Financial engineering problems hold considerable significance in the academic realm, where there remains a continued demand for efficient methods to scrutinize and analyze these models. Within this investigation, we delved into a fractional nonlinear coupled system for option pricing and volatility. The model we examined can be conceptualized as a fractional nonlinear coupled wave alternative to the governing system of Black-Scholes option pricing. This introduced a leveraging effect, wherein stock volatility aligns with stock returns. To generate novel solitonic wave structures in the system, the present article introduced a generalized Ricatti mapping method and new Kudryashov method. Graphical representations, both in 3D and 2D formats, were employed to elucidate the system's response to pulse propagation. These visualizations enabled the anticipation of appropriate parameter values that align with the observed data. Furthermore, a comparative analysis of solutions was presented for different fractional order values. Additionally, the article showcases the comparison of wave profiles through 2D graphs. The results of this investigation suggested that the proposed method served as a highly reliable and flexible alternative for problem-solving, preserving the physical attributes inherent in realistic processes. To sum up, the main objective of our work was to conceptualize a fractional nonlinear coupled wave system as an alternative to the Black-Scholes option pricing model and investigate its implications on stock volatility and returns. Additionally, we aimed to apply and analyze methods for generating solitonic wave structures and compare their solutions for different fractional order values.
Citation: Hamood Ur Rehman, Patricia J. Y. Wong, A. F. Aljohani, Ifrah Iqbal, Muhammad Shoaib Saleem. The fractional soliton solutions: shaping future finances with innovative wave profiles in option pricing system[J]. AIMS Mathematics, 2024, 9(9): 24699-24721. doi: 10.3934/math.20241203
[1] | C. W. Chukwu, Fatmawati . Modelling fractional-order dynamics of COVID-19 with environmental transmission and vaccination: A case study of Indonesia. AIMS Mathematics, 2022, 7(3): 4416-4438. doi: 10.3934/math.2022246 |
[2] | Moh. Mashum Mujur Ihsanjaya, Nanang Susyanto . A mathematical model for policy of vaccinating recovered people in controlling the spread of COVID-19 outbreak. AIMS Mathematics, 2023, 8(6): 14508-14521. doi: 10.3934/math.2023741 |
[3] | Salah Boulaaras, Ziad Ur Rehman, Farah Aini Abdullah, Rashid Jan, Mohamed Abdalla, Asif Jan . Coronavirus dynamics, infections and preventive interventions using fractional-calculus analysis. AIMS Mathematics, 2023, 8(4): 8680-8701. doi: 10.3934/math.2023436 |
[4] | Huda Alsaud, Muhammad Owais Kulachi, Aqeel Ahmad, Mustafa Inc, Muhammad Taimoor . Investigation of SEIR model with vaccinated effects using sustainable fractional approach for low immune individuals. AIMS Mathematics, 2024, 9(4): 10208-10234. doi: 10.3934/math.2024499 |
[5] | Mdi Begum Jeelani, Abeer S Alnahdi, Rahim Ud Din, Hussam Alrabaiah, Azeem Sultana . Mathematical model to investigate transmission dynamics of COVID-19 with vaccinated class. AIMS Mathematics, 2023, 8(12): 29932-29955. doi: 10.3934/math.20231531 |
[6] | Xiaoying Pan, Longkun Tang . A new model for COVID-19 in the post-pandemic era. AIMS Mathematics, 2024, 9(8): 21255-21272. doi: 10.3934/math.20241032 |
[7] | Qinyue Zheng, Xinwei Wang, Qiuwei Pan, Lei Wang . Optimal strategy for a dose-escalation vaccination against COVID-19 in refugee camps. AIMS Mathematics, 2022, 7(5): 9288-9310. doi: 10.3934/math.2022515 |
[8] | Peter Joseph Witbooi, Sibaliwe Maku Vyambwera, Mozart Umba Nsuami . Control and elimination in an SEIR model for the disease dynamics of COVID-19 with vaccination. AIMS Mathematics, 2023, 8(4): 8144-8161. doi: 10.3934/math.2023411 |
[9] | Maria M. Martignoni, Proton Rahman, Amy Hurford . Rotational worker vaccination provides indirect protection to vulnerable groups in regions with low COVID-19 prevalence. AIMS Mathematics, 2022, 7(3): 3988-4003. doi: 10.3934/math.2022220 |
[10] | Khalaf M. Alanazi . The asymptotic spreading speeds of COVID-19 with the effect of delay and quarantine. AIMS Mathematics, 2024, 9(7): 19397-19413. doi: 10.3934/math.2024945 |
Financial engineering problems hold considerable significance in the academic realm, where there remains a continued demand for efficient methods to scrutinize and analyze these models. Within this investigation, we delved into a fractional nonlinear coupled system for option pricing and volatility. The model we examined can be conceptualized as a fractional nonlinear coupled wave alternative to the governing system of Black-Scholes option pricing. This introduced a leveraging effect, wherein stock volatility aligns with stock returns. To generate novel solitonic wave structures in the system, the present article introduced a generalized Ricatti mapping method and new Kudryashov method. Graphical representations, both in 3D and 2D formats, were employed to elucidate the system's response to pulse propagation. These visualizations enabled the anticipation of appropriate parameter values that align with the observed data. Furthermore, a comparative analysis of solutions was presented for different fractional order values. Additionally, the article showcases the comparison of wave profiles through 2D graphs. The results of this investigation suggested that the proposed method served as a highly reliable and flexible alternative for problem-solving, preserving the physical attributes inherent in realistic processes. To sum up, the main objective of our work was to conceptualize a fractional nonlinear coupled wave system as an alternative to the Black-Scholes option pricing model and investigate its implications on stock volatility and returns. Additionally, we aimed to apply and analyze methods for generating solitonic wave structures and compare their solutions for different fractional order values.
[1] |
J. C. Zhao, M. Davison, R. M. Corless, Compact finite difference method for American option pricing, J. Comput. Appl. Math., 206 (2007), 306–321. https://doi.org/10.1016/j.cam.2006.07.006 doi: 10.1016/j.cam.2006.07.006
![]() |
[2] |
D. C. Lesmana, S. Wang, An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation under transaction costs, Appl. Math. Comput., 219 (2013), 8811–8828. https://doi.org/10.1016/j.amc.2012.12.077 doi: 10.1016/j.amc.2012.12.077
![]() |
[3] |
J. A. Rad, K. Parand, S. Abbasbandy, Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1178–1200. https://doi.org/10.1016/j.cnsns.2014.07.015 doi: 10.1016/j.cnsns.2014.07.015
![]() |
[4] |
F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637–654. https://doi.org/10.1142/9789814759588_0001 doi: 10.1142/9789814759588_0001
![]() |
[5] |
R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci., 4 (1973), 141–183. https://doi.org/10.1142/9789814759588_0002 doi: 10.1142/9789814759588_0002
![]() |
[6] |
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., 62 (2023), 1845–1856. https://doi.org/10.1007/s10614-022-10322-x doi: 10.1007/s10614-022-10322-x
![]() |
[7] |
P. P. Boyle, T. Vorst, Option replication in discrete time with transaction costs, J. Finance, 47 (1992), 271–293. https://doi.org/10.1111/j.1540-6261.1992.tb03986.x doi: 10.1111/j.1540-6261.1992.tb03986.x
![]() |
[8] |
H. E. Leland, Option pricing and replication with transactions costs, J. Finance, 40 (1985), 1283–1301. https://doi.org/10.1111/j.1540-6261.1985.tb02383.x doi: 10.1111/j.1540-6261.1985.tb02383.x
![]() |
[9] |
G. Barles, H. M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes equation, Finance Stoch., 2 (1998), 369–397. https://doi.org/10.1007/s007800050046 doi: 10.1007/s007800050046
![]() |
[10] |
S. Kusuoka, Limit theorem on option replication cost with transaction costs, Ann. Appl. Probab., 5 (1995), 198–221. https://doi.org/10.1214/aoap/1177004836 doi: 10.1214/aoap/1177004836
![]() |
[11] |
V. G. Ivancevic, Adaptive-wave alternative for the Black-Scholes option pricing model, Cogn. Comput., 2 (2010), 17–30. https://doi.org/10.1007/s12559-009-9031-x doi: 10.1007/s12559-009-9031-x
![]() |
[12] | A. S. Suresh, S. N. Prasath, A study to understand Elliott wave principle, Int. J. Eng. Res. Gen. Sci., 4 (2016), 352–363. |
[13] | A. W. Lo, Reconciling efficient markets with behavioral finance: the adaptive markets hypothesis, J. Invest. Consult., 7 (2005), 21–44. |
[14] |
S. C. Kak, Quantum neural computing, Adv. Imaging Electron Phys., 94 (1995), 259–313. https://doi.org/10.1016/S1076-5670(08)70147-2 doi: 10.1016/S1076-5670(08)70147-2
![]() |
[15] | O. González-Gaxiola, S. O. Edeki, O. O. Ugbebor, J. R. de Chávez, Solving the Ivancevic pricing model using the He's frequency amplitude formulation, Eur. J. Pure Appl. Math., 10 (2017), 631–637. |
[16] |
Q. L. Chen, H. M. Baskonus, W. Gao, E. Ilhan, Soliton theory and modulation instability analysis: the Ivancevic option pricing model in economy, Alexandria Eng. J., 61 (2022), 7843–7851. https://doi.org/10.1016/j.aej.2022.01.029 doi: 10.1016/j.aej.2022.01.029
![]() |
[17] |
K. K. Ali, M. A. Maaty, M. Maneea, Optimizing option pricing: exact and approximate solutions for the time-fractional Ivancevic model, Alexandria Eng. J., 84 (2023), 59–70. https://doi.org/10.1016/j.aej.2023.10.066 doi: 10.1016/j.aej.2023.10.066
![]() |
[18] | Z. Y. Yan, Financial rogue waves appearing in the coupled nonlinear volatility and option pricing model, 2011, arXiv: 1101.3107. |
[19] |
M. B. Riaz, A. R. Ansari, A. Jhangeer, M. Imran, C. K. Chan, The fractional soliton wave propagation of non-linear volatility and option pricing systems with a sensitive demonstration, Fractals Fract., 7 (2023), 1–28. https://doi.org/10.3390/fractalfract7110809 doi: 10.3390/fractalfract7110809
![]() |
[20] |
B. Acay, E. Bas, T. Abdeljawad, Non-local fractional calculus from different viewpoint generated by truncated M-derivative, J. Comput. Appl. Math., 366 (2020), 112410. https://doi.org/10.1016/j.cam.2019.112410 doi: 10.1016/j.cam.2019.112410
![]() |
[21] |
K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194 doi: 10.1006/jmaa.2000.7194
![]() |
[22] |
P. C. Ma, A. Najafi, J. F. Gomez-Aguilar, Sub mixed fractional Brownian motion and its application to finance, Chaos Solitons Fract., 184 (2024), 114968. https://doi.org/10.1016/j.chaos.2024.114968 doi: 10.1016/j.chaos.2024.114968
![]() |
[23] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006. |
[24] |
A. Goswami, J. Singh, D. Kumar, Sushila, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Phys. A, 524 (2019), 563–575. https://doi.org/10.1016/j.physa.2019.04.058 doi: 10.1016/j.physa.2019.04.058
![]() |
[25] |
A. Goswami, Sushila, J. Singh, D. Kumar, Numerical computation of fractional Kersten-Krasil'shchik coupled KdV-mKdV system occurring in multi-component plasmas, AIMS Math., 5 (2020), 2346–2368. https://doi.org/10.3934/math.2020155 doi: 10.3934/math.2020155
![]() |
[26] |
H. U. Rehman, M. I. Asjad, I. Iqbal, A. Akgül, Soliton solutions of space-time fractional Zoomeron differential equation, Int. J. Appl. Nonlinear Sci., 4 (2023), 29–46. https://doi.org/10.1504/IJANS.2023.133734 doi: 10.1504/IJANS.2023.133734
![]() |
[27] |
M. I. Asjad, N. Ullah, H. U. Rehman, T. N. Gia, Novel soliton solutions to the Atangana-Baleanu fractional system of equations for the ISALWs, Open Phys., 19 (2021), 770–779. https://doi.org/10.1515/phys-2021-0085 doi: 10.1515/phys-2021-0085
![]() |
[28] |
D. Shi, H. U. Rehman, I. Iqbal, M. Vivas-Cortez, M. S. Saleem, X. J. Zhang, Analytical study of the dynamics in the double-chain model of DNA, Results Phys., 52 (2023), 106787. https://doi.org/10.1016/j.rinp.2023.106787 doi: 10.1016/j.rinp.2023.106787
![]() |
[29] |
H. U. Rehman, A. U. Awan, E. M. Tag-ElDin, S. E. Alhazmi, M. F. Yassen, R. Haider, Extended hyperbolic function method for the (2+1)-dimensional nonlinear soliton equation, Results Phys., 40 (2022), 105802. https://doi.org/10.1016/j.rinp.2022.105802 doi: 10.1016/j.rinp.2022.105802
![]() |
[30] |
H. Zhao, J. G. Han, W. T. Wang, H. Y. An, Applications of extended hyperbolic function method for quintic discrete nonlinear Schrödinger equation, Commun. Theor. Phys., 47 (2007), 474. https://doi.org/10.1088/0253-6102/47/3/020 doi: 10.1088/0253-6102/47/3/020
![]() |
[31] |
H. U. Rehman, I. Iqbal, S. Subhi Aiadi, N. Mlaiki, M. S. Saleem, Soliton solutions of Klein-Fock-Gordon equation using Sardar subequation method, Mathematics, 10 (2022), 1–10. https://doi.org/10.3390/math10183377 doi: 10.3390/math10183377
![]() |
[32] |
I. Iqbal, H. U. Rehman, M. Mirzazadeh, M. S. Hashemi, Retrieval of optical solitons for nonlinear models with Kudryashov's quintuple power law and dual-form nonlocal nonlinearity, Opt. Quantum Electron., 55 (2023), 588. https://doi.org/10.1007/s11082-023-04866-x doi: 10.1007/s11082-023-04866-x
![]() |
[33] | H. U. Rehman, N. Ullah, M. A. Imran, Highly dispersive optical solitons using Kudryashov's method, Optik, 199 (2019), 163349. |
[34] |
P. N. Ryabov, D. I. Sinelshchikov, M. B. Kochanov, Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations, Appl. Math. Comput., 218 (2011), 3965–3972. https://doi.org/10.1016/j.amc.2011.09.027 doi: 10.1016/j.amc.2011.09.027
![]() |
[35] | M. A. Salam, M. S. Uddin, P. Dey, Generalized Bernoulli sub-ODE method and its applications, Ann. Pure Appl. Math., 10 (2015), 1–6. |
[36] | B. Zheng, A new Bernoulli sub-ODE method for constructing traveling wave solutions for two nonlinear equations with any order, U.P.B. Sci. Bull. Ser. A, 73 (2011), 85–94. |
[37] |
S. K. Liu, Z. T. Fu, S. D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69–74. https://doi.org/10.1016/S0375-9601(01)00580-1 doi: 10.1016/S0375-9601(01)00580-1
![]() |
[38] |
S. A. Allahyani, H. U. Rehman, A. U. Awan, E. M. Tag-ElDin, M. U. Hassan, Diverse variety of exact solutions for nonlinear Gilson-Pickering equation, Symmetry, 14 (2022), 1–15. https://doi.org/10.3390/sym14102151 doi: 10.3390/sym14102151
![]() |
[39] |
H. U. Rehman, N. Ullah, M. A. Imran, Optical solitons of Biswas-Arshed equation in birefringent fibers using extended direct algebraic method, Optik, 226 (2021), 165378. https://doi.org/10.1016/j.ijleo.2020.165378 doi: 10.1016/j.ijleo.2020.165378
![]() |
[40] |
A. Kurt, A. Tozar, O. Tasbozan, Applying the new extended direct algebraic method to solve the equation of obliquely interacting waves in shallow waters, J. Ocean Univ. China, 19 (2020), 772–780. https://doi.org/10.1007/s11802-020-4135-8 doi: 10.1007/s11802-020-4135-8
![]() |
[41] |
H. U. Rehman, M. S. Saleem, M. Zubair, S. Jafar, I. Latif, Optical solitons with Biswas-Arshed model using mapping method, Optik, 194 (2019), 163091. https://doi.org/10.1016/j.ijleo.2019.163091 doi: 10.1016/j.ijleo.2019.163091
![]() |
[42] |
X. Zeng, X. L. Yong, A new mapping method and its applications to nonlinear partial differential equations, Phys. Lett. A, 372 (2008), 6602–6607. https://doi.org/10.1016/j.physleta.2008.09.025 doi: 10.1016/j.physleta.2008.09.025
![]() |
[43] |
W. W. Mohammed, C. Cesarano, The soliton solutions for the (4+1)-dimensional stochastic Fokas equation, Math. Methods Appl. Sci., 46 (2023), 7589–7597. https://doi.org/10.1002/mma.8986 doi: 10.1002/mma.8986
![]() |
[44] |
W. W. Mohammed, F. M. Al-Askar, C. Cesarano, The analytical solutions of the stochastic mKdV equation via the mapping method, Mathematics, 10 (2022), 1–9. https://doi.org/10.3390/math10224212 doi: 10.3390/math10224212
![]() |
[45] |
M. U. Shahzad, H. U. Rehman, A. U. Awan, Z. Zafar, A. M. Hassan, I. Iqbal, Analysis of the exact solutions of nonlinear coupled Drinfeld-Sokolov-Wilson equation through ϕ6-model expansion method, Results Phys., 52 (2023), 106771. https://doi.org/10.1016/j.rinp.2023.106771 doi: 10.1016/j.rinp.2023.106771
![]() |
[46] |
N. Ahmed, M. Z. Baber, M. S. Iqbal, A. Annum, S. M. Ali, M. Ali, et al., Analytical study of reaction diffusion Lengyel-Epstein system by generalized Riccati equation mapping method, Sci. Rep., 13 (2023), 20033. https://doi.org/10.1038/s41598-023-47207-4 doi: 10.1038/s41598-023-47207-4
![]() |
[47] |
H. Naher, F. A. Abdullah, S. T. Mohyud-Din, Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation, AIP Adv., 3 (2013), 052104. https://doi.org/10.1063/1.4804433 doi: 10.1063/1.4804433
![]() |
[48] |
H. U. Rehman, G. S. Said, A. Amer, H. Ashraf, M. M. Tharwat, M. Abdel-Aty, et al., Unraveling the (4+1)-dimensional Davey-Stewartson-Kadomtsev-Petviashvili equation: exploring soliton solutions via multiple techniques, Alex. Eng. J., 90 (2024), 17–23. https://doi.org/10.1016/j.aej.2024.01.058 doi: 10.1016/j.aej.2024.01.058
![]() |
[49] |
H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math., 2011 (2011), 298628. https://doi.org/10.1155/2011/298628 doi: 10.1155/2011/298628
![]() |
[50] |
J. Zhang, X. L. Wei, Y. J. Lu, A generalized (G′/G)-expansion method and its applications, Phys. Lett. A, 372 (2008), 3653–3658. https://doi.org/10.1016/j.physleta.2008.02.027 doi: 10.1016/j.physleta.2008.02.027
![]() |
1. | Xin Liu, Kerim Fouli, Rui Kang, Martin Maier, Network-Coding-Based Energy Management for Next-Generation Passive Optical Networks, 2012, 30, 0733-8724, 864, 10.1109/JLT.2011.2182633 | |
2. | Konstantin Miller, Thorsten Biermann, Hagen Woesner, Holger Karl, 2010, Network Coding in Passive Optical Networks, 978-1-4244-7189-8, 1, 10.1109/NETCOD.2010.5487683 | |
3. | Z. Dulinski, M. Kantor, W. Krzysztofek, R. Stankiewicz, P. Cholda, 2010, Optimal Choice of Peers Based on BGP Information, 978-1-4244-6402-9, 1, 10.1109/ICC.2010.5502269 | |
4. | Zbigniew Dulinski, Rafal Stankiewicz, Piotr Wydrych, Miroslaw Kantor, Piotr Cholda, 2011, Cost-Driven Peer Rating Algorithm, 978-1-61284-232-5, 1, 10.1109/icc.2011.5962879 | |
5. | Dirk Haage, Ralph Holz, Heiko Niedermayer, Pavel Laskov, 2009, Chapter 24, 978-3-540-92665-8, 279, 10.1007/978-3-540-92666-5_24 | |
6. | Rafal Stankiewicz, Piotr Cholda, Jerzy Domzal, Robert Wojcik, Tobias Hosfeld, Thomas Zinner, Simon Oechsner, Frank Lehrieder, 2011, Influence of traffic management solutions on Quality of Experience for prevailing overlay applications, 978-1-4577-0915-9, 1, 10.1109/NGI.2011.5985866 | |
7. | Piotr Cholda, Jerzy Domzal, Robert Wojcik, Ratal Stankiewicz, Frank Lehrieder, Tobias HoBfeld, Simon Oechsner, Vlad Singeorzan, 2010, Performance evaluation of P2P caches: Flash-crowd case, 978-1-4244-8173-6, 108, 10.1109/ATNAC.2010.5680247 | |
8. | Matthias Wichtlhuber, Robert Reinecke, David Hausheer, An SDN-Based CDN/ISP Collaboration Architecture for Managing High-Volume Flows, 2015, 12, 1932-4537, 48, 10.1109/TNSM.2015.2404792 |