Exact Solutions of Partial Differential Equation of Black-Scholes Option Pricing Model using Partial Taylor Series Expansion Method

ABSTRACT

In this paper, a new series solution method called partial Taylor series expansion method is introduced for solving partial differential equations. The method is used in the present work to develop analytical solutions to Black–Scholes differential partial differential equations. The results of the solutions of the method are compared with the results of the exact analytical solutions and excellent agreements are achieved. Moreover, numerical examples for different options pricing are presented to establish the reliability, speed, accuracy, and ease of application of the proposed method.

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. J. 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.
• Gülkac¸ The homotopy perturbation method for the Black – Scholes equation, J. 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.
• Cen, Z., Le, A. (2011). A robust and accurate finite difference method for a generalized Black-Scholes equation. Journal of Computational and Applied Mathematics, 235 (2011), 3728-3733.
• Aminikhah,, Mehrdoust, F. On Approximate-Analytical Solution of Generalized Black-Scholes Equation. U.P.B. Sci. Bull., Series A, 77(4) (2015), 185-194

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