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The Convergence Rate of Option Prices in Trinomial Trees

Author

Listed:
  • Guillaume Leduc

    (Department of Mathematics and Statistics, American University of Sharjah, Sharjah P.O. Box 26666, United Arab Emirates)

  • Kenneth Palmer

    (Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan)

Abstract

We study the convergence of the binomial, trinomial, and more generally m -nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of 1 / n and 1 / n in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations.

Suggested Citation

  • Guillaume Leduc & Kenneth Palmer, 2023. "The Convergence Rate of Option Prices in Trinomial Trees," Risks, MDPI, vol. 11(3), pages 1-33, March.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:3:p:52-:d:1088867
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    References listed on IDEAS

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    1. Muroi, Yoshifumi & Suda, Shintaro, 2022. "Binomial tree method for option pricing: Discrete cosine transform approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 198(C), pages 312-331.
    2. Alona Bock & Ralf Korn, 2016. "Improving Convergence of Binomial Schemes and the Edgeworth Expansion," Risks, MDPI, vol. 4(2), pages 1-22, May.
    3. Yisong Tian, 1993. "A modified lattice approach to option pricing," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 13(5), pages 563-577, August.
    4. R. H. Liu, 2010. "Regime-Switching Recombining Tree For Option Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(03), pages 479-499.
    5. Erd.inc{c} Aky{i}ld{i}r{i}m & Yan Dolinsky & H. Mete Soner, 2012. "Approximating stochastic volatility by recombinant trees," Papers 1205.3555, arXiv.org, revised Jul 2014.
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    Cited by:

    1. Guillaume Leduc, 2024. "The Boyle–Romberg Trinomial Tree, a Highly Efficient Method for Double Barrier Option Pricing," Mathematics, MDPI, vol. 12(7), pages 1-15, March.

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