Analytical solutions of the nonlinear Ivancevic options pricing model

Document Type : Research Paper


University of the Witwatersrand, School of Mathematics, Johannesburg, South Africa


This paper studies the nonlinear quantum-probability based Schrodinger type, Ivancevic options pricing model using the method of Lie symmetries to determine its point symmetries, invariant analytical solutions and conversation laws. In our analysis, we consider a non-zero and zero adaptive market potential model. We demonstrate that this model is invariant under a five-dimensional Lie algebra for the former, and invariant under a seven-dimensional Lie algebra for the latter case. These symmetries allow for a progressive reduction of the equation and thus facilitate a solution. We obtain reductions, exact solutions and conservation laws for both the non-zero and zero adaptive market potential models. We show that many exact solutions are expressible in terms of two transcendental functions, the Fresnel sine and cosine integrals. Graphical solutions are provided in certain cases. This analysis and solutions to such a financial derivatives pricing model are unique, providing novel insights.


[1] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ. 81 (1973), 637–654.
[2] J.C. Hull, Options, futures, and other derivatives, Pearson, USA, 2006.
[3] R.M. Jena, S. Chakraverty, and D. Baleanu, A novel analytical technique for the solution of time-fractional Ivancevic option pricing model, Phys. A: Stat. Mech. Appl. 550 (2020), 124380.
[4] J. Voit, The statistical mechanics of financial markets, Springer, Berlin, 2001.
[5] O. Vukovic, Interconnectedness of Schrodinger and Black-Scholes Equation, J. Appl. Math. Phys. 3 (2015), no. 9, 1108–1113.
[6] A.H. Davison and S. Mamba, Symmetry methods for option pricing, Commun. Nonlinear Sci. Numer. Simul. 47 (2017), 421–425.
[7] Y. Chatibi, E.H.E Kinani, and A. Ouhadan, Lie symmetry analysis and conservation laws for the time fractional Black–Scholes equation, Int. J. Geom. Meth. Mod. Phys. 17 (2020), no. 01, 2050010.
[8] M. Bohner and Y. Zheng, On analytical solutions of the Black–Scholes equation, Appl. Math. Lett. 22 (2009), no. 3, 309–313.
[9] P. Sawangtong, K. Trachoo, W. Sawangtong, and B. Wiwattanapataphee, The analytical solution for the BlackScholes equation with two assets in the Liouville-Caputo fractional derivative sense, Mathematics 6 (2018), no. 8, 129.
[10] S.O. Edeki, O.O. Ugbebor, and E.A. Owoloko, Analytical solutions of the Black–Scholes pricing model for European option valuation via a projected differential transformation method, Entropy 17 (2015), no. 11, 7510–7521.
[11] F. Mehrdoust and M. Mirzazadeh, On analytical solution of the black-scholes equation by the first integral method, U.P.B. Sci. Bull., Series A. 76 (2014), no. 4, 85–90.
[12] V.G. Ivancevic, Adaptive-wave alternative for the Black-Scholes option pricing model, Cogn. Comput. 2 (2009), 17–30.
[13] U. Obaidullah and S. Jamal, On the formulaic solution of a (n+1)th order differential equation, Int. J. Appl. Comput. Math. 7 (2021) no. 58, 1–15.
[14] U. Obaidullah and S. Jamal, A computational procedure for exact solutions of Burgers’ hierarchy of nonlinear partial differential equations, J. Appl. Math. Comput. 65 (2021), 541-–551.
[15] S. Jamal and A. Paliathanasis, Approximate symmetries and similarity solutions for wave equations on liquid films, Appl. Anal. Discrete Math. 14 (2020), no. 2, 349–363.
[16] A.R. Adem, Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method, Comput. Math. Appl. 74 (2017), no. 8, 1897–1902.
[17] S. Mbusi, B. Muatjetjeja, and A.R. Adem, On the exact solutions and conservation laws of a generalized (1+2) dimensional Jaulent-Miodek equation with a power law nonlinearity, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 1721 1735.
[18] A.R. Adem, A (2+1)-dimensional Korteweg–de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws, Int. J. Mod. Phys. B. 30 (2016), 1640001.
[19] A.R. Adem and B. Muatjetjeja, Conservation laws and exact solutions for a 2D Zakharov–Kuznetsov equation, Appl. Math. Lett. 48 (2015), 109–117.
[20] M.C. Moroke, B. Muatjetjeja, and A.R. Adem, A generalized (2 + 1)-dimensional Calogaro–Bogoyavlenskii–Schiff equation: Symbolic computation, symmetry reductions, exact solutions, conservation laws, Int. J. Appl. Comput.Math. 7 (2021), 134.
[21] T.J. Podile, B. Muatjetjeja and A.R. Adem, Conservation laws and exact solutions of a generalized (2+1)- dimensional Bogoyavlensky-Konopelchenko equation, Int. J. Nonlinear Anal. Appl. 12 (2021), 709–718.
[22] U. Obaidullah and S. Jamal, pp-wave potential functions: A complete study using Noether symmetries, Int. J. Geom. Meth. Mod. Phys. 18 (2021), no. 7, 2150108.
[23] U. Obaidullah, S. Jamal, and G Shabbir Analytical field equation and wave function solutions of the Bianchi type I universe in vacuum f(R) gravity, Int. J. Geom. Meth. Mod. Phys. 19 (2022), no. 9, 2250136.
[24] S. Jamal and G. Shabbir, Potential Functions Admitted by Well-Known Spherically Symmetric Static Spacetimes, Rep. Math. Phys. 81 (2018), no. 02, 201-212.
[25] U. Obaidullah and S. Jamal, Classical solutions to Bianchi type II spacetimes in f(R) theory of gravity, Indian J. Phys. 96 (2022), no. 12, 3675—3688.
[26] O. Gonz´alez-Gaxiola, S. O. Edeki, O. O. Ugbebor, and J.R. Ch’avez, Solving the Ivancevic pricing model Using the He’s frequency amplitude formulation, Eur. J. Pure Appl. 10 (2017), no. 4, 631–637.
[27] S.O. Edeki, O.O. Ugbebor, and O. Gonz´alez-Gaxiola, Analytical solutions of the Ivancevic option pricing model with a nonzero adaptive market potential, Int. J. Appl. Math. Phys. 115 (2017), no. 1, 187–198.
[28] Y.Q. Chen, Y-H. Tang, J. Manafian, H. Rezazadeh, and M.S. Osman, Dark wave, rogue wave and perturbation solutions of Ivancevic option pricing model, Nonlinear Dyn. 105 (2021), 2539–2548.
Volume 14, Issue 3
March 2023
Pages 245-259
  • Receive Date: 07 March 2022
  • Revise Date: 29 June 2022
  • Accept Date: 12 July 2022