006-010, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. 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.
    4. 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.
    5. repec:spr:comgts:v:14:y:2017:i:4:d:10.1007_s10287-017-0283-8 is not listed on IDEAS
    6. 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.
    7. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    8. 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.
    9. 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.
    10. Härdle, Wolfgang & Hlávka, Zdenek, 2009. "Dynamics of state price densities," Journal of Econometrics, Elsevier, vol. 150(1), pages 1-15, May.
    11. Stefano Galluccio & Yann Le Cam, 2005. "Implied Calibration of Stochastic Volatility Jump Diffusion Models," Finance 0510028, EconWPA.
    12. 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.
    13. repec:eee:apmaco:v:258:y:2015:i:c:p:372-387 is not listed on IDEAS
    14. Johannes Rauch & Carol Alexander, 2016. "Tail Risk Premia for Long-Term Equity Investors," Papers 1602.00865, arXiv.org.
    15. 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.
    16. 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.
    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. Sylvain Corlay, 2013. "B-spline techniques for volatility modeling," Papers 1306.0995, arXiv.org, revised Jun 2015.
    19. Gaoyue Guo & Antoine Jacquier & Claude Martini & Leo Neufcourt, 2012. "Generalised arbitrage-free SVI volatility surfaces," Papers 1210.7111, arXiv.org, revised May 2016.
    20. 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.
    21. 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.
    22. Yanlin Qu & Randall R. Rojas, 2017. "Closed-form Solutions of Relativistic Black-Scholes Equations," Papers 1711.04219, arXiv.org.
    23. 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.

    More about this item

    Keywords

    Implied volatility surface; Local volatility; Cubic spline smoothing; No-arbitrage constraints;

    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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