In this paper, the Projected Differential Transform Method (PDTM) has been used to solve the Black Scholes differential equation for powered Modified Log Payoff (ML-Payoff) functions,$\max\{S^k\ln\big(\frac{S}{K}\big),0\}$ and $\max\{S^k\ln\big(\frac{K}{S}\big),0\}, (k\in \mathbb{R^{+}}\cup \{0\})$. It is the generalization of Black Scholes model for ML-Payoff functions. It can be seen that values from PDTM are quite accurate to the closed form solutions.
[1] Z. Cen, J. Huang, A. Xu, and A. Le, Numerical approximation of a time-fractional Black-Scholes equation, Computers & Mathematics with Applications, 75(8) (2018), 2874-2887.
[2] H. V. Dedania and S. J. Ghevariya, Option Pricing Formula for Modified Log-payoff Function, Inter. Journal of Mathematics and Soft Computing, 3(2) (2013), 129-140.
[3] H. V. Dedania and S. J. Ghevariya, Option Pricing Formulas for Fractional Polynomial Payoff Function, Inter- national Journal of Pure and Applied Mathematical Sciences, 6(1) (2013), 43-48.
[4] H. V. Dedania and S. J. Ghevariya, Graphical Interpretation of Various BSM Formulas, Global Journal of Pure and Applied Mathematics, 13(9) (2017), 6107-6112.
[5] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC Press, 2nd Ed., 2004.
[6] S. O. Edeki, R. M. Jena, O. P. Ogundile, and S. Chakraverty, PDTM for the solution of a time-fractional barrier option Black-Scholes model, Journal of Physics, Conference Series, 1734 (2021), 012055.
[7] S. O. Edeki, O. O. Ugberbor, and E. A. Owoloko, Analytical Solutions of the Black-Scholes Pricing Model for European Option Valuation via a Projected Differential Transform Method, Entropy, 17 (2015), 7510-7521.
[8] S. E. Fadugba and C. R. Nwozo, Valuation of European Call Options via the Fast Fourier Transform and the Improved Mellin Transform, Jr. of Math. Finance, 6 (2016), 338-359.
[9] S. J. Ghevariya, BSM Option Pricing Formulas Through Probabilistic Approach, Math. Today, 32 (2017), 31-34.
[10] S. J. Ghevariya, BSM European Put Option Pricing Formula for ML-Payoff Function with Mellin Transform, International Journal of Mathematics and its Applications, 6(2) (2018), 33-36.
[11] S. J. Ghevariya, BSM Model for ML-Payoff Function through PDTM, Asian-European Journal of Mathematics, 13(1) (2020), 2050024.
[12] S. J. Ghevariya and D. R. Thakkar, BSM model for the Generalized ML-Payoff, Journal of Applied Mathematics and Computational Mechanics, 18(4) (2019), 19-25.
[13] S. J. Ghevariya, An Improved Mellin Transform approach to BSM Formula of ML-Payoff Function, Journal of Interdisciplinary Mathematics, Taylor and Francis group, 22(6) (2019), 863-871.
[14] M. Hatami, D. D. Ganji, and M. Sheikholeslami, Differential Transformation Method for Mechanical Engineering Problems, Academic Press, 1st Ed., 2016.
[15] E. G. Haug, The Complete Guide to Option Pricing Formulas, McGraw-Hill, 2nd Ed., 2007.
[16] B. Jang, Solving linear and nonlinear initial value problems by the projected differential transform method, Com- puter Physics Communications, 181(5) (2010), 848-854.
[17] R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black-Scholes equation governing option pricing, Computers & Mathematics with Appl., 69(8) (2015), 777-797.
[18] R. Panini and R. P. Srivastav, Option pricing with Mellin Transform, Mathematical and Computer Modelling, 40 (2004), 43-56.
[19] P. Roul, A fourth order numerical method based on B-spline functions for Pricing Asian Options, Computers & Mathematics with Applications, 80(3) (2020), 504-521.
[20] P. Roul, A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation gov- erning European options, Applied Numerical Mathematics, 151 (2020), 472-493.
[21] P. Roul and V. M. K. Prasad Goura, A sixth order numerical method and its convergence for generalized Black- Scholes PDE, Journal of Comutational and Applied Mathematics, 377 (2020),112881.
[22] P. Roul and V. M. K. Prasad Goura, A new higher order compact finite difference method for generalized Black- Scholes partial differential equation: European call option, Journal of Comutational and Applied Mathematics, 363 (2020), 464-484.
[23] P. Wilmott, Paul Wilmott on Quantitative Finance, John Wiley & Sons, 2nd Ed., 2006.
[24] P. Wilmott, S. Howison and J. Dewynne, Mathematics of Financial Derivatives, Cambridge University Press, 2002.
[25] J. K. Zhou, Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
Ghevariya, S. (2022). PDTM approach to solve Black Scholes equation for powered ML-Payoff function. Computational Methods for Differential Equations, 10(2), 320-326. doi: 10.22034/cmde.2021.37944.1675
MLA
Sanjay J Ghevariya. "PDTM approach to solve Black Scholes equation for powered ML-Payoff function". Computational Methods for Differential Equations, 10, 2, 2022, 320-326. doi: 10.22034/cmde.2021.37944.1675
HARVARD
Ghevariya, S. (2022). 'PDTM approach to solve Black Scholes equation for powered ML-Payoff function', Computational Methods for Differential Equations, 10(2), pp. 320-326. doi: 10.22034/cmde.2021.37944.1675
VANCOUVER
Ghevariya, S. PDTM approach to solve Black Scholes equation for powered ML-Payoff function. Computational Methods for Differential Equations, 2022; 10(2): 320-326. doi: 10.22034/cmde.2021.37944.1675