A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models

Hijaz Ahmad; Muhammad Nawaz Khan; Imtiaz Ahmad; Mohamed Omri; Maged F. Alotaibi; Hijaz Ahmad; Muhammad Nawaz Khan; Imtiaz Ahmad; Mohamed Omri; Maged F. Alotaibi

doi:10.3934/math.20231003

AIMS Mathematics

2023, Volume 8, Issue 8: 19677-19698. doi: 10.3934/math.20231003

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Research article

A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models

1.
Near East University, Operational Research Center in Healthcare, Nicosia, 99138, TRNC Mersin 10, Turkey
2.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
3.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 3900186, Roma, Italy
4.
Department of Basic Sciences, University of Engineering and Technology Peshawar, Pakistan
5.
Institute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional, Kajang, Selangor, Malaysia
6.
Deanship of Scientific Research, King Abdulaziz University, Jeddah, Saudi Arabia
7.
Department of Physics, College of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Received: 07 January 2023 Revised: 24 April 2023 Accepted: 24 April 2023 Published: 12 June 2023
MSC : 26A33, 65D12, 91G20

The numerical solution of the time-fractional Black-Scholes model for European and American options is presented using a local meshless collocation approach based on hybrid Gaussian-cubic radial basis functions with polynomials is presented. The approach is then expanded to a nonlinear time-fractional model for an option with transaction costs in a market with low liquidity. The spatial derivatives of the models are discretized using the proposed meshless technique. Numerical experiments are carried out for the American option, European option, and nonlinear transaction cost option models. In order to evaluate the effectiveness and precision of the suggested meshless approach, $ L_{\infty} $ and $ L_{rel} $ error norms are utilized. Both call and put option volatility is explored. A non-uniform grid customized around the strike price region is also used to determine the prices of European call and American put options. The methods described in literature are compared with the numerical results.
- time-fractional Black-Scholes model,
- hybrid local meshless method,
- radial basis functions,
- polynomial
Citation: Hijaz Ahmad, Muhammad Nawaz Khan, Imtiaz Ahmad, Mohamed Omri, Maged F. Alotaibi. A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models[J]. AIMS Mathematics, 2023, 8(8): 19677-19698. doi: 10.3934/math.20231003

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Abstract

The numerical solution of the time-fractional Black-Scholes model for European and American options is presented using a local meshless collocation approach based on hybrid Gaussian-cubic radial basis functions with polynomials is presented. The approach is then expanded to a nonlinear time-fractional model for an option with transaction costs in a market with low liquidity. The spatial derivatives of the models are discretized using the proposed meshless technique. Numerical experiments are carried out for the American option, European option, and nonlinear transaction cost option models. In order to evaluate the effectiveness and precision of the suggested meshless approach, $ L_{\infty} $ and $ L_{rel} $ error norms are utilized. Both call and put option volatility is explored. A non-uniform grid customized around the strike price region is also used to determine the prices of European call and American put options. The methods described in literature are compared with the numerical results.

References

[1]	F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Pol. Econ., 81 (1973), 637–654. https://doi.org/10.1086/260062 doi: 10.1086/260062
[2]	P. Carr, L. Wu, The finite moment log stable process and option pricing, J. Finance, 58 (2003), 753–777. https://doi.org/10.1111/1540-6261.00544 doi: 10.1111/1540-6261.00544
[3]	A. Farhadi, M. Salehi, G. Erjaee, A new version of Black-Scholes equation presented by time-fractional derivative, Iran. J. Sci. Technol. Trans. Sci., 42 (2018), 2159–2166. https://doi.org/10.1007/s40995-017-0244-7 doi: 10.1007/s40995-017-0244-7
[4]	G. Jumarie, Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton's optimal portfolio, Comput. Math. Appl., 59 (2010), 1142–1164. https://doi.org/10.1016/j.camwa.2009.05.015 doi: 10.1016/j.camwa.2009.05.015
[5]	A. Cartea, D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A, 374 (2007), 749–763. https://doi.org/10.1016/j.physa.2006.08.071 doi: 10.1016/j.physa.2006.08.071
[6]	J. R. Liang, J. Wang, W. J. Zhang, W. Y. Qiu, F. Y. Ren, The solution to a bi-fractional Black-Scholes-Merton differential equation, Int. J. Pure Appl. Math., 58 (2010), 99–112.
[7]	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
[8]	N. Özdemir, M. Yavuz, Numerical solution of fractional Black-Scholes equation by using the multivariate padé approximation, Acta Phys. Pol., 132 (2017), 1050–1053. https://hdl.handle.net/20.500.12462/6405
[9]	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
[10]	W. Chen, X. Xu, S. P. Zhu, A predictor-corrector approach for pricing American options under the finite moment log-stable model, Appl. Numer. Math., 97 (2015), 15–29. https://doi.org/10.1016/j.apnum.2015.06.004 doi: 10.1016/j.apnum.2015.06.004
[11]	W. Chen, S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Comput. Math. Appl., 67 (2014), 77–90. https://doi.org/10.1016/j.camwa.2013.10.007 doi: 10.1016/j.camwa.2013.10.007
[12]	R. Company, L. Jódar, J. R. Pintos, A numerical method for European option pricing with transaction costs nonlinear equation, Math. Comput. Model., 50 (2009), 910–920. https://doi.org/10.1016/j.mcm.2009.05.019 doi: 10.1016/j.mcm.2009.05.019
[13]	S. O. Edeki, O. O. Ugbebor, E. A. Owoloko, Analytical solutions of a time-fractional nonlinear transaction-cost model for stock option valuation in an illiquid market setting driven by a relaxed Black-Scholes assumption, Cogent Math., 4 (2017), 1352118. https://doi.org/10.1080/23311835.2017.1352118 doi: 10.1080/23311835.2017.1352118
[14]	Y. Chen, L. Wei, S. Cao, F. Liu, Y. Yang, Y. Cheng, Numerical solving for generalized Black-Scholes-Merton model with neural finite element method, Digit. Signal Process., 131 (2022), 103757. https://doi.org/10.1016/j.dsp.2022.103757 doi: 10.1016/j.dsp.2022.103757
[15]	Y. Chen, Y. Li, M. Wu, F. Lu, M. Hou, Y. Yin, Differentiating Crohn's disease from intestinal tuberculosis using a fusion correlation neural network, Knowl. Based Syst., 244 (2022), 108570. https://doi.org/10.1016/j.knosys.2022.108570 doi: 10.1016/j.knosys.2022.108570
[16]	T. Muhammad, H. Ahmad, U. Farooq, A. Akgül, Computational investigation of magnetohydrodynamics boundary of Maxwell fluid across nanoparticle-filled sheet, Al-Salam J. Eng. Technol., 2 (2023), 88–97. https://doi.org/10.55145/ajest.2023.02.02.011 doi: 10.55145/ajest.2023.02.02.011
[17]	I. Ahmad, H. Ahmad, M. Inc, S. W. Yao, B. Almohsen, Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer, Therm. Sci., 24 (2020), 95–105. https://doi.org/10.2298/TSCI20S1095A doi: 10.2298/TSCI20S1095A
[18]	M. Nawaz, I. Ahmad, H. Ahmad, A radial basis function collocation method for space-dependent inverse heat problems, J. Appl. Comput. Mech., 6 (2020), 1187–1199. https://doi.org/10.22055/JACM.2020.32999.2123 doi: 10.22055/JACM.2020.32999.2123
[19]	M. N. Khan, I. Ahmad, A. Akgül, H. Ahmad, P. Thounthong, Numerical solution of time-fractional coupled Korteweg-de Vries and Klein-Gordon equations by local meshless method, Pramana, 95 (2021), 6. https://doi.org/10.1007/s12043-020-02025-5 doi: 10.1007/s12043-020-02025-5
[20]	C. Piret, E. Hanert, A radial basis functions method for fractional diffusion equations, J. Comput. Phys., 238 (2013), 71–81. https://doi.org/10.1016/j.jcp.2012.10.041 doi: 10.1016/j.jcp.2012.10.041
[21]	V. R. Hosseini, W. Chen, Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem., 38 (2014), 31–39. https://doi.org/10.1016/j.enganabound.2013.10.009 doi: 10.1016/j.enganabound.2013.10.009
[22]	H. R. Ghehsareh, S. H. Bateni, A. Zaghian, A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation, Eng. Anal. Bound. Elem., 61 (2015), 52–60. https://doi.org/10.1016/j.enganabound.2015.06.009 doi: 10.1016/j.enganabound.2015.06.009
[23]	A. Kumar, A. Bhardwaj, B. R. Kumar, A meshless local collocation method for time fractional diffusion wave equation, Comput. Math. Appl., 78 (2019), 1851–1861. https://doi.org/10.1016/j.camwa.2019.03.027 doi: 10.1016/j.camwa.2019.03.027
[24]	M. Dehghan, M. Abbaszadeh, A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear Sine-Gordon and Klein-Gordon equations, Eng. Anal. Bound. Elem., 50 (2015), 412–434. https://doi.org/10.1016/j.enganabound.2014.09.008 doi: 10.1016/j.enganabound.2014.09.008
[25]	A. Mohebbi, M. Abbaszadeh, M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics, Eng. Anal. Bound. Elem., 37 (2013), 475–485. https://doi.org/10.1016/j.enganabound.2012.12.002 doi: 10.1016/j.enganabound.2012.12.002
[26]	A. Mohebbi, M. Abbaszadeh, M. Dehghan, Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions (RBF) meshless method, Eng. Anal. Bound. Elem., 38 (2014), 72–82. https://doi.org/10.1016/j.enganabound.2013.09.015 doi: 10.1016/j.enganabound.2013.09.015
[27]	S. Wei, W. Chen, Y. C. Hon, Implicit local radial basis function method for solving two-dimensional time fractional diffusion equations, Therm. Sci., 19 (2015), 59–67. https://doi.org/10.2298/TSCI15S1S59W doi: 10.2298/TSCI15S1S59W
[28]	M. Aslefallah, E. Shivanian, Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions, Eur. Phys. J. Plus, 130 (2015), 47. https://doi.org/10.1140/epjp/i2015-15047-y doi: 10.1140/epjp/i2015-15047-y
[29]	S. Wei, W. Chen, Y. Zhang, H. Wei, R. M. Garrard, A local radial basis function collocation method to solve the variable-order time fractional diffusion equation in a two-dimensional irregular domain, Numer. Methods Partial. Differ. Equ., 34 (2018), 1209–1223. https://doi.org/10.1002/num.22253 doi: 10.1002/num.22253
[30]	Z. Avazzadeh, V. Hosseini, W. Chen, Radial basis functions and FDM for solving fractional diffusion-wave equation, Iran. J. Sci. Technol. Trans. Sci., 38 (2014), 205–212. https://doi.org/10.22099/IJSTS.2014.2260 doi: 10.22099/IJSTS.2014.2260
[31]	A. Golbabai, O. Nikan, T. Nikazad, Numerical analysis of time fractional Black-Scholes European option pricing model arising in financial market, Comput. Appl. Math., 38 (2019), 173. https://doi.org/10.1007/s40314-019-0957-7 doi: 10.1007/s40314-019-0957-7
[32]	P. K. Mishra, S. K. Nath, M. K. Sen, G. E. Fasshauer, Hybrid Gaussian-cubic radial basis functions for scattered data interpolation, Comput. Geosci., 22 (2018), 1203–1218. https://doi.org/10.1007/s10596-018-9747-3 doi: 10.1007/s10596-018-9747-3
[33]	G. Jumarie, Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations, Insur. Math. Econ., 42 (2008), 271–287. https://doi.org/10.1016/j.insmatheco.2007.03.001 doi: 10.1016/j.insmatheco.2007.03.001
[34]	M. Caputo, Linear models of dissipation whose Q is almost frequency {independent-II}, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[35]	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
[36]	Z. Cen, H. Jian, X. Aimin, L. Anbo, 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
[37]	A. Khaliq, D. Voss, S. Kazmi, A linearly implicit predictor-corrector scheme for pricing American options using a penalty method approach, J. Bank. Financ., 30 (2006), 489–502. https://doi.org/10.1016/j.jbankfin.2005.04.017 doi: 10.1016/j.jbankfin.2005.04.017
[38]	G. E. Fasshauer, A. Q. M. Khaliq, D. A. Voss, Using meshfree approximation for multi-asset American option problems, J. Chin. Inst. Eng., 27 (2004), 56. https://doi.org/10.1080/02533839.2004.9670904 doi: 10.1080/02533839.2004.9670904
[39]	M. K. Kadalbajooa, A. Kumar, L. Tripathia, Application of local radial basis function based finite difference method for American option problems, Int. J. Comput. Math., 8 (2016), 1608–1624. https://doi.org/10.1080/00207160.2014.950571 doi: 10.1080/00207160.2014.950571
[40]	S. A. Sarra, A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains, Appl. Math. Comput., 218 (2012), 9853–9865. https://doi.org/10.1016/j.amc.2012.03.062 doi: 10.1016/j.amc.2012.03.062
[41]	A. Yokuş, Comparison of caputo and conformable derivatives for time-fractional korteweg-de vries equation via the finite difference method, Int. J. Mod. Phys. B, 32 (2018), 1850365. https://doi.org/10.1142/S0217979218503654 doi: 10.1142/S0217979218503654

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