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Retrieving Risk-Neutral Densities Embedded in VIX Options: a Non-Structural Approach

Author

Listed:
  • Andrea Barletta

    (Aarhus University)

  • Paolo Santucci de Magistris

    (Aarhus University and CREATES)

  • Francesco Violante

    (Aarhus University and CREATES)

Abstract

We propose a non-structural pricing method to retrieve the risk-neutral density implied by options contracts on the CBOE VIX. The method is based on orthogonal polynomial expansions around a kernel density and yields the risk-neutral density of the underlying asset without the need for modeling its dynamics. The method imposes only mild regularity conditions on shape of the density. The approach can be thought of as an alternative to Hermite expansions where the kernel has positive support. .e family of Laguerre kernels is extended to include the GIG and the generalized Weibull densities, which, due to their flexible rate of decay, are better suited at modeling the density of the VIX. Based on this technique, we propose a simple and robust way to estimate the expansion coefficients by means of a principal components analysis. We show that the proposed methodology yields an accurate approximation of the risk-neutral density also when the no-arbitrage and efficient option prices are contaminated by measurement errors. A number of numerical illustrations support the adequacy and the flexibility of the proposed expansions in a large variety of cases.

Suggested Citation

  • Andrea Barletta & Paolo Santucci de Magistris & Francesco Violante, 2016. "Retrieving Risk-Neutral Densities Embedded in VIX Options: a Non-Structural Approach," CREATES Research Papers 2016-20, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2016-20
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    References listed on IDEAS

    as
    1. Mark Britten‐Jones & Anthony Neuberger, 2000. "Option Prices, Implied Price Processes, and Stochastic Volatility," Journal of Finance, American Finance Association, vol. 55(2), pages 839-866, April.
    2. Filipović, Damir & Mayerhofer, Eberhard & Schneider, Paul, 2013. "Density approximations for multivariate affine jump-diffusion processes," Journal of Econometrics, Elsevier, vol. 176(2), pages 93-111.
    3. Jin E. Zhang & Yingzi Zhu, 2006. "VIX futures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 26(6), pages 521-531, June.
    4. Coutant, Sophie & Jondeau, Eric & Rockinger, Michael, 2001. "Reading PIBOR futures options smiles: The 1997 snap election," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1957-1987, November.
    5. Jondeau, Eric & Rockinger, Michael, 2001. "Gram-Charlier densities," Journal of Economic Dynamics and Control, Elsevier, vol. 25(10), pages 1457-1483, October.
    6. Charles J. Corrado & Tie Su, 1996. "Skewness And Kurtosis In S&P 500 Index Returns Implied By Option Prices," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 19(2), pages 175-192, June.
    7. Charles J. Corrado & Tie Su, 1996. "S&P 500 index option tests of Jarrow and Rudd's approximate option valuation formula," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 16(6), pages 611-629, September.
    8. Massimiliano Caporin & Eduardo Rossi & Paolo Santucci de Magistris, 2014. "Chasing volatility - A persistent multiplicative error model with jumps," CREATES Research Papers 2014-29, Department of Economics and Business Economics, Aarhus University.
    9. Tim Bollerslev & Viktor Todorov, 2011. "Tails, Fears, and Risk Premia," Journal of Finance, American Finance Association, vol. 66(6), pages 2165-2211, December.
    10. Song, Zhaogang & Xiu, Dacheng, 2016. "A tale of two option markets: Pricing kernels and volatility risk," Journal of Econometrics, Elsevier, vol. 190(1), pages 176-196.
    11. Rama Cont & Thomas Kokholm, 2013. "A Consistent Pricing Model For Index Options And Volatility Derivatives," Post-Print hal-00801536, HAL.
    12. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    13. Robert JARROW & Andrew RUDD, 2008. "Approximate Option Valuation For Arbitrary Stochastic Processes," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 1, pages 9-31, World Scientific Publishing Co. Pte. Ltd..
    14. Damiano Brigo & Fabio Mercurio, 2002. "Lognormal-Mixture Dynamics And Calibration To Market Volatility Smiles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(04), pages 427-446.
    15. Mencía, Javier & Sentana, Enrique, 2013. "Valuation of VIX derivatives," Journal of Financial Economics, Elsevier, vol. 108(2), pages 367-391.
    16. Viktor Todorov & George Tauchen, 2011. "Volatility Jumps," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(3), pages 356-371, July.
    17. Trino-Manuel Ñíguez & Javier Perote, 2012. "Forecasting Heavy-Tailed Densities with Positive Edgeworth and Gram-Charlier Expansions," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 74(4), pages 600-627, August.
    18. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    19. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    20. Yingzi Zhu & Jin E. Zhang, 2007. "Variance Term Structure And Vix Futures Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(01), pages 111-127.
    21. Charles J. Corrado & Tie Su, 1996. "Skewness And Kurtosis In S&P 500 Index Returns Implied By Option Prices," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 19(2), pages 175-192, June.
    22. Rompolis, Leonidas S. & Tzavalis, Elias, 2008. "Recovering Risk Neutral Densities from Option Prices: A New Approach," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 43(4), pages 1037-1053, December.
    23. Todorov, Viktor & Tauchen, George & Grynkiv, Iaryna, 2014. "Volatility activity: Specification and estimation," Journal of Econometrics, Elsevier, vol. 178(P1), pages 180-193.
    24. Zhiguang Wang & Robert T. Daigler, 2011. "The performance of VIX option pricing models: Empirical evidence beyond simulation," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 31(3), pages 251-281, March.
    25. Rama Cont, 2006. "Model uncertainty and its impact on the pricing of derivative instruments," Post-Print halshs-00002695, HAL.
    26. Ait-Sahalia, Yacine & Lo, Andrew W., 2000. "Nonparametric risk management and implied risk aversion," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 9-51.
    27. Peter Christoffersen & Kris Jacobs & Karim Mimouni, 2010. "Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns, and Option Prices," The Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 3141-3189, August.
    28. Andersen, Torben G. & Fusari, Nicola & Todorov, Viktor, 2015. "The risk premia embedded in index options," Journal of Financial Economics, Elsevier, vol. 117(3), pages 558-584.
    29. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    30. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    31. Zhang, Lan & Mykland, Per A. & Aït-Sahalia, Yacine, 2011. "Edgeworth expansions for realized volatility and related estimators," Journal of Econometrics, Elsevier, vol. 160(1), pages 190-203, January.
    32. Dilip B. Madan & Frank Milne, 1994. "Contingent Claims Valued And Hedged By Pricing And Investing In A Basis," Mathematical Finance, Wiley Blackwell, vol. 4(3), pages 223-245, July.
    33. Xiu, Dacheng, 2014. "Hermite polynomial based expansion of European option prices," Journal of Econometrics, Elsevier, vol. 179(2), pages 158-177.
    34. Christian Bayer & Jim Gatheral & Morten Karlsmark, 2013. "Fast Ninomiya--Victoir calibration of the double-mean-reverting model," Quantitative Finance, Taylor & Francis Journals, vol. 13(11), pages 1813-1829, November.
    35. repec:bla:jfinan:v:53:y:1998:i:2:p:499-547 is not listed on IDEAS
    36. Monteiro, Ana Margarida & Tutuncu, Reha H. & Vicente, Luis N., 2008. "Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity," European Journal of Operational Research, Elsevier, vol. 187(2), pages 525-542, June.
    37. George J. Jiang & Yisong S. Tian, 2005. "The Model-Free Implied Volatility and Its Information Content," The Review of Financial Studies, Society for Financial Studies, vol. 18(4), pages 1305-1342.
    38. Rama Cont, 2006. "Model Uncertainty And Its Impact On The Pricing Of Derivative Instruments," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 519-547, July.
    39. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
    40. Bujar Huskaj & Marcus Nossman, 2013. "A Term Structure Model for VIX Futures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(5), pages 421-442, May.
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    More about this item

    Keywords

    VIX options; orthogonal expansions; non-structural modeling; principal components;
    All these keywords.

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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