Copyright © AIMS Press
Citation: Jie Gu, Lijuan Nong, Qian Yi, An Chen. Two high-order compact difference schemes with temporal graded meshes for time-fractional Black-Scholes equation[J]. Networks and Heterogeneous Media, 2023, 18(4): 1692-1712. doi: 10.3934/nhm.2023074
| [1] | N. Abdi, H. Aminikhah, A. R. Sheikhani, High-order compact finite difference schemes for the time-fractional Black-Scholes model governing European options,Chaos Solitons Fractals,162 (2022), 112423. https://doi.org/10.1016/j.chaos.2022.112423 doi:10.1016/j.chaos.2022.112423 |
| [2] | 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 doi:10.1016/j.camwa.2018.01.016 |
| [3] | C. M. Chen, F. Liu, I. Turner, V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion,J. Comput. Phys.,227 (2007), 886–897. https://doi.org/10.1016/j.jcp.2007.05.012 doi:10.1016/j.jcp.2007.05.012 |
| [4] | W. Chen, X. Xu, S. P. Zhu, Analytically pricing double barrier options based on a time-fractional Black–Scholes equation,Comput. Math. Appl.,69 (2015), 1407–1419. https://doi.org/10.1016/j.camwa.2015.03.025 doi:10.1016/j.camwa.2015.03.025 |
| [5] | H. Ding, C. Li, High‐order compact difference schemes for the modified anomalous subdiffusion equation,Numer. Methods Partial Differ. Equ.,32 (2016), 213–242. https://doi.org/10.1002/num.21992 doi:10.1002/num.21992 |
| [6] | R. L. Du, Z. Z. Sun, H. Wang, Temporal second-order finite difference schemes for variable-order time-fractional wave equations,SIAM J. Numer. Anal.,60 (2022), 104–132. https://doi.org/10.1137/19m1301230 doi:10.1137/19m1301230 |
| [7] | J. Gu, L. Nong, Q. Yi, A. Chen, Compact difference schemes with temporal uniform/non-uniform meshes for time-fractional Black–Scholes equation,Fractal Fract.,7 (2023), 340. https://doi.org/10.3390/fractalfract7040340 doi:10.3390/fractalfract7040340 |
| [8] | Y. Huang, Q. Li, R. Li, F. Zeng, L. Guo, A unified fast memory-saving time-stepping method for fractional operators and its applications,Numer. Math. Theor. Meth. Appl.,15 (2022), 679–714. https://doi.org/10.4208/nmtma.oa-2022-0023 doi:10.4208/nmtma.oa-2022-0023 |
| [9] | B. Jin, R. Lazarov, Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview,Comput. Methods Appl. Mech. Engrg.,346 (2019), 332–358. https://doi.org/10.1016/j.cma.2018.12.011 doi:10.1016/j.cma.2018.12.011 |
| [10] | K. Kazmi, A second order numerical method for the time-fractional Black–Scholes European option pricing model,J. Comput. Appl. Math.,418 (2023), 114647. https://doi.org/10.1016/j.cam.2022.114647 doi:10.1016/j.cam.2022.114647 |
| [11] | C. Li, Q. Yi, A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations,J. Comput. Phys.,316 (2016), 614–631. https://doi.org/10.1016/j.jcp.2016.04.039 doi:10.1016/j.jcp.2016.04.039 |
| [12] | J. R. Liang, J. Wang, W. J. Zhang, W. Y. Qiu, F. Y. Ren, Option pricing of a bi-fractional Black–Merton–Scholes model with the Hurst exponent H in $[\frac{1}{2}, 1]$,Appl. Math. Lett.,23 (2010) 859–863. https://doi.org/10.1016/j.aml.2010.03.022 doi:10.1016/j.aml.2010.03.022 |
| [13] | Y. Liu, J. Roberts, Y. Yan, Detailed error analysis for a fractional Adams method with graded meshes,Numer. Algorithms,78 (2018), 1195–1216. https://doi.org/10.1007/s11075-017-0419-5 doi:10.1007/s11075-017-0419-5 |
| [14] | P. Lyu, S. Vong, A high-order method with a temporal nonuniform mesh for a time-fractional Benjamin–Bona–Mahony equation,J. Sci. Comput.,80 (2019), 1607–1628. https://doi.org/10.1007/s10915-019-00991-6 doi:10.1007/s10915-019-00991-6 |
| [15] | H. Mesgarani, M. Bakhshandeh, Y. E. Aghdam, J. F. Gómez-Aguilar, The convergence analysis of the numerical calculation to price the time-fractional Black–Scholes model,Comput. Econ., 2022. https://doi.org/10.1007/s10614-022-10322-x doi:10.1007/s10614-022-10322-x |
| [16] | L. Nong, A. Chen, Numerical schemes for the time-fractional mobile/immobile transport equation based on convolution quadrature,J. Appl. Math. Comput.,68 (2022), 199–215. https://doi.org/10.1007/s12190-021-01522-z doi:10.1007/s12190-021-01522-z |
| [17] | L. Nong, A. Chen, J. Cao, Error estimates for a robust finite element method of two-term time-fractional diffusion-wave equation with nonsmooth data,Math. Model. Nat. Phenom.,16 (2021), 12. https://doi.org/10.1051/mmnp/2021007 doi:10.1051/mmnp/2021007 |
| [18] | H. Qin, D. Li, Z. Zhang, A novel scheme to capture the initial dramatic evolutions of nonlinear subdiffusion equations,J. Sci. Comput.,89 (2021), 65. https://doi.org/10.1007/s10915-021-01672-z doi:10.1007/s10915-021-01672-z |
| [19] | P. Roul, A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options,Appl. Numer. Math.,151 (2020), 472–493. https://doi.org/10.1016/j.apnum.2019.11.004 doi:10.1016/j.apnum.2019.11.004 |
| [20] | P. Roul, Design and analysis of a high order computational technique for time‐fractional Black–Scholes model describing option pricing,Math. Methods Appl. Sci.,45 (2022), 5592–5611. https://doi.org/10.1002/mma.8130 doi:10.1002/mma.8130 |
| [21] | F. Soleymani, S. Zhu, Error and stability estimates of a time-fractional option pricing model under fully spatial–temporal graded meshes,J. Comput. Appl. Math.,425 (2023) 115075. https://doi.org/10.1016/j.cam.2023.115075 doi:10.1016/j.cam.2023.115075 |
| [22] | M. Stynes, E. O'Riordan, J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation,SIAM J. Numer. Anal.,55 (2017), 1057–1079. https://doi.org/10.1137/16m1082329 doi:10.1137/16m1082329 |
| [23] | Z. Tian, S. Zhai, H. Ji, Z. Weng, A compact quadratic spline collocation method for the time-fractional Black–Scholes model,J. Appl. Math. Comput.,66 (2021), 327–350. https://doi.org/10.1007/s12190-020-01439-z doi:10.1007/s12190-020-01439-z |
| [24] | S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing,IMA J. Numer. Anal.,24 (2004), 699–720. https://doi.org/10.1093/imanum/24.4.699 doi:10.1093/imanum/24.4.699 |
| [25] | X. Xu, M. Chen, Discovery of subdiffusion problem with noisy data via deep learning,J. Sci. Comput.,92 (2022), 23. https://doi.org/10.1007/s10915-022-01879-8 doi:10.1007/s10915-022-01879-8 |
| [26] | B. Yin, Y. Liu, H. Li, F. Zeng, A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equations,Appl. Numer. Math.,165 (2021), 56–82. https://doi.org/10.1016/j.apnum.2021.02.007 doi:10.1016/j.apnum.2021.02.007 |
| [27] | W. Yuan, D. Li, C. Zhang, Linearized transformed $L1$ Galerkin FEMs with unconditional convergence for nonlinear time fractional Schrödinger equations,Numer. Math. Theory Methods Appl.,16 (2023), 348–369. https://doi.org/10.4208/nmtma.oa-2022-0087 doi:10.4208/nmtma.oa-2022-0087 |
| [28] | W. Yuan, C. Zhang, and D. Li, Linearized fast time-stepping schemes for time-space fractional Schrödinger equations,Physica D,454 (2023), 133865. https://doi.org/10.1016/j.physd.2023.133865 doi:10.1016/j.physd.2023.133865 |
| [29] | H. Zhang, F. Liu, I. Turner, Q. Yang, Numerical solution of the time fractional Black–Scholes model governing European options,Comput. Math. Appl.,71 (2016), 1772–1783. https://doi.org/10.1016/j.camwa.2016.02.007 doi:10.1016/j.camwa.2016.02.007 |
| [30] | J. Zhou, X. M. Gu, Y. L. Zhao, H. Li, A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black–Scholes model,Int. J. Comput. Math., (2023), 1–23. https://doi.org/10.1080/00207160.2023.2254412 doi:10.1080/00207160.2023.2254412 |
Article views(1354)PDF downloads(116)Cited by(3)
Figures(2) / Tables(3)
DownLoad: 
