Analytical solution of time-fractional N-dimensional Black-Scholes equation using LHPM
DOI:
https://doi.org/10.23755/rm.v48i0.1230Keywords:
Financial derivatives, European options, n-dimensional Black-Scholes Equation, Liouville-Caputo Fractional Derivative, Laplace homotopy perturbation method.Abstract
A famous Black-Scholes differential equation is used for pricing options in financial world which represents financial derivatives more significantly. Option is one of the crucial financial derivatives. Sawangtong P., Trachoo K., Sawangtong W. and Wiwattanapataphee B. obtained analytical solution of Black-Scholes equation with two assets in the Liouville-Caputo time-fractional derivative sense using Laplace homotopy perturbation method (LHPM). The aim of this paper is to derive solution of Liouville-Caputo time-fractional Black-Scholes equation with n assets using LHPM. Numerical results shows that our approach gives very accurate results and our formulas are quite close to the plain vanilla options.References
H. V. Dedania and S. J. Ghevariya. Option Pricing Formula for Modified Logpayoff Function. Inter. Journal of Mathematics and Soft Computing, 3(2):129–140, 2013a.
H. V. Dedania and S. J. Ghevariya. Option Pricing Formulas for Fractional Polynomial Payoff Function. Inter. Jr. of Pure and Applied Mathematical Sciences,6(1):43–48, 2013b.
S. E. Fadugba and C. R. Nwozo. Valuation of European Call Options via the Fast Fourier Transform and the Improved Mellin Transforml. Jr. of Math. Finance, 6:338–359, 2016.
S. J. Ghevariya. An improved Mellin transform approach to BSM formula of ML-payoff function. Journal of Interdisciplinary Mathematics, 22(6):863–871, 2019.
S. J. Ghevariya. BSM Model for ML-Payoff Function through PDTM. Asian European Jr. of Mathematics, 13(1):2050024 (6 pages), 2020.
S. J. Ghevariya. PDTM Approach to Solve Black Scholes Equation for Powered MLPayoff Function. Computational Methods for Differential Equations, 10(2):320–326, 2021.
S. J. Ghevariya. Homotopy Perturbation Method to Solve Black Scholes Differential Equation for ML-Payoff Function. Journal of Interdisciplinary Mathematics, 25(2):553–561, 2022a.
Sanjay. J. Ghevariya. PDTM approach to solve Black Scholes equation for powered ML-Payoff function. Computational Methods for Differential Equation,10(2):320–326, 2022b.
T. Guillaue. On the multidimensionl Black-Scholes partial differential equation. Annals of Operations Research, 281:229–251, 2019.
H. Zhang, F. Liu, I. Turner and Q. Yang. Numerical solution of the time fractional Black-Scholes model governing European options. Computers and Mathematics with Applications, 71(9):1772–1783, 2016.
E. G. Haug. The Complete Guide to Option Pricing Formulas. McGraw-Hill, 2007.
J. H. He. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3-4):257–262, 1999.
J. H. He. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation, 135(1):73–79, 2003.
J. Kim, T. Kim, J. Jo, Y. Choi, S. Lee, H. Hwang, M. Yoo and D. Jeong. A practical finite difference method for the three-dimensional Black-Scholes equation. European Journal of Operational Research, 252(1):183–190, 2016.
J. Joonglee and K. Yongsik. Comparision of numerical schemes on multidimensional Black-Scholes equations. Bull. Korean Math. Soc., 6:2035–2051, 2013.
K. Trachoo, W. Sawangtong and P. Sawangtong. Laplace transform homotopy perturbation method for the two dimensional Black Scholes model with European call option. Mathematical and computational applicatios, 22(1):1–11, 2017.
Y. Khan and Q. Wu. Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput. Math.Appl., 61(8):1963–1967, 2011.
F. Mehrdoust and A. R. Najafi. Pricing european options under fractional black scholes model with a weak payoff function. Computational Economics, 52: 685–706, 2017.
L. Meng and W. Wang. Comparison of Black-Scholes formula with fractional Black-Scholes formula in the foreign exchange option market with changing volatility. Asia-Pacific Financial Markets, 17(2):99–111, 2010.
K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. Jhon Willey & Sons, USA, 2003.
P. Sawangtong, K. Trachoo, W. Sawangtong and B. Wiwattanapataphee. The Analytic Solution for the Black-Scholes Equation with Two Assets in the Liouville Caputo Fractional Derivative Sense. Mathematics, 6(8):129, 2018.
P. Wilmott, S. Howison and J. Dewynne. Option Pricing: Mathematical Models and Computation. Oxford Financial Press, England, 1993.
P. Wilmott, S. Howison and J. Dewynne. The Mathematics of Financial Derivatives. Cambridge University Press, 2002.
R. Panini and R. P. Srivastav. Option pricing with Mellin Transform. Mathematical and Computer Modelling, 40:43–56, 2004.
I. Podlubny. Fractional Differential Equations. Academic press, San Diego, 1999.
D. Prathumwan and K. Trachoo. On the solution of two-dimensional fractional Black-Scholes equation for European put option. Advances in Difference Equations, 146:9, 2020.
R. J. Haber, P. J. Schonbucher and P. Wilmott. Pricing Parisan Options. Journal of Derivatives, 6(3):71–79, 1999.
S. J. Ghevariya, C. N. Patel and S. E. Fadugba. Laplace Transform Homotopy Perturbation Method for Black-Scholes Multi-Assets Model. Eurasian Journal of Mathematical and Computer Applications, 10(4):62–72, 2022.
S. Kumar, D. Kumar and J. Singh. Numerical computation of fractional Black Scholes equation arising in financial market. Egyptian Journal of Basic and Applied Sciences, 1(3-4):177–183, 2014.
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, 17:7510–7521, 2015.
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, page 1734, 2021.
L. Song and W. Wang. Solution of the fractional Black Scholes option pricing model by finite difference method. Abstract and Applied Analysis, page 10, 2013.
J. H. Yoon. Mellin Transform method for European option pricing with Hull White stochastic interest rate. Journal of Applied Mathematics, page 7, 2014.
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