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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 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 & Continuous Dynamical Systems - S, 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.   Google Scholar [2] L. Martínez Alonso, On the Noether map, Letters in Mathematical Physics, 3 (1979), 419-424.  doi: 10.1007/BF00397216.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.   Google Scholar [2] L. Martínez Alonso, On the Noether map, Letters in Mathematical Physics, 3 (1979), 419-424.  doi: 10.1007/BF00397216.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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