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Dynamics of State Price Densities

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  • Wolfgang Härdle
  • Zdenek Hlavka

Abstract

State price densities (SPD) are an important element in applied quantitative finance. In a Black-Scholes model they are lognormal distributions with constant volatility parameter. In practice volatility changes and the distribution deviates from log-normality. We estimate SPDs using EUREX option data on the DAX index via a nonparametric estimator of the second derivative of the (European) call price function. The estimator is constrained so as to satisfy no-arbitrage constraints and it corrects for intraday covariance structure. Given a low dimensional representation of this SPD we study its dynamic for the years 1995–2003. We calculate a prediction corridor for the DAX for a 45 day forecast. The proposed algorithm is simple, it allows calculation of future volatility and can be applied to hedging exotic options.

Suggested Citation

  • Wolfgang Härdle & Zdenek Hlavka, 2005. "Dynamics of State Price Densities," SFB 649 Discussion Papers SFB649DP2005-021, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2005-021
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    References listed on IDEAS

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    1. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, April.
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    3. 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.
    4. 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.
    5. 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.
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    8. Huynh, Kim & Kervella, Pierre & Zheng, Jun, 2002. "Estimating state-price densities with nonparametric regression," SFB 373 Discussion Papers 2002,40, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    9. Diebold, Francis X & Gunther, Todd A & Tay, Anthony S, 1998. "Evaluating Density Forecasts with Applications to Financial Risk Management," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 863-883, November.
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    11. Matthias Fengler, 2009. "Arbitrage-free smoothing of the implied volatility surface," Quantitative Finance, Taylor & Francis Journals, vol. 9(4), pages 417-428.
    12. Bondarenko, Oleg, 2003. "Estimation of risk-neutral densities using positive convolution approximation," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 85-112.
    13. 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.
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    16. L. Randall Wray & Stephanie Bell, 2004. "Introduction," Chapters,in: Credit and State Theories of Money, chapter 1 Edward Elgar Publishing.
    17. 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.
    18. Philippe Robert-Demontrond & R. Ringoot, 2004. "Introduction," Post-Print halshs-00081823, HAL.
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    Citations

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    Cited by:

    1. Hans Buehler, 2006. "Expensive martingales," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 207-218.
    2. 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.
    3. Ostap Okhrin & Stefan Trück, 2015. "Editorial to the special issue on Applicable semiparametrics of computational statistics," Computational Statistics, Springer, vol. 30(3), pages 641-646, September.
    4. Karl Härdle, Wolfgang & López-Cabrera, Brenda & Teng, Huei-Wen, 2015. "State price densities implied from weather derivatives," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 106-125.
    5. Hlavka, Zdenek, 2006. "Fast algorithm for nonparametric arbitrage-free SPD estimation," Computational Statistics & Data Analysis, Elsevier, vol. 51(4), pages 2339-2349, December.
    6. repec:hal:journl:peer-00834423 is not listed on IDEAS
    7. Matthias Fengler, 2009. "Arbitrage-free smoothing of the implied volatility surface," Quantitative Finance, Taylor & Francis Journals, vol. 9(4), pages 417-428.
    8. Diks, Cees & Panchenko, Valentyn & van Dijk, Dick, 2011. "Likelihood-based scoring rules for comparing density forecasts in tails," Journal of Econometrics, Elsevier, vol. 163(2), pages 215-230, August.
    9. Zdenek Hlavka & Michal Pesta, 2006. "Constrained General Regression in Pseudo-Sobolev Spaces with Application to Option Pricing," SFB 649 Discussion Papers SFB649DP2006-069, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

    More about this item

    Keywords

    option pricing; state price density estimation; nonlinear least squares; confidence intervals;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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