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Nonparametric Option Pricing under Shape Restrictions

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Author Info
Yacine Ait-Sahalia
Jefferson Duarte

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Abstract

Frequently, economic theory places shape restrictions on functional relationships between economic variables. This paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate the state price density (SPD), or risk-neutral density, implicit in the market prices of options. The option pricing function must be monotonic and convex. Simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day-to-day option pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P500 option prices, we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999, unlike the constrained estimator we propose.

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Paper provided by National Bureau of Economic Research, Inc in its series NBER Working Papers with number 8944.

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Date of creation: May 2002
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Handle: RePEc:nbr:nberwo:8944

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Find related papers by JEL classification:
G12 - Financial Economics - - General Financial Markets - - - Asset Pricing
C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Semiparametric and Nonparametric Methods

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Ait-Sahalia, Yacine & Wang, Yubo & Yared, Francis, 2001. "Do option markets correctly price the probabilities of movement of the underlying asset?," Journal of Econometrics, Elsevier, vol. 102(1), pages 67-110, May. [Downloadable!] (restricted)
  2. Andreas Gottschling & Christian Haefke & Halbert White, 2000. "Closed Form Integration of Artificial Neural Networks with Some Applications to Finance," Econometric Society World Congress 2000 Contributed Papers 1080, Econometric Society. [Downloadable!]
    Other versions:
  3. Varian, Hal R, 1982. "The Nonparametric Approach to Demand Analysis," Econometrica, Econometric Society, vol. 50(4), pages 945-73, July. [Downloadable!] (restricted)
  4. Jarrow, Robert & Rudd, Andrew, 1982. "Approximate option valuation for arbitrary stochastic processes," Journal of Financial Economics, Elsevier, vol. 10(3), pages 347-369, November. [Downloadable!] (restricted)
  5. E. Mammen, 1999. "Smoothing Splines and Shape Restrictions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics, Finnish Statistical Society, Norwegian Statistical Association and Swedish Statistical Association, vol. 26(2), pages 239-252. [Downloadable!] (restricted)
    Other versions:
  6. Ait-Sahalia, Yacine, 1996. "Nonparametric Pricing of Interest Rate Derivative Securities," Econometrica, Econometric Society, vol. 64(3), pages 527-60, May. [Downloadable!] (restricted)
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  7. Diewert, W. E. & Parkan, C., 1985. "Tests for the consistency of consumer data," Journal of Econometrics, Elsevier, vol. 30(1-2), pages 127-147. [Downloadable!] (restricted)
  8. Yacine Ait-Sahalia & Andrew W. Lo, 1995. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," NBER Working Papers 5351, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  9. 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. [Downloadable!] (restricted)
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  10. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October. [Downloadable!] (restricted)
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  20. 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 637-54, May-June. [Downloadable!] (restricted)
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Full references

Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Abel Rodriguez & Enrique ter Horst, 2008. "Measuring expectations in options markets: An application to the SP500 index," Quantitative Finance Papers 0901.0033, arXiv.org. [Downloadable!]
  2. Xibin Zhang & Robert D. Brooks & Maxwell L. King, 2007. "A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation," Monash Econometrics and Business Statistics Working Papers 11/07, Monash University, Department of Econometrics and Business Statistics. [Downloadable!]
  3. Bertholon, H. & Monfort, A. & Pegoraro, F., 2007. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Documents de Travail 188, Banque de France. [Downloadable!]
    Other versions:
  4. Vladislav Kargin, 2003. "Consistent Estimation of Pricing Kernels from Noisy Price Data," Quantitative Finance Papers math/0310223, arXiv.org. [Downloadable!]
  5. Vladislav Kargin, 2003. "Consistent Estimation of Pricing Kernels from Noisy Price Data," Finance 0311001, EconWPA. [Downloadable!]
  6. Li, Minqiang, 2008. "Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern," MPRA Paper 11530, University Library of Munich, Germany. [Downloadable!]
  7. Jianqing Fan, 2004. "A selective overview of nonparametric methods in financial econometrics," Quantitative Finance Papers math/0411034, arXiv.org. [Downloadable!]
  8. W. Härdle & A. Yatchew, . "Dynamic Nonparametric State Price Density Estimation Using Constrained Least Squares and the Bootstrap," Sonderforschungsbereich 373 2002-16, Humboldt Universitaet Berlin.
  9. Laurini, Márcio P. & Moura, Marcelo, 2007. "Constrained Smoothing Splines for the Term Structure of Interest Rates," Ibmec Working Papers wpe_98, Ibmec Working Paper, Ibmec São Paulo. [Downloadable!]
  10. Wolff, Hendrik & Heckelei, Thomas & Mittelhammer, Ron C., 2004. "Imposing Monotonicity And Curvature On Flexible Functional Forms," 2004 Annual meeting, August 1-4, Denver, CO 20256, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association). [Downloadable!]
  11. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December. [Downloadable!] (restricted)
  12. Selim Elekdag & Prakash Kannan, 2009. "Incorporating Market Information into the Construction of the Fan Chart," IMF Working Papers 09/178, International Monetary Fund. [Downloadable!]
  13. Ming Yuan, 2009. "State price density estimation via nonparametric mixtures," Quantitative Finance Papers 0910.1430, arXiv.org. [Downloadable!]
  14. Matthias R. Fengler, 2005. "Arbitrage-Free Smoothing of the Implied Volatility Surface," SFB 649 Discussion Papers SFB649DP2005-019, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany. [Downloadable!]
  15. Laurini, Márcio P., 2007. "Imposing No-Arbitrage Conditions In Implied Volatility Surfaces Using Constrained Smoothing Splines," Ibmec Working Papers wpe_87, Ibmec Working Paper, Ibmec São Paulo. [Downloadable!]
  16. 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. [Downloadable!]
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