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A Bilinear Pseudo-spectral Method for Solving Two-asset European and American Pricing Options

Author

Listed:
  • M. Khasi

    (Iran University of Science and Technology)

  • J. Rashidinia

    (Iran University of Science and Technology)

Abstract

This paper presents a bilinear Chebyshev pseudo-spectral method to compute European and American option prices under the two-asset Black–Scholes and Heston models. We expand a function and its derivatives into their Chebyshev series, so the differentiation matrices that act on the Chebyshev coefficients are sparse and better conditioned. First, the equation is spatially discretized using a bilinear pseudo-spectral method created by Chebyshev polynomials and a first–order matrix differential equation (MDE) is obtained. Then, using the Kronecker product, this equation is converted to a system of first–order ODEs. For solving the European options, by using an eigenvalue decomposition method, the arising system will be analytically solved. Therefore, the arising errors are because of spatial discretization and quadrature errors. For solving the American options, the approach is combined with the operator splitting method or penalty method to obtain the temporal discretization. By avoiding some transforms to convert the equation to a constant coefficient equation without mixed derivatives, we will obtain more accurate solutions. Also, transforming the European option into a set of separated ODEs by an eigenvalue decomposition causes to reduce the computational complexity. We also consider the hedge ratios which show the sensitivity of an option to the stock prices. Several numerical examples are included to show the accuracy and efficiency of the proposed approach. The results show that the spectral convergence can be achieved for models with smooth functions.

Suggested Citation

  • M. Khasi & J. Rashidinia, 2024. "A Bilinear Pseudo-spectral Method for Solving Two-asset European and American Pricing Options," Computational Economics, Springer;Society for Computational Economics, vol. 63(2), pages 893-918, February.
    • Handle: RePEc:kap:compec:v:63:y:2024:i:2:d:10.1007_s10614-023-10364-9
    DOI: 10.1007/s10614-023-10364-9
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    References listed on IDEAS

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    1. Sinem Kozp{i}nar & Murat Uzunca & Bulent Karasozen, 2016. "Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements," Papers 1606.08381, arXiv.org, revised Mar 2020.
    2. Donald MacKenzie, 2008. "An Engine, Not a Camera: How Financial Models Shape Markets," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262633671, December.
    3. Xie, Fei & He, Zhijian & Wang, Xiaoqun, 2019. "An importance sampling-based smoothing approach for quasi-Monte Carlo simulation of discrete barrier options," European Journal of Operational Research, Elsevier, vol. 274(2), pages 759-772.
    4. Ahmad Golbabai & Omid Nikan, 2020. "A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model," Computational Economics, Springer;Society for Computational Economics, vol. 55(1), pages 119-141, January.
    5. Tinne Haentjens & Karel J. in 't Hout, 2015. "ADI Schemes for Pricing American Options under the Heston Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(3), pages 207-237, July.
    6. Maximilian Gaß & Kathrin Glau & Mirco Mahlstedt & Maximilian Mair, 2018. "Chebyshev interpolation for parametric option pricing," Finance and Stochastics, Springer, vol. 22(3), pages 701-731, July.
    7. Zhang, Ling & Lai, Yongzeng & Zhang, Shuhua & Li, Lin, 2019. "Efficient control variate methods with applications to exotic options pricing under subordinated Brownian motion models," The North American Journal of Economics and Finance, Elsevier, vol. 47(C), pages 602-621.
    8. Kozpınar, Sinem & Uzunca, Murat & Karasözen, Bülent, 2020. "Pricing European and American options under Heston model using discontinuous Galerkin finite elements," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 568-587.
    9. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    10. Reza Mollapourasl & Ali Fereshtian & Michèle Vanmaele, 2019. "Radial Basis Functions with Partition of Unity Method for American Options with Stochastic Volatility," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 259-287, January.
    11. 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.
    12. Margrabe, William, 1978. "The Value of an Option to Exchange One Asset for Another," Journal of Finance, American Finance Association, vol. 33(1), pages 177-186, March.
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