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Efficient High-Order Numerical Methods for Pricing of Options

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  • Mojtaba Hajipour
  • Alaeddin Malek

Abstract

In this paper we present efficient high-order methods based on weighted essentially non-oscillatory (WENO) technique and backward differencing formula (BDF) to solve the European and American put options of the Black–Scholes equation. In order to achieve high-order convergent and prevent the appearance of spurious solutions close to non-smooth points, the WENO method is imposed for the spatial discretization. To achieve the high-order accuracy in non-smooth points as well as smooth points, a grid stretching transformation is employed. For the numerical solution of American put option, a predictor–corrector scheme based on WENO and BDF is constructed. The high efficiency of proposed methods for the solution of European and American put options is demonstrated numerically. Comparisons are made with the available methods in the literature. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Mojtaba Hajipour & Alaeddin Malek, 2015. "Efficient High-Order Numerical Methods for Pricing of Options," Computational Economics, Springer;Society for Computational Economics, vol. 45(1), pages 31-47, January.
    • Handle: RePEc:kap:compec:v:45:y:2015:i:1:p:31-47
    DOI: 10.1007/s10614-013-9405-8
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    References listed on IDEAS

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    1. Zhongdi Cen & Anbo Le & Aimin Xu, 2012. "A Second-Order Difference Scheme for the Penalized Black–Scholes Equation Governing American Put Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 40(1), pages 49-62, June.
    2. Patrick Jaillet & Damien Lamberton & Bernard Lapeyre, 1990. "Variational inequalities and the pricing of American options," Post-Print hal-01667008, HAL.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. Kaushik Amin & Ajay Khanna, 1994. "Convergence Of American Option Values From Discrete‐ To Continuous‐Time Financial Models1," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 289-304, October.
    5. Courtadon, Georges, 1982. "A More Accurate Finite Difference Approximation for the Valuation of Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 17(5), pages 697-703, December.
    6. Boyle, Phelim P., 1977. "Options: A Monte Carlo approach," Journal of Financial Economics, Elsevier, vol. 4(3), pages 323-338, May.
    7. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    8. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    9. Song-Ping Zhu, 2006. "An exact and explicit solution for the valuation of American put options," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 229-242.
    10. Guy Barles & Julien Burdeau & Marc Romano & Nicolas Samsoen, 1995. "Critical Stock Price Near Expiration," Mathematical Finance, Wiley Blackwell, vol. 5(2), pages 77-95, April.
    11. Barone-Adesi, Giovanni & Whaley, Robert E, 1987. "Efficient Analytic Approximation of American Option Values," Journal of Finance, American Finance Association, vol. 42(2), pages 301-320, June.
    12. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    13. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
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    Cited by:

    1. Darae Jeong & Minhyun Yoo & Junseok Kim, 2018. "Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions," Computational Economics, Springer;Society for Computational Economics, vol. 51(4), pages 961-972, April.
    2. Luca Vincenzo Ballestra, 2018. "Fast and accurate calculation of American option prices," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 399-426, November.

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