Journal of the Korean Society for Industrial and Applied Mathematics

  • 제26권2호
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  • Pages.121-137
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  • 2022
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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AN EFFICIENT METHOD FOR SOLVING TWO-ASSET TIME FRACTIONAL BLACK-SCHOLES OPTION PRICING MODEL

  • DELPASAND, R. (DEPARTMENT OF APPLIED MATHEMATICS AND MAHANI, MATHEMATICAL RESEARCH CENTER FACULTY OF MATHEMATICS AND COMPUTER SHAHID BAHONAR UNIVERSITY OF KERMAN) ;
  • HOSSEINI, M.M. (DEPARTMENT OF APPLIED MATHEMATICS AND MAHANI, MATHEMATICAL RESEARCH CENTER FACULTY OF MATHEMATICS AND COMPUTER SHAHID BAHONAR UNIVERSITY OF KERMAN)
  • 투고 : 2022.05.15
  • 심사 : 2022.06.15
  • 발행 : 2022.06.25
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초록

In this paper, we investigate an efficient hybrid method for solving two-asset time fractional Black-Scholes partial differential equations. The proposed method is based on the Crank-Nicolson the radial basis functions methods. We show that, this method is convergent and we obtain good approximations for solution of our problems. The numerical results show high accuracy of the proposed method without needing high computational cost.

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참고문헌

  1. F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Political. Econ, 81 (1973), 637-659. https://doi.org/10.1086/260062
  2. R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci, 4 (1973), 141-183. https://doi.org/10.2307/3003143
  3. K. in't Hout,R. Volk, Numerical solution of a two-asset option valuation PDE by ADI finite difference discretization, AIP Conference Proceedings, AIP, Proceedings of ICNAAM, Rhodes, Greece 2014.
  4. D. Joeng, J. Kim, In. Suk Wee, An accurate and efficient numerical method for Black-Scholes equations, Commun. Korean. Soc, 24 (2009), 617-628. https://doi.org/10.4134/CKMS.2009.24.4.617
  5. L. V. Ballestra, G. Pacelli, Pricing European and American options with two stocastic factors: a highly efficient radial basis function approach, J. Econom. Dynam. Control, 37 (2013), 1142-1167. https://doi.org/10.1016/j.jedc.2013.01.013
  6. J. A. Rad, K. Parand, L. V. Ballestra, Pricing European and American options by radial basis point interpolation, Appl. Math. Comput, 251 (2015), 363-377. https://doi.org/10.1016/j.amc.2014.11.016
  7. V. Shcherbakov. E. Larsson, Radial basis function partition for unity methods for pricing vanilla basket options, Comput. Math. Appl, 71 (2016), 185-200. https://doi.org/10.1016/j.camwa.2015.11.007
  8. Sh. Zhang, Radial basis functions method for valuing options: A multinomial tree approach, J. Comput. Appl. Math, 319 (2017), 97-107. https://doi.org/10.1016/j.cam.2016.12.036
  9. J. Sabatier, O. P. Agrawal, J. A. Ttenreiro Machado, Advances in fractional calculus, Theoretical Developments and Applications in Physics and Engineering, Springer, 2007.
  10. A. A. Kilbas, H. M. Srivastara, J. J. Trujillo, Theory and application of factional differential equations, Elsevier, Amsterdam, 2006.
  11. I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.
  12. H. Sheng, Y. Chen, T. Qiu,Fractional processes and fractional-order signal processing: Thechniques and applications, Springer-Verlag, London, 2011.
  13. W. Wyss, The fractional Black-Scholes equation, Fract. Calc. Appl. Theory. Appl., 3 (2000), 51-61.
  14. G. Jumaire, Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. application to fractional Black-Scholes equation, Insur. Math. Econ., 42 (2008), 271-285. https://doi.org/10.1016/j.insmatheco.2007.03.001
  15. R. Kalantari, S. Shahmorad, A stable and convergent finite difference method for fractional Black-Scholes model of American put option pricing, Comput. Econ, 53 (2019), 191-205. https://doi.org/10.1007/s10614-017-9734-0
  16. M. N. Kolvea, L. G. Vulkov, Numerical solution of time fractional Black-Scholes equation, Comput. Appl. Math., 36 (2017), 1699-1715. https://doi.org/10.1007/s40314-016-0330-z
  17. L. Song, W. Wang, Solution of the fractional Black-Scholes option pricing model by finite difference method, Abstr. Appl. Anal., 2013 (2013), 1-10.
  18. H. Zhang, F. Liu, I. Turner, S. Chen, The numerical simulation of the tempered fractional Black-Scholes equation for European double barrier option, Appl. Math. Model., 40 (2016), 5819-5834. https://doi.org/10.1016/j.apm.2016.01.027
  19. S. Hagh, M. Hussain, Selection of shape parameter in radial basis functions for solution of time fractional Black-Scholes models, Appl. Math. Comput., 335 (2018), 248-263. https://doi.org/10.1016/j.amc.2018.04.045
  20. H. Zhang, F. Liu, I. Turner, Q. Yang, Numerical solution of the time fractional Black-Scholes model governinig European options, Comput. Math. Appl., 71 (2016), 1772-1783. https://doi.org/10.1016/j.camwa.2016.02.007
  21. H. Zhang, F. Liu, S. Chen, V. Anh, J. Chen, Fast numerical simulation of a new time-space fractional option pricing model governing European call option, Appl. Math. Comput, 339 (2018), 186-198. https://doi.org/10.1016/j.amc.2018.06.030
  22. Z. Cen, J. Huang, A. Xu, A.Le, Numerical approximation of a time fractional Black-Scholes equation, Comput. Math. Appl., 75 (2018), 2874-2887. https://doi.org/10.1016/j.camwa.2018.01.016
  23. S. Kumar, D. Kumar, J. Singh, Numerical computation of fractional Black-Scholes equation arising in financial market, Eghypt. J. Basic. Appl. Sci, 1 (2014), 177-183.
  24. P. Phachoo, A. Luadsong, N. Aschariyaphotha, The meshless local Petrov-Galerkin based on moving kriging interpolation for solving fractional Black-Scholes model, J. King. Saud. Uni. Sci, 28 (2016), 111-117. https://doi.org/10.1016/j.jksus.2015.08.004
  25. P. Roul, A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European option, Appl. Numer. Math., 151 (2020), 472-493. https://doi.org/10.1016/j.apnum.2019.11.004
  26. M. Rezaei, A. R. Yazdanian, A. Ashrafi, S. M. Mahmoudi, Numerical pricing based on fractional BlackScholes equations with time-dependent parameters under the CEY models double barriers option, Comput. Math. Appl., 90 (2021), 104-111. https://doi.org/10.1016/j.camwa.2021.02.021
  27. X. An, F. Liu, M. Zheng, Y. Anh, I. W. Turner, A space-time spectral method for time-fractional Black-Scholes equations, Appl. Numer. Math, 165 (2021), 152-166. https://doi.org/10.1016/j.apnum.2021.02.009
  28. S. E. Fadugba, Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation, Chaos. Solitons. Fract, 141 (2020), 110351. https://doi.org/10.1016/j.chaos.2020.110351
  29. P. Roul, V. M. K. Prased Goura, A compact finite difference scheme for fractional Black-Scholes option pricing model, Appl. Numer. Math, 166 (2021), 40-60. https://doi.org/10.1016/j.apnum.2021.03.017
  30. R. Delpasand, M. M. Hosseini, An efficient hybrid numerical method for the two-asset Black-Scholes PDE, J. Korean. Soc. Ind. Appl. Math, 25 (2021), 93-106. https://doi.org/10.12941/JKSIAM.2021.25.093
  31. R. Doostaki, M. M. Hosseini, Option pricing by the Legendre Wavelets method, Comput. Econ, 59 (2022), 749-773. https://doi.org/10.1007/s10614-021-10100-1
  32. A. Golbabai, E. Mohebianfar, A new method for evaluating options based on multiquadric RBF-FD method. Appl. Math. Comput, 308 (2017), 130-141. https://doi.org/10.1016/j.amc.2017.03.019
  33. S. Kim, C. Lee, W. Lee, S. Kwak, D. Jeong, J. Kim, Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black-Scholes Equation. Journal of Function Spaces, 2021 (2021), 1-11.
  34. W. R. Madych, Miscellaneaus error bounds for multiquadratic and related interpolants, Comput. Math. Appl, 24 (1992), 121-130. https://doi.org/10.1016/0898-1221(92)90175-H
  35. M. Buhmann, N. Oyn, Spectral convergence of multiquadratic interpolation, Proc. Edinb. Math. Soc, 36 (1993), 319-333. https://doi.org/10.1017/S0013091500018411
  36. H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005.
  37. W. Margrabe, The value of an option to exchnge one-asset for another, J. Finac, 33 (1978), 177-186. https://doi.org/10.1111/j.1540-6261.1978.tb03397.x
  38. H. Johnson, Options on the minimum or the maximum of several assets, J. Finac. Quant. Anal, 22 (1987), 277-283. https://doi.org/10.2307/2330963
  39. M. Rubinste, Somewhere over the rainbow, Risk Magazine, 4 (1995), 63-66.
  40. R. Stulz, Options in the minimum or the maximum of two risky assets, J. Finac. Econ, 10 (1982), 161-185 https://doi.org/10.1016/0304-405X(82)90011-3