Abstract
In this paper, the approximate solution u(x, t) of the temporal fractional Black–Scholes model involving the time derivative in the Caputo sense with initial and boundary conditions has been studied. This equation has the main part in defining the European option in the financial activities. Time discretization is performed by linear interpolation with a temporally \(\tau ^{2-\alpha }\) order accuracy, and the Chebyshev collocation is based on the orthogonal polynomials used for spatial discretization. Additionally, the convergence and stability analysis of the specified methods are considered. Finally, the numerical solutions of some examples were obtained and compared with their analytical solutions that demonstrate the high accuracy and feasibility of the proposed approach.
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Acknowledgements
The authors would like to express their sincere thanks to the referees for their careful review of this manuscript and their useful suggestions which led to an improved version. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.
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HM: Conceptualization, methodology, validation, formal analysis, investigation; MB: Conceptualization, methodology, validation; YEA: Conceptualization, methodology, validation, formal analysis, investigation; JFG-A: Conceptualization, methodology, validation, writing-review and editing. All authors have read and agreed to the published version of the manuscript.
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Mesgarani, H., Bakhshandeh, M., Aghdam, Y.E. et al. The Convergence Analysis of the Numerical Calculation to Price the Time-Fractional Black–Scholes Model. Comput Econ 62, 1845–1856 (2023). https://doi.org/10.1007/s10614-022-10322-x
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DOI: https://doi.org/10.1007/s10614-022-10322-x
Keywords
- Time-fractional Black–Scholes equation
- Chebyshev polynomials of the third kind
- Linear interpolation
- Collocation method
- Convergence analysis