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A Multinomial Approximation For American Option Prices In Lévy Process Models

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  • Ross A. Maller
  • David H. Solomon
  • Alex Szimayer

Abstract

This paper gives a tree‐based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American‐type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path‐dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity.

Suggested Citation

  • Ross A. Maller & David H. Solomon & Alex Szimayer, 2006. "A Multinomial Approximation For American Option Prices In Lévy Process Models," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 613-633, October.
    • Handle: RePEc:bla:mathfi:v:16:y:2006:i:4:p:613-633
    DOI: 10.1111/j.1467-9965.2006.00286.x
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    References listed on IDEAS

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    Cited by:

    1. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    2. Boris Buchmann & Benjamin Kaehler & Ross Maller & Alexander Szimayer, 2015. "Multivariate Subordination using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing," Papers 1502.03901, arXiv.org, revised Oct 2016.
    3. Buchmann, Boris & Kaehler, Benjamin & Maller, Ross & Szimayer, Alexander, 2017. "Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2208-2242.
    4. Xu Guo & Yutian Li, 2016. "Valuation of American options under the CGMY model," Quantitative Finance, Taylor & Francis Journals, vol. 16(10), pages 1529-1539, October.
    5. Hatem Ben‐Ameur & Rim Chérif & Bruno Rémillard, 2020. "Dynamic programming for valuing American options under a variance‐gamma process," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 40(10), pages 1548-1561, October.
    6. Kleinert, Florian & van Schaik, Kees, 2015. "A variation of the Canadisation algorithm for the pricing of American options driven by Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3234-3254.
    7. Aleš Černý, 2007. "Optimal Continuous‐Time Hedging With Leptokurtic Returns," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 175-203, April.
    8. Warren J. Hahn & James S. Dyer, 2011. "A Discrete Time Approach for Modeling Two-Factor Mean-Reverting Stochastic Processes," Decision Analysis, INFORMS, vol. 8(3), pages 220-232, September.
    9. Xiaotong Lian & Yingda Song, 2021. "Pricing and calibration of the futures options market: A unified approximation," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(7), pages 1074-1091, July.
    10. Florian Kleinert & Kees van Schaik, 2013. "A variation of the Canadisation algorithm for the pricing of American options driven by L\'evy processes," Papers 1304.4534, arXiv.org.
    11. Kourouvakalis, Stylianos, 2008. "Méthodes numériques pour la valorisation d'options swings et autres problèmes sur les matières premières," Economics Thesis from University Paris Dauphine, Paris Dauphine University, number 123456789/116 edited by Geman, Hélyette.

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