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Density Functionals, With An Option-Pricing Application

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  • Abadir, Karim M.
  • Rockinger, Michael

Abstract

We present a method of estimating density-related functionals, without prior knowledge of the density s functional form. The approach revolves around the specification of an explicit formula for a new class of distributions that encompasses many of the known cases in statistics, including the normal, gamma, inverse gamma, and mixtures thereof. The functionals are based on a couple of hypergeometric functions. Their parameters can be estimated, and the estimates then reveal both the functional form of the density and the parameters that determine centering, scaling, etc. The function to be estimated always leads to a valid density, by design, namely, one that is nonnegative everywhere and integrates to 1. Unlike fully nonparametric methods, our approach can be applied to small datasets. To illustrate our methodology, we apply it to finding risk-neutral densities associated with different types of financial options. We show how our approach fits the data uniformly very well. We also find that our estimated densities functional forms vary over the dataset, so that existing parametric methods will not do uniformly well.We thank Hans-J rg B ttler, Ale ern , Tony Culyer, Les Godfrey, David Hendry, Sam Kotz, Steve Lawford, Peter Phillips, Bas Werker, and three anonymous referees for their comments. We also thank for their feedback the participants at the seminars and conferences where this paper has been invited, in particular the 1998 CEPR Finance Network Workshop, the 1998 METU conference, the 1998 FORC (Warwick) conference Options: Recent Advances, Money Macro Finance Group, the Swiss National Bank, Imperial College, Tilburg University, Universit Libre de Bruxelles, the University of Oxford, Southampton University, and UMIST. The first author acknowledges support from the ESRC (UK) grant R000239538. The second author acknowledges help from the HEC Foundation and the European Community TMR grant Financial Market Efficiency and Economic Efficiency. This paper was written when the second author was affiliated with HEC-Paris.

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Bibliographic Info

Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 19 (2003)
Issue (Month): 05 (October)
Pages: 778-811

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Handle: RePEc:cup:etheor:v:19:y:2003:i:05:p:778-811_19

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Cited by:
  1. Kristensen, Dennis & Mele, Antonio, 2011. "Adding and subtracting Black-Scholes: A new approach to approximating derivative prices in continuous-time models," Journal of Financial Economics, Elsevier, vol. 102(2), pages 390-415.
  2. Guglielmo Caporale & Mario Cerrato, 2010. "Using Chebyshev Polynomials to Approximate Partial Differential Equations," Computational Economics, Society for Computational Economics, vol. 35(3), pages 235-244, March.
  3. Giacomini, Raffaella & Gottschling, Andreas & Haefke, Christian & White, Halbert, 2007. "Mixtures of t-distributions for Finance and Forecasting," Economics Series 216, Institute for Advanced Studies.
  4. Guglielmo Maria Caporale & Mario Cerrato, 2008. "Chebyshev polynomial approximation to approximate partial differential equations," Working Papers 2008_16, Business School - Economics, University of Glasgow.
  5. Guillermo Benavides Perales & Israel Felipe Mora Cuevas, 2008. "Parametric vs. non-parametric methods for estimating option implied risk-neutral densities: the case of the exchange rate Mexican peso – US dollar," Ensayos Revista de Economia, Universidad Autonoma de Nuevo Leon, Facultad de Economia, vol. 0(1), pages 33-52, May.
  6. Ruijun Bu & Kaddour Hadri, 2005. "Estimating the Risk Neutral Probability Density Functions Natural Spline versus Hypergeometric Approach Using European Style Options," Research Papers 200510, University of Liverpool Management School.
  7. Karim M. Abadir, 2013. "Lies, Damned Lies, and Statistics? Examples From Finance and Economics," Central European Journal of Economic Modelling and Econometrics, CEJEME, vol. 5(4), pages 231-248, December.
  8. Ruijun Bu & Ludovic Giet & Kaddour Hadri & Michel Lubrano, 2009. "Modeling Multivariate Interest Rates using Time-Varying Copulas and Reducible Stochastic Differential Equations," Working Papers halshs-00408014, HAL.
  9. Rompolis, Leonidas S., 2010. "Retrieving risk neutral densities from European option prices based on the principle of maximum entropy," Journal of Empirical Finance, Elsevier, vol. 17(5), pages 918-937, December.
  10. A. Monteiro & R. Tütüncü & L. Vicente, 2011. "Estimation of risk-neutral density surfaces," Computational Management Science, Springer, vol. 8(4), pages 387-414, November.
  11. Peter Christoffersen & Kris Jacobs & Bo Young Chang, 2011. "Forecasting with Option Implied Information," CREATES Research Papers 2011-46, School of Economics and Management, University of Aarhus.
  12. 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.

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