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A note on skewness and kurtosis adjusted option pricing models under the Martingale restriction

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  • Emmanuel Jurczenko
  • Bertrand Maillet
  • Bogdan Negrea

Abstract

Several authors have proposed series expansion methods to price options when the risk-neutral density is asymmetric and leptokurtic. Among these, Corrado and Su (1996) provide an intuitive pricing formula based on a Gram-Charlier Type A series expansion. However, their formula contains a typographic error that can be significant. However, their formula contains a typographic error that can be significant. Brown and Robinson (2002) correct their pricing formula and provide and example of economic significance under plausible market conditions. The purpose of this comment is to slightly modify their pricing formula to provide consistency with a martingale restriction. We also compare the sensitivities of option prices to shifts in skewness and kurtosis using parameter values from sCorrado and Su (19ssss96) and Brown and Robinson (2002), and market data from the French options market. We show that differences between the original, corrected and our modified versions of the Corrado and Su (1996) original model are minor on the whole sample, but could be economically significant in specific cases, namely for the maturity and far-from-the-money options when markets are turbulent.

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  • Emmanuel Jurczenko & Bertrand Maillet & Bogdan Negrea, 2004. "A note on skewness and kurtosis adjusted option pricing models under the Martingale restriction," Quantitative Finance, Taylor & Francis Journals, vol. 4(5), pages 479-488.
    • Handle: RePEc:taf:quantf:v:4:y:2004:i:5:p:479-488
    DOI: 10.1080/14697680400000032
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    1. Andreou, Panayiotis C. & Charalambous, Chris & Martzoukos, Spiros H., 2010. "Generalized parameter functions for option pricing," Journal of Banking & Finance, Elsevier, vol. 34(3), pages 633-646, March.
    2. Chateau, Jean-Pierre D., 2011. "Contribution à la réglementation de Bâle-3 : de la consistance interne du continuum du crédit commercial en marquant à la « valeur de modèle » le risque de crédit des engagements de crédit," L'Actualité Economique, Société Canadienne de Science Economique, vol. 87(4), pages 445-479, décembre.
    3. Monica Billio & Bertrand Maillet & Loriana Pelizzon, 2022. "A meta-measure of performance related to both investors and investments characteristics," Annals of Operations Research, Springer, vol. 313(2), pages 1405-1447, June.
    4. Ciprian Necula & Gabriel Drimus & Walter Farkas, 2019. "A general closed form option pricing formula," Review of Derivatives Research, Springer, vol. 22(1), pages 1-40, April.
    5. Lin, Shin-Hung & Huang, Hung-Hsi & Li, Sheng-Han, 2015. "Option pricing under truncated Gram–Charlier expansion," The North American Journal of Economics and Finance, Elsevier, vol. 32(C), pages 77-97.
    6. Dongdong Hu & Hasanjan Sayit & Svetlozar T. Rachev, 2021. "Moment Matching Method for Pricing Spread Options with Mean-Variance Mixture L\'evy Motions," Papers 2109.02872, arXiv.org, revised Feb 2024.
    7. Jiménez, Inés & Mora-Valencia, Andrés & Perote, Javier, 2022. "Has the interaction between skewness and kurtosis of asset returns information content for risk forecasting?," Finance Research Letters, Elsevier, vol. 49(C).
    8. Cortés, Lina M. & Mora-Valencia, Andrés & Perote, Javier, 2020. "Retrieving the implicit risk neutral density of WTI options with a semi-nonparametric approach," The North American Journal of Economics and Finance, Elsevier, vol. 54(C).
    9. Ozge Sezgin Alp, 2016. "The Performance of Skewness and Kurtosis Adjusted Option Pricing Model in Emerging Markets: A case of Turkish Derivatives Market," International Journal of Finance & Banking Studies, Center for the Strategic Studies in Business and Finance, vol. 5(3), pages 70-84, April.
    10. Chateau, John-Peter D., 2009. "Marking-to-model credit and operational risks of loan commitments: A Basel-2 advanced internal ratings-based approach," International Review of Financial Analysis, Elsevier, vol. 18(5), pages 260-270, December.
    11. De Clerk, Luke & Savel’ev, Sergey, 2022. "AI algorithms for fitting GARCH parameters to empirical financial data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).
    12. Sofiane Aboura & Didier Maillard, 2016. "Option Pricing Under Skewness and Kurtosis Using a Cornish–Fisher Expansion," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 36(12), pages 1194-1209, December.
    13. Dongdong Hu & Hasanjan Sayit & Frederi Viens, 2023. "Pricing basket options with the first three moments of the basket: log-normal models and beyond," Papers 2302.08041, arXiv.org, revised Feb 2023.
    14. Chevallier, Julien & Ielpo, Florian & Mercier, Ludovic, 2009. "Risk aversion and institutional information disclosure on the European carbon market: A case-study of the 2006 compliance event," Energy Policy, Elsevier, vol. 37(1), pages 15-28, January.
    15. Inés Jiménez & Andrés Mora-Valencia & Javier Perote, 2022. "Dynamic selection of Gram–Charlier expansions with risk targets: an application to cryptocurrencies," Risk Management, Palgrave Macmillan, vol. 24(1), pages 81-99, March.
    16. Schlögl, Erik, 2013. "Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order," Journal of Economic Dynamics and Control, Elsevier, vol. 37(3), pages 611-632.
    17. Emmanuel Jurczenko & Bertrand Maillet & Paul Merlin, 2008. "Efficient Frontier for Robust Higher-order Moment Portfolio Selection," Post-Print halshs-00336475, HAL.
    18. Maria Grazia Zoia & Gianmarco Vacca & Laura Barbieri, 2020. "Modeling Multivariate Financial Series and Computing Risk Measures via Gram–Charlier-Like Expansions," Risks, MDPI, vol. 8(4), pages 1-21, November.
    19. Chateau, John-Peter D., 2007. "Beyond Basel-2 simplified standardized approach: Credit risk valuation of short-term loan commitments," International Review of Financial Analysis, Elsevier, vol. 16(5), pages 412-433.

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