Journal of Applied and Pure Mathematics

  • 제6권3_4호
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  • Pages.141-154
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  • 2024
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  • 2671-4000(pISSN)
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  • 2636-1612(eISSN)
한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
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THE DYNAMICS OF EUROPEAN-STYLE OPTION PRICING IN THE FINANCIAL MARKET UTILIZING THE BLACK-SCHOLES MODEL WITH TWO ASSETS, SUPPORTED BY VARIATIONAL ITERATION TECHNIQUE

  • 투고 : 2024.02.06
  • 심사 : 2024.05.18
  • 발행 : 2024.07.30
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초록

This article offers a thorough exploration of a modified Black-Scholes model featuring two assets. The determination of option prices is accomplished through the Black-Scholes partial differential equation, leveraging the variational iteration method. This approach represents a semi-analytical technique that incorporates the use of Lagrange multipliers. The Lagrange multiplier emerges as a beacon of efficiency, adeptly streamlining the computational intricacies, and elevating the model's efficacy to unprecedented heights. For better understanding of the presented system, a graphical and tabular interpretation is presented with the help of Maple software.

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과제정보

The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions, which improved the quality of this paper.

참고문헌

  1. O.I. Abiodun, A. Jantan, A.E. Omolara, K.V. Dada, N.A. Mohamed, H. Arshad, State-of-the-art in artificial neural network applications: A survey, Heliyon 4 (2018), e00938. https://doi.org/10.1016/j.heliyon.2018.e00938
  2. G. Adomian, Solving frontier problems of physics: the decomposition method, Springer Science and Business Media, 60, 2013.
  3. A. Bermudez, M.R. Nogueiras, C. V'azquez, Numerical solution of variational inequalities for pricing Asian options by higher order Lagrange-Galerkin methods, Appl. Numer. Math. 56 (2006), 1256-1270. https://doi.org/10.1016/j.apnum.2006.03.026
  4. F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973), 637-654.
  5. B. Bin-Mohsin, M.A. Noor, K.I. Noor, R. Latif, Resolvent dynamical systems and mixed variational inequalities, J. Nonl. Sci. Appl. 10 (2017), 2925-2933. https://doi.org/10.22436/jnsa.010.06.07
  6. R. Cavoretto, A. De Rossi, E. Perracchione, Optimal selection of local approximants in RBF-PU interpolation, J. Sci. Comput. 74 (2018), 1-22. https://doi.org/10.1007/s10915-017-0418-7
  7. M. Contreras, A. Llanquihu'en, M. Villena, On the solution of the multi-asset black-scholes model: Correlations, eigenvalues and geometry, (2015). arXiv preprint arXiv:1510.02768
  8. T. Corzo, M. Prat, E. Vaquero, Behavioral Finance in Joseph de la Vega's Confusion de Confusiones, J. Behav. Finance 15 (2014), 341-350. https://doi.org/10.1080/15427560.2014.968722
  9. C. de Lamare Bastian-Pinto, Modeling generic mean reversion processes with a symmetrical binomial lattice-applications to real options, Procedia Comput. Sci. 55 (2015), 764-773. https://doi.org/10.1016/j.procs.2015.07.160
  10. A.N. Fall, S.N. Ndiaye, N. Sene, Black-Scholes option pricing equations described by the Caputo generalized fractional derivative, Chaos Solitons Fractals 125 (2019), 108-118. https://doi.org/10.1016/j.chaos.2019.05.024
  11. J.-H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul. 6 (2005), 207-208.
  12. M.S. Joshi, The concepts and practice of mathematical finance, Cambridge Univ. Press, 1 2003.
  13. J. Kim, T. Kim, J. Jo, Y. Choi, S. Lee, H. Hwang, M. Yoo, D. Jeong, A practical finite difference method for the three-dimensional Black-Scholes equation, Eur. J. Oper. Res. 252 (2016), 183-190. https://doi.org/10.1016/j.ejor.2015.12.012
  14. N. Khan, Q.M.U. Hassan, E.U. Haq, M.Y. Khan, K. Ayub, J. Ayub, Analytical technique with Lagrange multiplier for solving specific nonlinear differential equations, J. Sci. Arts 21 (2021), 5-14. https://doi.org/10.46939/J.Sci.Arts-21.1-a01
  15. M. Kadalbajoo, L.P. Tripathi, A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation, Math. Comput. Modelling 55 (2012), 1483-1505. https://doi.org/10.1016/j.mcm.2011.10.040
  16. M.N. Koleva, 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. 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.
  18. M.D. Marcozzi, S. Choi, C.S. Chen, On the use of boundary conditions for variational formulations arising in financial mathematics, Appl. Math. Comput. 124 (2001), 197-214.
  19. K. Nonlaopon, F.A. Shah, K. Ahmed, and G. Farid, A generalized iterative scheme with computational results concerning the systems of linear equations, AIMS Math. 8 (2023), 6504-6519.
  20. M.A. Noor and F.A. Shah, A family of iterative schemes for finding zeros of nonlinear equations, Appl. Math. Inf. Sci. 8 (2014), 2367-2373. https://doi.org/10.12785/amis/080532
  21. P. Pharochoo, A. Luadsong, N. Aschariyaphotha, A numerical study of the European option by the MLPG method with moving kriging interpolation, SpringerPlus 5 (2016), 1-14. https://doi.org/10.1186/s40064-015-1659-2
  22. K.R. Raslan, H.A. Baghdady, A Comparison between the Variational Iteration Method and Adomian Decomposition Method, Gen 20 (2014), 125-135.
  23. H. Rouhparvar, M. Salamatbakhsh, Analytical Solution Of The Black-Scholes Equation By Using Variational Iteration Method, Appl. Math. E-Notes 13 (2013), 243-
  24. P. Pharochoo, A. Luadsong, N. Aschariyaphotha, A numerical study of the European option by the MLPG method with moving kriging interpolation, SpringerPlus 5 (2016), 1-14. https://doi.org/10.1186/s40064-015-1659-2
  25. H. Rouhparvar, M. Salamatbakhsh, Analytical Solution Of The Black-Scholes Equation By Using Variational Iteration Method, Appl. Math. E-Notes 13 (2013), 243-256.
  26. A. Santos, P. de Carvalho, F. Silva, Analyzing the financial performance of Brazilian investment funds using decision trees and neural networks, Comput. Ind. Eng. 128 (2019), 570-580.
  27. F.A. Shah and M.A. Noor, Application of decomposition technique and efficient methods for the approximate solution of nonlinear equations, U.P.B. Sci. Bull. A, Series A 79 (2017), 171-180.
  28. F.A. Shah, M. Darus, I. Faisal, and M.A. Shafiq, Decomposition technique and a family of efficient schemes for nonlinear equations, Discrete Dynam. Nat. Sci. 2017 (2017), 3794357.
  29. F.A. Shah and M.A. Noor, Some numerical methods for solving nonlinear equations by using decomposition technique, Appl. Math. Comput. 251 (2015), 378-386.
  30. F.A. Shah and M.A. Noor, Higher order iterative schemes for nonlinear equations using decomposition technique, Appl. Math. Comput. 266 (2015), 414-423.
  31. F.A. Shah, M.A. Noor, and M. Waseem, Some Steffensen-type iterative schemes for the approximate solution of nonlinear equations, Miskolc Math. Notes 22 (2021), 939-949. https://doi.org/10.18514/MMN.2021.2787
  32. F.A. Shah, E.U. Haq, M.A. Noor, and M. Waseem, Some novel schemes by using multiplicative calculus for nonlinear equations, TWMS J. Appl. Eng. Math. 13 (2023), 723-733.
  33. F.A. Shah, M.A. Noor, Variational iteration technique and some methods for the approximate solution of nonlinear equations, Appl. Math. Inf. Sci. Lett. 3 (2009), 85-93.
  34. F.A. Shah, M.A. Noor, Variational iteration technique for solving nonlinear equations, J. Appl. Math. Comput. 31 (2009), 247-254. https://doi.org/10.1007/s12190-008-0207-4
  35. F.A. Shah and M. Waseem, Quadrature based innovative techniques concerning nonlinear equations having unknown multiplicity, Examples and Counterexamples 6 (2024), 100150.
  36. R. Singh, M. Kumar, Application of Adomian Decomposition Method for Solving Black-Scholes Equation, J. Comput. Math. 35 (2016), 112-126.
  37. E. Tuncer, A. Yildirim, Solution of the Black-Scholes partial differential equation using the Modified Variational Iteration Method, Appl. Math. Comput. 236 (2014), 53-63.
  38. J. Wang, Q. Wang, L. Liu, Numerical solutions of the Black-Scholes model with varying parameters, Comput. Math. Appl. 64 (2012), 293-307.
  39. A.-M. Wazwaz, A comparison between the variational iteration method and Adomian decomposition method, Journal of Computational and Applied Mathematics 207 (2007), 129-136. https://doi.org/10.1016/j.cam.2006.07.018
  40. T. Zamir, F.A. Shah, M. Shoaib, and A. Ullah, Design of intelligent computing networks for a two-phase fluid flow with dusty particles hanging above a stretched cylinder, Computers and Concrete 32 (2023), 399-410. https://doi.org/10.12989/CAC.2023.32.4.399
  41. Y. Zhang, X. Zhao, C. Zhang, A computational approach for the Black-Scholes equation using RBF interpolation, Appl. Math. Comput. 271 (2016), 305-318.
  42. A. Zulfiqar, J. Ahmad, Q.M.U. Hassan, Analytical study of fractional Newell-Whitehead-Segel equation using an efficient method, J. Sci. Arts 19 (2019), 839-850.