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Convergence of numerical methods for valuing path-dependent options using interpolation

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  • P. Forsyth
  • K. Vetzal
  • R. Zvan

Abstract

One method for valuing path-dependent options is the augmented state space approach described in Hull and White (1993) and Barraquand and Pudet (1996), among others. In certain cases, interpolation is required because the number of possible values of the additional state variable grows exponentially. We provide a detailed analysis of the convergence of these algorithms. We show that it is possible for the algorithm to be non-convergent, or to converge to an incorrect answer, if the interpolation scheme is selected in appropriately. We concentrate on Asian options, due to their popularity and because of some errors in the previous literature. Copyright Kluwer Academic Publishers 2002

Suggested Citation

  • 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.
    • Handle: RePEc:kap:revdev:v:5:y:2002:i:3:p:273-314
    DOI: 10.1023/A:1020823700228
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    References listed on IDEAS

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    Cited by:

    1. Massimo Costabile & Ivar Massabó & Emilio Russo, 2011. "A binomial approximation for two-state Markovian HJM models," Review of Derivatives Research, Springer, vol. 14(1), pages 37-65, April.
    2. Shevchenko, Pavel V. & Luo, Xiaolin, 2017. "Valuation of variable annuities with Guaranteed Minimum Withdrawal Benefit under stochastic interest rate," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 104-117.
    3. Emilio Russo, 2020. "A Discrete-Time Approach to Evaluate Path-Dependent Derivatives in a Regime-Switching Risk Model," Risks, MDPI, vol. 8(1), pages 1-22, January.
    4. 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.
    5. H. A. Windcliff & P. A. Forsyth & K. R. Vetzal, 2006. "Numerical Methods and Volatility Models for Valuing Cliquet Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(4), pages 353-386.
    6. Emilio Russo & Alessandro Staino, 2018. "A Lattice-Based Model For Evaluating Bonds And Interest-Sensitive Claims Under Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(04), pages 1-18, June.
    7. Dai, Min & Li, Peifan & Zhang, Jin E., 2010. "A lattice algorithm for pricing moving average barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 34(3), pages 542-554, March.
    8. Massimo Costabile & Ivar Massabó & Emilio Russo, 2006. "An adjusted binomial model for pricing Asian options," Review of Quantitative Finance and Accounting, Springer, vol. 27(3), pages 285-296, November.
    9. Vicky Henderson & Kamil Klad'ivko & Michael Monoyios & Christoph Reisinger, 2017. "Executive stock option exercise with full and partial information on a drift change point," Papers 1709.10141, arXiv.org, revised Jul 2020.
    10. Tian-Shyr Dai & Yuh-Yuan Fang & Yuh-Dauh Lyuu, 2005. "Analytics for geometric average trigger reset options," Applied Economics Letters, Taylor & Francis Journals, vol. 12(13), pages 835-840.
    11. Tian-Shyr Dai & Jr-Yan Wang & Hui-Shan Wei, 2008. "Adaptive placement method on pricing arithmetic average options," Review of Derivatives Research, Springer, vol. 11(1), pages 83-118, March.
    12. Pavel V. Shevchenko & Xiaolin Luo, 2016. "Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate," Papers 1602.03238, arXiv.org, revised Jan 2017.
    13. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    14. Alex Backwell & Thomas A. McWalter & Peter H. Ritchken, 2022. "On buybacks, dilutions, dividends, and the pricing of stock‐based claims," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 273-308, January.
    15. Nabeel Butt, 2019. "On Discrete Probability Approximations for Transaction Cost Problems," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 26(3), pages 365-389, September.
    16. Kyoung-Sook Moon & Yunju Jeong & Hongjoong Kim, 2016. "An Efficient Binomial Method for Pricing Asian Options," ECONOMIC COMPUTATION AND ECONOMIC CYBERNETICS STUDIES AND RESEARCH, Faculty of Economic Cybernetics, Statistics and Informatics, vol. 50(2), pages 151-164.
    17. 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, arXiv.org, revised Dec 2014.
    18. Cheung, Yan-Leung & Cheung, Yin-Wong & He, Angela W.W. & Wan, Alan T.K., 2010. "A trading strategy based on Callable Bull/Bear Contracts," Pacific-Basin Finance Journal, Elsevier, vol. 18(2), pages 186-198, April.
    19. Xiaolin Luo & Pavel Shevchenko, 2014. "Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Optimal Withdrawal Strategy," Papers 1410.8609, arXiv.org.
    20. Jérôme Lelong & Antonino Zanette, 2010. "Tree methods," Post-Print hal-00776713, HAL.

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