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On option pricing models in the presence of heavy tails

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  • Michel Vellekoop
  • Hans Nieuwenhuis

Abstract

We propose a modification of the option pricing framework derived by Borland which removes the possibilities for arbitrage within this framework. It turns out that such arbitrage possibilities arise due to an incorrect derivation of the martingale transformation in the non-Gaussian option models which are used in that paper. We show how a similar model can be built for the asset price processes which excludes arbitrage. However, the correction causes the pricing formulas to be less explicit than the ones in the original formulation, since the stock price itself is no longer a Markov process. Practical option pricing algorithms will therefore have to resort to Monte Carlo methods or partial differential equations and we show how these can be implemented. An extra parameter, which needs to be specified before the model can be used, will give market makers some extra freedom when fitting their model to market data.

Suggested Citation

  • Michel Vellekoop & Hans Nieuwenhuis, 2007. "On option pricing models in the presence of heavy tails," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 563-573.
    • Handle: RePEc:taf:quantf:v:7:y:2007:i:5:p:563-573
    DOI: 10.1080/14697680601077967
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    References listed on IDEAS

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    1. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
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    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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    7. 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 637-654, May-June.
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    Cited by:

    1. Giacomo Bormetti & Sofia Cazzaniga, 2011. "Multiplicative noise, fast convolution, and pricing," Papers 1107.1451, arXiv.org.
    2. Zhao, Pan & Pan, Jian & Yue, Qin & Zhang, Jinbo, 2021. "Pricing of financial derivatives based on the Tsallis statistical theory," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    3. Allan Jonathan da Silva & Jack Baczynski & Leonardo Fagundes de Mello, 2023. "Hedging Interest Rate Options with Reinforcement Learning: an investigation of a heavy-tailed distribution," Business and Management Studies, Redfame publishing, vol. 9(2), pages 1-14, December.
    4. Borland, Lisa, 2016. "Exploring the dynamics of financial markets: from stock prices to strategy returns," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 59-74.
    5. Ahmad Hajihasani & Ali Namaki & Nazanin Asadi & Reza Tehrani, 2020. "Non-Extensive Value-at-Risk Estimation During Times of Crisis," Papers 2005.09036, arXiv.org, revised Jan 2021.
    6. Trindade, Marco A.S. & Floquet, Sergio & Filho, Lourival M. Silva, 2020. "Portfolio theory, information theory and Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    7. Giacomo Bormetti & Sofia Cazzaniga, 2014. "Multiplicative noise, fast convolution and pricing," Quantitative Finance, Taylor & Francis Journals, vol. 14(3), pages 481-494, March.
    8. Moretto, Enrico & Pasquali, Sara & Trivellato, Barbara, 2016. "Option pricing under deformed Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 446(C), pages 246-263.
    9. P. Peirano & D. Challet, 2012. "Baldovin-Stella stochastic volatility process and Wiener process mixtures," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 85(8), pages 1-12, August.
    10. Namaki, A. & Koohi Lai, Z. & Jafari, G.R. & Raei, R. & Tehrani, R., 2013. "Comparing emerging and mature markets during times of crises: A non-extensive statistical approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(14), pages 3039-3044.
    11. Sosa-Correa, William O. & Ramos, Antônio M.T. & Vasconcelos, Giovani L., 2018. "Investigation of non-Gaussian effects in the Brazilian option market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 525-539.

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