Analysis of Black-Scholes Option Pricing Differential Equations Powered by Log-Payoff function using Method of Partial Taylor Series Expansion

ABSTRACT In this paper, a new method called partial Taylor series expansion method is applied to Black–Scholes differential equation for Log-Payoff function. The analytical method was used to develop solutions for the call and put options pricing for the classical Black–Scholes equations for the Log Payoff functions. The new method displayed high level of simplicity and low cost of computation. Moreover, the reliability, speed, accuracy, and ease of application of the proposed method show its potential for wide areas of applications in science, financial mathematics, and engineering. References John C. Hull. Option, Futures and Other Derivatives, Global edition, Eighth edition, Pearson Education Limited. (2015-2016). Lutfi Mardianto, , Aditya Putra Pratama, A. P, Soemarsono, A. R., Hakam, A., Rokhmati, E., Putri, M. Comparison of Numerical Methods on Pricing of European Put Options. International Journal of Computing Science and Applied Mathematics 2019, 5(1), 2019 Black, F.; Scholes, M.S. The pricing of options and corporate liabilities. Political Econ. 1973, 81, 637-654 Merton, C. Theory of rational option pricing. Bell J. Econ. Manage. Sci. 1973, 4, 141-183 Black,, M. Scholes, M. The pricing of options and corporate liabilities. J. Pol. Econ. 81 (1973) 637-659 Barles, Soner, H. M. Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance Stoch. 2 (1998) 369-397 Company, , Jódar, L., Pintos, J. R. Numerical analysis and computing for option pricing models in illiquid markets. Math. Comput. Modelling 2010, 52 (7–8), 1066–1073 Wilmott,, Howison, S., Dewynne, J. The Mathematics of Financial Derivatives: A Student Introduction,Cambridge University Press, Cambridge, 1995. Cen, Z.; Le, A. A robust and accurate finite difference method for a generalized Black-Scholes equation. Comput. Appl. Math. 2011, 235, 3728–3733 Cho, C.; Kim, T.; Kwon, Y. Estimation of local volatilities in a generalized Black-Scholes model. Math. Comput. 2005, 162, 1135–1149 Kangro, R.; Nicolaides, R. Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 2000, 38, 1357–1368 Ashyralyev, A.; Erdogan, A.S.; Tekalan, S.N. An investigation on finite difference method for the first order partial differential equation with the nonlocal boundary condition. Comput. Math. 2019, 18, 247–260 Anguelov, R.; Lubuma, J.M. Contributions to the mathematics of the nonstandard finite difference method and applications. Methods Partial Differ. Equ. 2001, 17, 518–543 Mickens, R.E. Exact solutions to a finite difference model of a nonlinear reaction-advection equation: Implications for numerical Numer. Methods Part. Differ. Equ. 1989, 5, 313–325 Mickens, R.E. Nonstandard Finite Difference Models of Differential Equations; World Scientific: Singapore, 1994. Mehdizadeh Khalsaraei, M.; Shokri, A.; Mohammadnia, Z.; Sedighi, H.M. Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black–Scholes Equation. Math. 2021, 2012, 6679484 Mickens, R.E. A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion. Math. Appl. 2003, 45, 429–436 Mickens, R.E. Nonstandard finite difference schemes for reaction diffusion equations having linear advection. Methods Partial Differ. Equ. 2000, 4, 361–364 Mickens, R.E.; Jordan, P.M. A new positivity-preserving nonstandard finite difference scheme for the DWE. Methods Partial Differ. Equ. 2005, 21, 976–985 Cen, Z.; Le, A.; Xu, A. Exponential time integration and second-order difference scheme for a generalized Black-Scholes equation. Appl. Math. Art. 2012, 2012, 796814 Kadalbajoo, M.K.; Tripathi, L.P.; Kumar, A. A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation. Comput. Model. 2012, 55, 1483–1505 Valkov, R. Fitted finite volume method for a generalized Black-Scholes equation transformed on finite interval. Algorithms 2014, 65, 195–220 Huang, J.; Cen, Z. Cubic spline method for a generalized Black-Scholes equation. Probl. Eng. Art. 2014, 2014, 484362 Mohammadi, R. Quintic B-spline collocation approach for solving generalized Black-Scholes equation governing option pricing. Math. Appl. 2015, 69, 777–794. Gülkaç, V. The homotopy perturbation method for the Black – Scholes equation. Stat. Comput. Simul. 2010, 80(12), 1349-1354 Kumar, , Yildirim, A., Khan, Y., Jafari, H., Sayevand, K. and I. Wei, I. Analytical Solution of Fractional Black-Scholes European Option Pricing Equation by Using Laplace Transform. Journal of Fractional Calculus and Applications 2012, 2(8), 1-9 Allahviranloo, T. and S. S. Behzadi,S. The use of iterative methods for solving Black-Scholes equation. Int. J. Ind. Math. 2013, 5(1), 1-11 Edeki, O. Ugbebor, O.O and Owoloka, E.A. Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method. Entropy . 2015, 17, 7510-7521 Edeki, O. Ugbebor, O.O and Ogundile, O.O. Analytical Solutions of a Continuous Arithmetic Asian Model for Option Pricing using Projected Differential Transform Method. Engineering Letters 2019, 27(2), 303-310 Biazar, J. and F. Goldoust, The Adomian Decomposition Method for the Black-Scholes Equation. 3rd Int. Conf. Appl. Math. Pharm. Scincences, Singapore, 2013, 321-323 Gonz´alez-Gaxiola,, Ru´ız de Ch´avez J. and Santiago, J.A. A Nonlinear Option Pricing Model Through the Adomian Decomposition Method. Int. J. Appl. Comput. Math. 2016, 2, 435-467 Yavuz, M. and Ozdemir, A Quantitative Approach to Fractional Option Pricing Problems with Decomposition Series. Konuralp Journal of Mathematics, 2018, 6(1), 102-109 Putri, R. M., Mardianto, L., Hakam, A., Imron, C., Susanto, H. Removing non-smoothness in solving Black-Scholes equation using aperturbation method. Physics Letters A 402 (2021) 127367 Sumiati, , Rusyaman, E. and Sukono. Black-Scholes Equation Solution Using Laplace-Adomian Decomposition Method. IAENG International Journal of Computer Science , 2019, 46(4) (2019) 21 Corliss, G., Chang, Y. F. (1982). Solving ordinary differential equations using Taylor series. ACM Trans. Math. Software 8 (2) (1982) 114–144 Chang, Y. -F., Corliss, G. ATOMFT: solving ODEs and DAEs using Taylor series. Math. Appl. 28 (10-12) (1994) 209–233 Pryce J. D. Solving high-index DAEs by Taylor series. Algorithms 19 (1–4) (1998) 195–211 Barrio, R. Performance of the Taylor series method for ODEs/DAEs. Math. Comput. 163 (2) (2005) 525–545 Nedialkov, N. S., Pryce, J. D. Solving differential-algebraic equations by Taylor series. I. Computing Taylor coefficients. BIT 45 (3) (2005) 561–591 Nedialkov, N. S., Pryce, J. D. Solving differential-algebraic equations by Taylor series. II. Computing the system Jacobian. BIT 47 (1) (2007) 121–135 Nedialkov, N. S., Pryce, J. D. Solving differential algebraic equations by Taylor series. III. The DAETS code. Numer. Anal. Ind. Appl. Math. 3 (1-2) (2008) 61–80 Jorba, Á., Zou., M. A software package for the numerical integration of ODEs by means of high-order Taylor methods. Math. 14 (1) (2005) 99–117 Makino, K., Berz, M. Taylor models and other validated functional inclusion methods. J. Pure Appl. Math. 6 (3) (2003) 239–316 Barrio, R. Performance of the Taylor series method for ODEs/DAEs. Math. Comput. 163 (2005) 525-545 Ren, Y, Zhang, B., Qiao, H. A simple Taylor-series expansion method for a class of second kind integral equations. Journal of Computational and Applied Mathematics 110(1 ) (1999) 15 15-24 Abbasbandy S. and Bervillier, C. Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations. Math. Comput. 218 (2011) 2178 Kanwal, R. P. and Liu, K. C. A Taylor expansion approach for solving integral equations. J. Math. Ed. Sci. Technol 20 (1989) 411-414 Huang, L., Li, X. F. Zhao, Y. Duan. X. Y. Approximate solution of fractional integro-differential equations by Taylor expansion method. Math. Appl. 62 (2011) 1127-1134 Nedialkov, N. S. and Pryce, J. D. Solving differential-algebraic equations by Taylor series (II): computing the system Jacobian. BIT Numer. Math. 47 (2007) 121-135 Goldfine, A. Taylor series methods for the solution of Volterra integral and integro-differential equations. Comput. 31 (1977) 691-708 Wilmott, P. Paul Wilmott on Quantitative Finance. 2nd ed., John Wiley & Sons, 2006. Wilmott, P., Howison, S., & Dewynne, J. Mathematics of Financial Derivatives. Cambridge University Press, 2002. Download all article in PDF WSN 172 (2022) 118-130

Analysis of Black-Scholes Option Pricing Differential Equations Powered by Log-Payoff function using Method of Partial Taylor Series Expansion

Author: Gbeminiyi M. Sobamowo, 172 (2022) 118-130

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Analysis of Black-Scholes Option Pricing Differential Equations Powered by Log-Payoff function using Method of Partial Taylor Series Expansion

Author: Gbeminiyi M. Sobamowo, 172 (2022) 118-130

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