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Asymptotics of the price oscillations of a European call option in a tree model

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  • Francine Diener
  • MARC Diener

Abstract

It is well known that the price of a European vanilla option computed in a binomial tree model converges toward the Black‐Scholes price when the time step tends to zero. Moreover, it has been observed that this convergence is of order 1/n in usual models and that it is oscillatory. In this paper, we compute this oscillatory behavior using asymptotics of Laplace integrals, giving explicitly the first terms of the asymptotics. This allows us to show that there is no asymptotic expansion in the usual sense, but that the rate of convergence is indeed of order 1/n in the case of usual binomial models since the second term (in ) vanishes. The next term is of type C2(n)/n, with C2(n) some explicit bounded function of n that has no limit when n tends to infinity.

Suggested Citation

  • Francine Diener & MARC Diener, 2004. "Asymptotics of the price oscillations of a European call option in a tree model," Mathematical Finance, Wiley Blackwell, vol. 14(2), pages 271-293, April.
    • Handle: RePEc:bla:mathfi:v:14:y:2004:i:2:p:271-293
    DOI: 10.1111/j.0960-1627.2004.00192.x
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    References listed on IDEAS

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    1. Imme van den Berg, 2000. "Principles of Infinitesimal Stochastic and Financial Analysis," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4468, January.
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    Cited by:

    1. Ralf Korn & Stefanie Müller, 2013. "The optimal-drift model: an accelerated binomial scheme," Finance and Stochastics, Springer, vol. 17(1), pages 135-160, January.
    2. Mark Joshi & Mike Staunton, 2012. "On the analytical/numerical pricing of American put options against binomial tree prices," Quantitative Finance, Taylor & Francis Journals, vol. 12(1), pages 17-20, December.
    3. Mark Joshi, 2009. "Achieving smooth asymptotics for the prices of European options in binomial trees," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 171-176.
    4. Karl Grosse-Erdmann & Fabien Heuwelyckx, 2015. "The pricing of lookback options and binomial approximation," Papers 1502.02819, arXiv.org.
    5. Atul Chandra & Peter R. Hartley & Gopalan Nair, 2022. "Multiple Volatility Real Options Approach to Investment Decisions Under Uncertainty," Decision Analysis, INFORMS, vol. 19(2), pages 79-98, June.
    6. San-Lin Chung & Pai-Ta Shih, 2007. "Generalized Cox-Ross-Rubinstein Binomial Models," Management Science, INFORMS, vol. 53(3), pages 508-520, March.
    7. Kyoung-Sook Moon & Hongjoong Kim, 2013. "A multi-dimensional local average lattice method for multi-asset models," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 873-884, May.
    8. Jean-Christophe Breton & Youssef El-Khatib & Jun Fan & Nicolas Privault, 2021. "A q-binomial extension of the CRR asset pricing model," Papers 2104.10163, arXiv.org, revised Feb 2023.
    9. Leduc, Guillaume, 2012. "Arbitrarily Fast CRR Schemes," MPRA Paper 42094, University Library of Munich, Germany, revised 20 Oct 2012.
    10. Jérôme Lelong & Antonino Zanette, 2010. "Tree methods," Post-Print hal-00776713, HAL.
    11. Gongqiu Zhang & Lingfei Li, 2019. "Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior," Operations Research, INFORMS, vol. 67(2), pages 407-427, March.
    12. Guillaume Leduc & Merima Nurkanovic Hot, 2020. "Joshi’s Split Tree for Option Pricing," Risks, MDPI, vol. 8(3), pages 1-26, August.
    13. Fabien Heuwelyckx, 2013. "Convergence of European Lookback Options with Floating Strike in the Binomial Model," Papers 1302.2312, arXiv.org, revised Oct 2013.
    14. Wael Bahsoun & Pawel Góra & Silvia Mayoral & Manuel Morales, 2006. "Random Dynamics and Finance: Constructing Implied Binomial Trees from a Predetermined Stationary Den," Faculty Working Papers 13/06, School of Economics and Business Administration, University of Navarra.
    15. Alona Bock & Ralf Korn, 2016. "Improving Convergence of Binomial Schemes and the Edgeworth Expansion," Risks, MDPI, vol. 4(2), pages 1-22, May.
    16. N. El Karoui & Y. Jiao, 2009. "Stein’s method and zero bias transformation for CDO tranche pricing," Finance and Stochastics, Springer, vol. 13(2), pages 151-180, April.
    17. Elisa Appolloni & Andrea Ligori, 2014. "Efficient tree methods for pricing digital barrier options," Papers 1401.2900, arXiv.org, revised Jan 2014.
    18. Karl Grosse-Erdmann & Fabien Heuwelyckx, 2016. "The pricing of lookback options and binomial approximation," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 39(1), pages 33-67, April.

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