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Binomial models for option valuation - examining and improving convergence


  • Dietmar Leisen
  • Matthias Reimer


Binomial models, which describe the asset price dynamics of the continuous-time model in the limit, serve for approximate valuation of options, especially where formulas cannot be derived analytically due to properties of the considered option type. To evaluate results, one inevitably must understand the convergence properties. In the literature we find various contributions proving convergence of option prices. We examine convergence behaviour and convergence speed. Unfortunately, even in the case of European call options, distorted results occur when calculating prices along the iteration of tree refinements. These convergence patterns are examined and order of convergence one is proven for the Cox-Ross-Rubinstein model as well as for two alternative tree parameter selections from the literature. Furthermore, we define new binomial models, where the calculated option prices converge smoothly to the Black-Scholes solution, and we achieve order of convergence two with much smaller initial error. Notably, only the formulas to determine the up- and down-factors change. Finally, following a recent approach from the literature, all tree approaches are compared with respect to speed and accuracy, calculating the relative root-mean-squared error of approximate option values for a sample of randomly selected parameters across a set of refinements. Here, on average, the same degree of accuracy is achieved 1400 times faster with the new binomial models. We also give some insights into the peculiarities in the valuation of the American put option. Inspecting the numerical results, the approximation of American-type options with the new models exhibits order of convergence one, but with a smaller initial error than with previously existing binomial models, giving the same accuracy on average ten-times faster than previous binomial methods.

Suggested Citation

  • Dietmar Leisen & Matthias Reimer, 1996. "Binomial models for option valuation - examining and improving convergence," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(4), pages 319-346.
  • Handle: RePEc:taf:apmtfi:v:3:y:1996:i:4:p:319-346
    DOI: 10.1080/13504869600000015

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    References listed on IDEAS

    1. Nelson, Daniel B & Ramaswamy, Krishna, 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models," Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 393-430.
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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. Barone-Adesi, Giovanni & Bermudez, Ana & Hatgioannides, John, 2003. "Two-factor convertible bonds valuation using the method of characteristics/finite elements," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1801-1831, August.
    3. D. Andricopoulos, Ari & Widdicks, Martin & Newton, David P. & Duck, Peter W., 2007. "Extending quadrature methods to value multi-asset and complex path dependent options," Journal of Financial Economics, Elsevier, vol. 83(2), pages 471-499, February.
    4. Dietmar P.J. Leisen, 1997. "The Random-Time Binomial Model," Finance 9711005, EconWPA, revised 29 Nov 1998.
    5. David Heath & Stefano Herzel, 2002. "Efficient option valuation using trees," Applied Mathematical Finance, Taylor & Francis Journals, vol. 9(3), pages 163-178.
    6. repec:pal:assmgt:v:17:y:2016:i:6:d:10.1057_s41260-016-0024-5 is not listed on IDEAS
    7. San-Lin Chung & Pai-Ta Shih, 2007. "Generalized Cox-Ross-Rubinstein Binomial Models," Management Science, INFORMS, vol. 53(3), pages 508-520, March.
    8. Leisen, Dietmar P. J., 1999. "The random-time binomial model," Journal of Economic Dynamics and Control, Elsevier, vol. 23(9-10), pages 1355-1386, September.
    9. Andricopoulos, Ari D. & Widdicks, Martin & Duck, Peter W. & Newton, David P., 2003. "Universal option valuation using quadrature methods," Journal of Financial Economics, Elsevier, vol. 67(3), pages 447-471, March.
    10. Muroi, Yoshifumi & Suda, Shintaro, 2013. "Discrete Malliavin calculus and computations of greeks in the binomial tree," European Journal of Operational Research, Elsevier, vol. 231(2), pages 349-361.
    11. Leduc, Guillaume, 2012. "European Option General First Order Error Formula," MPRA Paper 42015, University Library of Munich, Germany, revised 01 Oct 2012.
    12. P. Forsyth & K. Vetzal & R. Zvan, 2002. "Convergence of numerical methods for valuing path-dependent options using interpolation," Review of Derivatives Research, Springer, vol. 5(3), pages 273-314, October.
    13. Leisen, Dietmar P. J., 1998. "Pricing the American put option: A detailed convergence analysis for binomial models," Journal of Economic Dynamics and Control, Elsevier, vol. 22(8-9), pages 1419-1444, August.
    14. Alona Bock & Ralf Korn, 2016. "Improving Convergence of Binomial Schemes and the Edgeworth Expansion," Risks, MDPI, Open Access Journal, vol. 4(2), pages 1-22, May.
    15. Pier Giuseppe Giribone & Simone Ligato, 2016. "Flexible-forward pricing through Leisen–Reimer trees: Implementation and performance comparison with traditional Markov chains," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(02), pages 1-21, June.
    16. Xiaolin Luo & Pavel V. Shevchenko, 2014. "Fast and Simple Method for Pricing Exotic Options using Gauss-Hermite Quadrature on a Cubic Spline Interpolation," Papers 1408.6938,, revised Dec 2014.


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