# American Institute of Mathematical Sciences

October  2020, 13(10): 2841-2851. doi: 10.3934/dcdss.2020122

## Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries

 a. Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, 53200, Pakistan b. Department of Mathematics, School of Science and Engineering, Lahore University of Management Sciences, LUMS, Lahore Cantt 54792, Pakistan

* Corresponding author: Imran Naeem

Received  January 2019 Revised  May 2019 Published  October 2020 Early access  October 2019

We analyze the local conservation laws, auxiliary (potential) systems, potential symmetries and a class of new exact solutions for the Black-Scholes model time-dependent parameters (BST model). First, we utilize the computer package GeM to construct local conservation laws of the BST model for three different forms of multipliers. We obtain two conserved vectors for the second-order multipliers of form $\Lambda(x,u,u_x,u_{xx})$. We define two potential variables $v$ and $w$ corresponding to the conserved vectors. We construct two singlet potential systems involving a single potential variable $v$ or $w$ and one couplet potential system involving both potential variables $v$ and $w$. Moreover, a spectral potential system is constructed by introducing a new potential variable $p_\alpha$ which is a linear combination of potential variables $v$ and $w$. The potential symmetries of BST model are derived by computing the point symmetries of its potential systems. Both singlet potential systems provide three potential symmetries. The couplet potential system yields three potential symmetries and no potential symmetries exist for the spectral potential system. We utilize the potential symmetries of singlet potential systems to construct three new solutions of BST model.

Citation: Rehana Naz, Imran Naeem. Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2841-2851. doi: 10.3934/dcdss.2020122
##### References:
 [1] I. S. Akhatov, R. K. Gazizov and N. K. Ibragimov, Nonlocal symmetries: A heuristic approach, Journal of Soviet Mathematics, 55 (1991), 1401-1450. [2] L. Martínez Alonso, On the Noether map, Letters in Mathematical Physics, 3 (1979), 419-424.  doi: 10.1007/BF00397216. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [4] G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168. Springer, New York, 2010. doi: 10.1007/978-0-387-68028-6. [5] A. F. Cheviakov and R. Naz, A recursion formula for the construction of local conservation laws of differential equations, Journal of Mathematical Analysis and Applications, 448 (2017), 198-212.  doi: 10.1016/j.jmaa.2016.10.042. [6] G. W. Bluman, G. J. Reid and S. Kumei, New classes of symmetries for partial differential equations, Journal of Mathematical Physics, 29 (1988), 806-811.  doi: 10.1063/1.527974. [7] G. Bluman, Use and construction of potential symmetries, Mathematical and Computer Modelling, 18 (1993), 1-14.  doi: 10.1016/0895-7177(93)90211-G. [8] G. Bluman and A. F. Cheviakov, Framework for potential systems and non-local symmetries: Algorithmic approach, Journal of Mathematical Physics, 46 (2005), 123506, 19 pp. doi: 10.1063/1.2142834. [9] G. Bluman, A. F. Cheviakov and N. M. Ivanova, Framework for nonlocally related partial differential equation systems and nonlocal symmetries: Extension, simplification, and examples, Journal of Mathematical Physics, 47 (2006), 113505, 23 pp. doi: 10.1063/1.2349488. [10] A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Computer Physics Communications, 176 (2007), 48-61.  doi: 10.1016/j.cpc.2006.08.001. [11] A. F. Cheviakov, Computation of fluxes of conservation laws, Journal of Engineering Mathematics, 66 (2010), 153-173.  doi: 10.1007/s10665-009-9307-x. [12] R. M. Edelstein and K. S. Govinder, Conservation laws for the Black-Scholes equation, Nonlinear Analysis: Real World Applications, 10 (2009), 3372-3380.  doi: 10.1016/j.nonrwa.2008.10.064. [13] M. L. Gandarias, Potential symmetries of a porous medium equation, Journal of Physics A: Mathematical and General, 29 (1996), 5919-5934.  doi: 10.1088/0305-4470/29/18/021. [14] M. L. Gandarias, New potential symmetries for some evolution equations, Physica A: Statistical Mechanics and its Applications, 387 (2008), 2234-2242.  doi: 10.1016/j.physa.2007.12.013. [15] R. K. Gazizov and N. H. Ibragimov, Lie symmetry analysis of differential equations in finance, Nonlinear Dynamics, 17 (1998), 387-407.  doi: 10.1023/A:1008304132308. [16] W. Hereman, M. Colagrosso, R. Sayers, A. Ringler, B. Deconinck, M. Nivala and M. Hickman, Continuous and discrete homotopy operators and the computation of conservation laws, Differential Equations with Symbolic Computation, Trends Math., Birkhäuser, Basel, (2005), 255–290. doi: 10.1007/3-7643-7429-2_15. [17] I. S. Krasil'shchik and A. M. Vinogradov, Nonlocal symmetries and the theory of coverings: An addendum to Vinogradov's "Local symmetries and conservation laws", Acta Applicandae Mathematica, 2 (1984), 79-96.  doi: 10.1007/BF01405492. [18] I. S. Krasil'Shchik and A. M. Vinogradov, Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäklund transformations, Acta Appl. Math., 15 (1989), 161-209.  doi: 10.1007/BF00131935. [19] S. Lie, On integration of a class of linear partial differential equations by means of definite integrals Archiv for Mathematik og Naturvidenskab, Gesammelte Abhadlundgen, 6 (1881), 328-368. [20] F. M. Mahomed, Complete invariant characterization of scalar linear $(1+1)$ parabolic equations, J. Nonlinear Math. Phys., 15 (2008), 112-123.  doi: 10.2991/jnmp.2008.15.s1.10. [21] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [22] R. C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143. [23] R. Naz and A. G. Johnpillai, Exact solutions via invariant approach for Black-Scholes model with time-dependent parameters, Mathematical Methods in the Applied Sciences, 41 (2018), 4417-4427.  doi: 10.1002/mma.4903. [24] R. Naz, F. M. Mahomed and D. P. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Applied Mathematics and Computation, 205 (2008), 212-230.  doi: 10.1016/j.amc.2008.06.042. [25] R. Naz, I. L. Freire and I. Naeem, Comparison of different approaches to construct first integrals for ordinary differential equations, AAbstract and Applied Analysis, (2014), Art. ID 978636, 15 pp. doi: 10.1155/2014/978636. [26] R. Naz, F. M. Mahomed and A. Chaudhry, A partial Hamiltonian approach for current value Hamiltonian systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3600-3610.  doi: 10.1016/j.cnsns.2014.03.023. [27] R. Naz, A. Chaudhry and F. M. Mahomed, Closed-form solutions for the Lucas zawa model of economic growth via the partial Hamiltonian approach, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 299-306.  doi: 10.1016/j.cnsns.2015.06.033. [28] R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-Linear Mechanics, 86 (2016), 1-6. [29] R. Naz and I. Naeem, The artificial hamiltonian, first integrals, and closed-form solutions of dynamical systems for epidemics, Zeitschrift f Naturforschung A, 73 (2018), 323-330. [30] R. O. Popovych and N. M. Ivanova, Hierarchy of conservation laws of diffusion-convection equations, Journal of Mathematical Physics, 46 (2005), 043502, 22 pp. doi: 10.1063/1.1865813. [31] E. Pucci and G. Saccomandi, Potential symmetries and solutions by reduction of partial differential equations, Journal of Physics A: Mathematical and General, 26 (1993), 681-690.  doi: 10.1088/0305-4470/26/3/025. [32] T. M. Rocha Filho and A. Figueiredo, [SADE] a Maple package for the symmetry analysis of differential equations, Computer Physics Communications, 182 (2011), 467-476. [33] M. R. Rodrigo and R. S. Mamon, An alternative approach to solving the Black choles equation with time-varying parameters, Applied Mathematics Letters, 19 (2006), 398-402.  doi: 10.1016/j.aml.2005.06.012. [34] A. Sjöberg and F. M. Mahomed, Non-local symmetries and conservation laws for one-dimensional gas dynamics equations, Applied Mathematics and Computation, 150 (2004), 379-397.  doi: 10.1016/S0096-3003(03)00259-5. [35] K. M. Tamizhmani, K. Krishnakumar and P. G. L. Leach, Algebraic resolution of equations of the Black-Scholes type with arbitrary time-dependent parameters, Applied Mathematics and Computation, 247 (2014), 115-124.  doi: 10.1016/j.amc.2014.08.087. [36] K. T. Vu, G. F. Jefferson and J. Carminati, Finding higher symmetries of differential equations using the MAPLE package DESOLVII, Computer Physics Communications, 183 (2012), 1044-1054.  doi: 10.1016/j.cpc.2012.01.005. [37] T. Wolf, A comparison of four approaches to the calculation of conservation laws, European Journal of Applied Mathematics, 13 (2002), 129-152.  doi: 10.1017/S0956792501004715. [38] T. Wolf, A. Brand and M. Mohammadzadeh, Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions, Journal of Symbolic Computation, 27 (1999), 221-238.  doi: 10.1006/jsco.1998.0250.

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##### References:
 [1] I. S. Akhatov, R. K. Gazizov and N. K. Ibragimov, Nonlocal symmetries: A heuristic approach, Journal of Soviet Mathematics, 55 (1991), 1401-1450. [2] L. Martínez Alonso, On the Noether map, Letters in Mathematical Physics, 3 (1979), 419-424.  doi: 10.1007/BF00397216. [3] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [4] G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168. Springer, New York, 2010. doi: 10.1007/978-0-387-68028-6. [5] A. F. Cheviakov and R. Naz, A recursion formula for the construction of local conservation laws of differential equations, Journal of Mathematical Analysis and Applications, 448 (2017), 198-212.  doi: 10.1016/j.jmaa.2016.10.042. [6] G. W. Bluman, G. J. Reid and S. Kumei, New classes of symmetries for partial differential equations, Journal of Mathematical Physics, 29 (1988), 806-811.  doi: 10.1063/1.527974. [7] G. Bluman, Use and construction of potential symmetries, Mathematical and Computer Modelling, 18 (1993), 1-14.  doi: 10.1016/0895-7177(93)90211-G. [8] G. Bluman and A. F. Cheviakov, Framework for potential systems and non-local symmetries: Algorithmic approach, Journal of Mathematical Physics, 46 (2005), 123506, 19 pp. doi: 10.1063/1.2142834. [9] G. Bluman, A. F. Cheviakov and N. M. Ivanova, Framework for nonlocally related partial differential equation systems and nonlocal symmetries: Extension, simplification, and examples, Journal of Mathematical Physics, 47 (2006), 113505, 23 pp. doi: 10.1063/1.2349488. [10] A. F. Cheviakov, GeM software package for computation of symmetries and conservation laws of differential equations, Computer Physics Communications, 176 (2007), 48-61.  doi: 10.1016/j.cpc.2006.08.001. [11] A. F. Cheviakov, Computation of fluxes of conservation laws, Journal of Engineering Mathematics, 66 (2010), 153-173.  doi: 10.1007/s10665-009-9307-x. [12] R. M. Edelstein and K. S. Govinder, Conservation laws for the Black-Scholes equation, Nonlinear Analysis: Real World Applications, 10 (2009), 3372-3380.  doi: 10.1016/j.nonrwa.2008.10.064. [13] M. L. Gandarias, Potential symmetries of a porous medium equation, Journal of Physics A: Mathematical and General, 29 (1996), 5919-5934.  doi: 10.1088/0305-4470/29/18/021. [14] M. L. Gandarias, New potential symmetries for some evolution equations, Physica A: Statistical Mechanics and its Applications, 387 (2008), 2234-2242.  doi: 10.1016/j.physa.2007.12.013. [15] R. K. Gazizov and N. H. Ibragimov, Lie symmetry analysis of differential equations in finance, Nonlinear Dynamics, 17 (1998), 387-407.  doi: 10.1023/A:1008304132308. [16] W. Hereman, M. Colagrosso, R. Sayers, A. Ringler, B. Deconinck, M. Nivala and M. Hickman, Continuous and discrete homotopy operators and the computation of conservation laws, Differential Equations with Symbolic Computation, Trends Math., Birkhäuser, Basel, (2005), 255–290. doi: 10.1007/3-7643-7429-2_15. [17] I. S. Krasil'shchik and A. M. Vinogradov, Nonlocal symmetries and the theory of coverings: An addendum to Vinogradov's "Local symmetries and conservation laws", Acta Applicandae Mathematica, 2 (1984), 79-96.  doi: 10.1007/BF01405492. [18] I. S. Krasil'Shchik and A. M. Vinogradov, Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäklund transformations, Acta Appl. Math., 15 (1989), 161-209.  doi: 10.1007/BF00131935. [19] S. Lie, On integration of a class of linear partial differential equations by means of definite integrals Archiv for Mathematik og Naturvidenskab, Gesammelte Abhadlundgen, 6 (1881), 328-368. [20] F. M. Mahomed, Complete invariant characterization of scalar linear $(1+1)$ parabolic equations, J. Nonlinear Math. Phys., 15 (2008), 112-123.  doi: 10.2991/jnmp.2008.15.s1.10. [21] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [22] R. C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143. [23] R. Naz and A. G. Johnpillai, Exact solutions via invariant approach for Black-Scholes model with time-dependent parameters, Mathematical Methods in the Applied Sciences, 41 (2018), 4417-4427.  doi: 10.1002/mma.4903. [24] R. Naz, F. M. Mahomed and D. P. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Applied Mathematics and Computation, 205 (2008), 212-230.  doi: 10.1016/j.amc.2008.06.042. [25] R. Naz, I. L. Freire and I. Naeem, Comparison of different approaches to construct first integrals for ordinary differential equations, AAbstract and Applied Analysis, (2014), Art. ID 978636, 15 pp. doi: 10.1155/2014/978636. [26] R. Naz, F. M. Mahomed and A. Chaudhry, A partial Hamiltonian approach for current value Hamiltonian systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3600-3610.  doi: 10.1016/j.cnsns.2014.03.023. [27] R. Naz, A. Chaudhry and F. M. Mahomed, Closed-form solutions for the Lucas zawa model of economic growth via the partial Hamiltonian approach, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 299-306.  doi: 10.1016/j.cnsns.2015.06.033. [28] R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-Linear Mechanics, 86 (2016), 1-6. [29] R. Naz and I. Naeem, The artificial hamiltonian, first integrals, and closed-form solutions of dynamical systems for epidemics, Zeitschrift f Naturforschung A, 73 (2018), 323-330. [30] R. O. Popovych and N. M. Ivanova, Hierarchy of conservation laws of diffusion-convection equations, Journal of Mathematical Physics, 46 (2005), 043502, 22 pp. doi: 10.1063/1.1865813. [31] E. Pucci and G. Saccomandi, Potential symmetries and solutions by reduction of partial differential equations, Journal of Physics A: Mathematical and General, 26 (1993), 681-690.  doi: 10.1088/0305-4470/26/3/025. [32] T. M. Rocha Filho and A. Figueiredo, [SADE] a Maple package for the symmetry analysis of differential equations, Computer Physics Communications, 182 (2011), 467-476. [33] M. R. Rodrigo and R. S. Mamon, An alternative approach to solving the Black choles equation with time-varying parameters, Applied Mathematics Letters, 19 (2006), 398-402.  doi: 10.1016/j.aml.2005.06.012. [34] A. Sjöberg and F. M. Mahomed, Non-local symmetries and conservation laws for one-dimensional gas dynamics equations, Applied Mathematics and Computation, 150 (2004), 379-397.  doi: 10.1016/S0096-3003(03)00259-5. [35] K. M. Tamizhmani, K. Krishnakumar and P. G. L. Leach, Algebraic resolution of equations of the Black-Scholes type with arbitrary time-dependent parameters, Applied Mathematics and Computation, 247 (2014), 115-124.  doi: 10.1016/j.amc.2014.08.087. [36] K. T. Vu, G. F. Jefferson and J. Carminati, Finding higher symmetries of differential equations using the MAPLE package DESOLVII, Computer Physics Communications, 183 (2012), 1044-1054.  doi: 10.1016/j.cpc.2012.01.005. [37] T. Wolf, A comparison of four approaches to the calculation of conservation laws, European Journal of Applied Mathematics, 13 (2002), 129-152.  doi: 10.1017/S0956792501004715. [38] T. Wolf, A. Brand and M. Mohammadzadeh, Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions, Journal of Symbolic Computation, 27 (1999), 221-238.  doi: 10.1006/jsco.1998.0250.
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