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On the Rate of Convergence of Discrete-Time Contingent Claims

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  • Steve Heston
  • Guofu Zhou

Abstract

This paper characterizes the rate of convergence of discrete-time multinomial option prices. We show that the rate of convergence depends on the smoothness of option payoff functions, and is much lower than commonly believed because option payoff functions are often of all-or-nothing type and are not continuously differentiable. To improve the accuracy, we propose two simple methods, an adjustment of the discrete-time solution prior to maturity and smoothing of the payoff function, which yield solutions that converge to their continuous-time limit at the maximum possible rate enjoyed by smooth payoff functions. We also propose an intuitive approach that systematically derives multinomial models by matching the moments of a normal distribution. A highly accurate trinomial model also is provided for interest rate derivatives. Numerical examples are carried out to show that the proposed methods yield fast and accurate results. Copyright Blackwell Publishers, Inc. 2000.

Suggested Citation

  • Steve Heston & Guofu Zhou, 2000. "On the Rate of Convergence of Discrete-Time Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 53-75.
  • Handle: RePEc:bla:mathfi:v:10:y:2000:i:1:p:53-75
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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. Luca Barzanti & Corrado Corradi & Martina Nardon, 2006. "On the efficient application of the repeated Richardson extrapolation technique to option pricing," Working Papers 147, Department of Applied Mathematics, Università Ca' Foscari Venezia.
    3. Lee, Kiseop & Xu, Mingxin, 2007. "Parameter estimation from multinomial trees to jump diffusions with k means clustering," MPRA Paper 3307, University Library of Munich, Germany, revised 26 Apr 2007.
    4. Chuang-Chang Chang & Jun-Biao Lin & Wei-Che Tsai & Yaw-Huei Wang, 2012. "Using Richardson extrapolation techniques to price American options with alternative stochastic processes," Review of Quantitative Finance and Accounting, Springer, vol. 39(3), pages 383-406, October.
    5. Simona Sanfelici, 2004. "Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 27(2), pages 125-151, December.
    6. Lo-Bin Chang & Ken Palmer, 2007. "Smooth convergence in the binomial model," Finance and Stochastics, Springer, vol. 11(1), pages 91-105, January.
    7. Evis Këllezi & Nick Webber, 2004. "Valuing Bermudan options when asset returns are Levy processes," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 87-100.
    8. Raahauge, Peter, 2004. "Higher-Order Finite Element Solutions of Option Prices," Working Papers 2004-5, Copenhagen Business School, Department of Finance.
    9. Primbs, James A. & Yamada, Yuji, 2006. "A moment computation algorithm for the error in discrete dynamic hedging," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 519-540, February.
    10. Katarzyna Toporek, 2012. "Simple is better. Empirical comparison of American option valuation methods," Ekonomia journal, Faculty of Economic Sciences, University of Warsaw, vol. 29.
    11. San-Lin Chung & Pai-Ta Shih, 2007. "Generalized Cox-Ross-Rubinstein Binomial Models," Management Science, INFORMS, vol. 53(3), pages 508-520, March.
    12. J. X. Jiang & R. H. Liu & D. Nguyen, 2016. "A Recombining Tree Method For Option Pricing With State-Dependent Switching Rates," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(02), pages 1-26, March.
    13. Mark Broadie & Yusaku Yamamoto, 2003. "Application of the Fast Gauss Transform to Option Pricing," Management Science, INFORMS, vol. 49(8), pages 1071-1088, August.
    14. Nicola Bruti Liberati & Eckhard Platen, 2004. "On the Efficiency of Simplified Weak Taylor Schemes for Monte Carlo Simulation in Finance," Research Paper Series 114, Quantitative Finance Research Centre, University of Technology, Sydney.
    15. Chung, San-Lin & Shih, Pai-Ta, 2009. "Static hedging and pricing American options," Journal of Banking & Finance, Elsevier, vol. 33(11), pages 2140-2149, November.
    16. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    17. repec:eee:apmaco:v:321:y:2018:i:c:p:401-421 is not listed on IDEAS
    18. Hörfelt, Per, 2003. "A probabilistic interpretation of the [theta]-method," Statistics & Probability Letters, Elsevier, vol. 62(2), pages 117-122, April.
    19. Chang, Chuang-Chang & Lin, Jun-Biao, 2010. "The valuation of contingent claims using alternative numerical methods," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 20(5), pages 490-508, December.
    20. repec:eee:apmaco:v:252:y:2015:i:c:p:418-437 is not listed on IDEAS
    21. Windcliff, H. & Vetzal, K. R. & Forsyth, P. A. & Verma, A. & Coleman, T. F., 2003. "An object-oriented framework for valuing shout options on high-performance computer architectures," Journal of Economic Dynamics and Control, Elsevier, vol. 27(6), pages 1133-1161, April.
    22. Khaliq, A.Q.M. & Voss, D.A. & Yousuf, M., 2007. "Pricing exotic options with L-stable Pade schemes," Journal of Banking & Finance, Elsevier, vol. 31(11), pages 3438-3461, November.

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