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CAPM and Option Pricing with Elliptical Disbributions


  • Mahmoud Hamada
  • Emiliano A. Valdez


In this paper, we offer an alternative proof of the Capital Asset Pricing Model when the returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption in modelling asset returns. The class of elliptical distributions,which includes the more familiar Normal distribution, provides flexibility in modelling the thickness of tails associated with the possibility that asset returns take extreme values with non-negligible probabilities. Within this framework, we prove a new version of Stein's lemma for elliptical distribution and use this result to derive the CAPM when returns are elliptical. We also derive a closed form solution of call option prices when the underlying is elliptically distributed. We use the probability distortion function approach based on the dual utility theory of choice under uncertainty.

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  • Mahmoud Hamada & Emiliano A. Valdez, 2004. "CAPM and Option Pricing with Elliptical Disbributions," Research Paper Series 120, Quantitative Finance Research Centre, University of Technology, Sydney.
  • Handle: RePEc:uts:rpaper:120

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    References listed on IDEAS

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    2. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 26(01), pages 71-92, May.
    3. Landsman, Zinoviy, 2002. "Credibility theory: a new view from the theory of second order optimal statistics," Insurance: Mathematics and Economics, Elsevier, vol. 30(3), pages 351-362, June.
    4. Mahmoud Hamada & Michael Sherris, 2003. "Contingent claim pricing using probability distortion operators: methods from insurance risk pricing and their relationship to financial theory," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(1), pages 19-47.
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    6. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    7. N. H. Bingham & Rudiger Kiesel, 2002. "Semi-parametric modelling in finance: theoretical foundations," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 241-250.
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    Cited by:

    1. Kume, Alfred & Hashorva, Enkelejd, 2012. "Calculation of Bayes premium for conditional elliptical risks," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 632-635.
    2. Polonik, Wolfgang & Yao, Qiwei, 2008. "Testing for multivariate volatility functions using minimum volume sets and inverse regression," Journal of Econometrics, Elsevier, vol. 147(1), pages 151-162, November.
    3. Landsman, Zinoviy & Neslehová, Johanna, 2008. "Stein's Lemma for elliptical random vectors," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 912-927, May.
    4. Lundtofte, Frederik & Wilhelmsson, Anders, 2011. "Idiosyncratic Risk and Higher-Order Cumulants," Working Papers 2011:33, Lund University, Department of Economics.
    5. Fan, Jianqing & Han, Fang & Liu, Han & Vickers, Byron, 2016. "Robust inference of risks of large portfolios," Journal of Econometrics, Elsevier, vol. 194(2), pages 298-308.
    6. SADEFO KAMDEM Jules, 2004. "VaR and ES for Linear Portfolios with mixture of Generalized Laplace Distributed Risk Factors," Risk and Insurance 0406001, EconWPA.
    7. Jianqing Fan & Yuan Liao & Han Liu, 2016. "An overview of the estimation of large covariance and precision matrices," Econometrics Journal, Royal Economic Society, vol. 19(1), pages 1-32, February.
    8. Nikolay Gospodinov & Raymond Kan & Cesare Robotti, 2012. "Analytical solution for the constrained Hansen-Jagannathan distance under multivariate ellipticity," FRB Atlanta Working Paper 2012-18, Federal Reserve Bank of Atlanta.
    9. Francisco Blasques & Andre Lucas & Erkki Silde, 2013. "Stationarity and Ergodicity Regions for Score Driven Dynamic Correlation Models," Tinbergen Institute Discussion Papers 13-097/IV/DSF59, Tinbergen Institute.
    10. Nikolay Gospodinov & Raymond Kan & Cesare Robotti, 2010. "On the Hansen-Jagannathan distance with a no-arbitrage constraint," FRB Atlanta Working Paper 2010-04, Federal Reserve Bank of Atlanta.
    11. Gospodinov, Nikolay & Kan, Raymond & Robotti, Cesare, 2016. "On the properties of the constrained Hansen–Jagannathan distance," Journal of Empirical Finance, Elsevier, vol. 36(C), pages 121-150.

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