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Option Pricing by the Legendre Wavelets Method

Author

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  • Reza Doostaki

    (Shahid Bahonar University of Kerman
    Shahid Bahonar University of Kerman)

  • Mohammad Mehdi Hosseini

    (Shahid Bahonar University of Kerman
    Shahid Bahonar University of Kerman)

Abstract

This paper presents the numerical solution of the Black–Scholes partial differential equation (PDE) for the evaluation of European call and put options. The proposed method is based on the finite difference and Legendre wavelets aproximation scheme. We derive a matrix structure for the Legendre wavelets integral operator which has been widely used so far. Moreover, in order to use the payoff function, another operational matrix is derived. By the proposed combined method, the solving Black–Scholes PDE problem reduces to those of solving a Sylvester equation. The proposed algorithms show that in compared to literature methods, the proposed method is easy to be implemented and have high execution speed. Furthermore, we prove that the obtained Sylvester equation has a unique solution. In addition, the effect of the finite difference space step size to the computational accuracy is studied. For having suitable solution, the numerical solutions show that there is no need to select very small step size. Also only a small number of basis functions in the Legendre wavelets series is needed. The numerical results demonstrate efficiency and capability of the proposed method.

Suggested Citation

  • Reza Doostaki & Mohammad Mehdi Hosseini, 2022. "Option Pricing by the Legendre Wavelets Method," Computational Economics, Springer;Society for Computational Economics, vol. 59(2), pages 749-773, February.
    • Handle: RePEc:kap:compec:v:59:y:2022:i:2:d:10.1007_s10614-021-10100-1
    DOI: 10.1007/s10614-021-10100-1
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    References listed on IDEAS

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    1. Liu, Xiaoquan & Cao, Yi & Ma, Chenghu & Shen, Liya, 2019. "Wavelet-based option pricing: An empirical study," European Journal of Operational Research, Elsevier, vol. 272(3), pages 1132-1142.
    2. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
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    4. Fukang Yin & Junqiang Song & Xiaoqun Cao & Fengshun Lu, 2013. "Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-11, February.
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    7. Ma, Chenghu, 2006. "Intertemporal recursive utility and an equilibrium asset pricing model in the presence of Levy jumps," Journal of Mathematical Economics, Elsevier, vol. 42(2), pages 131-160, April.
    8. Roger Lord & Christian Kahl, 2006. "Optimal Fourier Inversion in Semi-analytical Option Pricing," Tinbergen Institute Discussion Papers 06-066/2, Tinbergen Institute, revised 05 Jun 2007.
    9. Liming Feng & Vadim Linetsky, 2008. "Pricing Discretely Monitored Barrier Options And Defaultable Bonds In Lévy Process Models: A Fast Hilbert Transform Approach," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 337-384, July.
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    Cited by:

    1. Černá, Dana & Fiňková, Kateřina, 2024. "Option pricing under multifactor Black–Scholes model using orthogonal spline wavelets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 309-340.

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