On The Optimization of DC-Pension Fund Investment by Lie Symmetry Analysis
Keywords:
DC pension, Lie symmetry, CRRA utility function maximization, optimal portfolio strategy
Abstract
We applied the works inspired by Kumei and Bluman in the determination and evaluation of time-dependent financial investments especially in the DC-Pension fund optimization. We established the applicability of Lie symmetry analysis and the reductions techniques of the Lie-symmetry to transform (2+1) dimensional nonlinear pde into a system of (1+1) linear equation with its solutions and obtained the optimal strategy.
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References
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[2] Fischer Black, Myron Scholes, The valuation of option contracts and a test of market efficiency, J.Finance 27 (1972) 399–417.
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[14] Goard, J. Exact and approximate solutions for options with time-dependent stochastic volatility.Appl.Math. Model. 38, 2771–2780 (2014)
[15] Lo C F, Hui C H. Valuation of financial derivatives with time-dependent parameters: Lie-algebraicapproach. Quant Finance, 2001, 1: 73-78
[16] Lo C F, Hui C H. Pricing multi-asset financial derivatives with time-dependent parameters- Lie algebraic approach. Int J math Sci, 2002, 32: 401-410
[17] Lo C F, Hui C H. Lie algebraic approach for pricing moving barrier options with time- dependent parameters. J. Math Anal Appl, 2006, 323: 1455-1464.
[18] Leach P G L, O’ Hara J G, Sinkaka W. Symmetry-based solution of a model for a Combination of a risky investment and riskless Investment. J. Math Anal Appl, 2007, 334: 368-381.
[19] Sinkaka W, Leach P G L, O’Hara J G. An Optimal system and group-invariant solutions of the Cox-Ingersoll-Ross pricing equation. Appl Math Comput, 2001, 201: 95-107
[20] T. Zariphopoulou, Optimal investment and consumption models with nonlinear stock dynamics, Math. Methods Oper. Res. 50 (1999) 271–296.
[21] R.C. Merton, Lifetime portfolio selection under uncertainty: The continuous time case, Rev.Econom. Statist. 51(1969) 247–257.
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[23] Sherring, A.K. Head, G.E. Prince, Dimsym and LIE: Symmetry determining packages,
Math. Comput. Modelling 25 (1997) 153–164.
[24] Cheviakov, A.F.: GeM software package for computation of symmetries and conservation lawsof differential equations. Comput. Phys. Commun. 176, 48–61 (2007)
[25] Cheviakov, A.F.: Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models. Comput. Phys. Commun. 220, 56–73 (2017)
[26] M.C. Nucci, Interactive REDUCE programs for calculating classical, nonclassical and
Lie–Bäcklund symmetries for differential equations, preprint, Georgia Institute of Technology, Math 062090-051, 1990.
[27] M.C. Nucci, Interactive REDUCE programs for calculating Lie point, nonclassical, Lie–Bäcklund, and approximate symmetries of differential equations: Manual and floppy disk, in: N.H. Ibragimov (Ed.), New Trends in Theoretical Development and Computational Methods, in: CRC Handbook of Lie Group Analysis of Differential Equations, vol. III, CRC Press, Boca Raton, 1996.
[28] G.W.Bluman, A.F. Cheviakov, S.C. Anco, Application of Symmetry Methods to Partial Differential Equations, Springer,2010
[29] J.F. Ganghoffer, I. Mladenov, Similarity and Similarity Methods, Springer, 2014.
[30] G.W. Bluman, Symmetry and integration Methods for Differential Equations, Springer, New York, 2002
[31] Sinkala, W., Leach, P.G.L., O’Hara, J.G.: An optimal system and group-invariant solutions of the Cox-Ingersoll-Ross pricing equation. Appl. Math. Comput. 201, 95– 107 (2008)
[32] Kumei, S., Bluman,G.W.: When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173 (1982)
[33] Ovsiannikov, L.V. Group Analysis of Differential Equations. Academic Press, New York (1982)
Finance, Nonlinear Dynamics 17: 387-407, 1998
[2] Fischer Black, Myron Scholes, The valuation of option contracts and a test of market efficiency, J.Finance 27 (1972) 399–417.
[3] Fischer Black, Myron Scholes, The pricing of options and corporate liabilities, J. Political Economy 81 (1973) 637–659.
[4] C. Robert Merton, Theory of rational option pricing, Bell J. Econom. Management Sci. 4 (1973) 141–183.
[5] S.M. Myeni, P.G.L. Leach, The complete symmetry group of evolution equations, preprint, School of Mathematical Sciences, University of KwaZulu-Natal, 2005.
[6] S.M. Myeni, P.G.L. Leach, Nonlocal symmetries and the complete symmetry group of 1 + 1 evolution equations, J. Nonlinear Math. Phys. 13 (2006) 377–392
[7] K. Andriopoulos, P.G.L. Leach, G.P. Flessas, Complete symmetry groups of ordinary differential equations and their integrals: Some basic considerations, J. Math. Anal. Appl. 262 (2001) 256–273.
[8] K. Andriopoulos, P.G.L. Leach, The economy of complete symmetry groups for linear higher-dimensional systems, J. Nonlinear Math. Phys. 9 (Second Suppl.) (2002) 10–23.
[9] K. Andriopoulos, P.G.L. Leach, The complete symmetry group of the generalised hyperladder problem, J. Math. Anal. Appl. 293 (2004) 633–644.
[10] [32] M.C. Nucci, P.G.L. Leach, Jacobi’s last multiplier and the complete symmetry group of the Euler–Poinsot system, J. Nonlinear Math. Phys. 9 (Second Suppl.) (2002) 110–121
[11] [5] F.E. Benth, K.H. Karlsen, A note on Merton’s portfolio selection problem for the Schwartzmean-reversion model, Stoch. Anal. Appl. 23 (2005) 687–704
[12] Olver P.J. Applications of Lie Groups to Differential Equations. New York: Academic Press, 1982.
[13] [24] G.M.Mubarakzyanov, Classification of real structures of five-dimensional Lie algebras, Izv. Vyssh. Uchebn. Zaved. Mat. 34 (1963) 99–106.
[14] Goard, J. Exact and approximate solutions for options with time-dependent stochastic volatility.Appl.Math. Model. 38, 2771–2780 (2014)
[15] Lo C F, Hui C H. Valuation of financial derivatives with time-dependent parameters: Lie-algebraicapproach. Quant Finance, 2001, 1: 73-78
[16] Lo C F, Hui C H. Pricing multi-asset financial derivatives with time-dependent parameters- Lie algebraic approach. Int J math Sci, 2002, 32: 401-410
[17] Lo C F, Hui C H. Lie algebraic approach for pricing moving barrier options with time- dependent parameters. J. Math Anal Appl, 2006, 323: 1455-1464.
[18] Leach P G L, O’ Hara J G, Sinkaka W. Symmetry-based solution of a model for a Combination of a risky investment and riskless Investment. J. Math Anal Appl, 2007, 334: 368-381.
[19] Sinkaka W, Leach P G L, O’Hara J G. An Optimal system and group-invariant solutions of the Cox-Ingersoll-Ross pricing equation. Appl Math Comput, 2001, 201: 95-107
[20] T. Zariphopoulou, Optimal investment and consumption models with nonlinear stock dynamics, Math. Methods Oper. Res. 50 (1999) 271–296.
[21] R.C. Merton, Lifetime portfolio selection under uncertainty: The continuous time case, Rev.Econom. Statist. 51(1969) 247–257.
[22] A.K. Head, LIE, a PC program for Lie analysis of differential equations, Comput. Phys. Comm. 77 (1993) 241–248.
[23] Sherring, A.K. Head, G.E. Prince, Dimsym and LIE: Symmetry determining packages,
Math. Comput. Modelling 25 (1997) 153–164.
[24] Cheviakov, A.F.: GeM software package for computation of symmetries and conservation lawsof differential equations. Comput. Phys. Commun. 176, 48–61 (2007)
[25] Cheviakov, A.F.: Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models. Comput. Phys. Commun. 220, 56–73 (2017)
[26] M.C. Nucci, Interactive REDUCE programs for calculating classical, nonclassical and
Lie–Bäcklund symmetries for differential equations, preprint, Georgia Institute of Technology, Math 062090-051, 1990.
[27] M.C. Nucci, Interactive REDUCE programs for calculating Lie point, nonclassical, Lie–Bäcklund, and approximate symmetries of differential equations: Manual and floppy disk, in: N.H. Ibragimov (Ed.), New Trends in Theoretical Development and Computational Methods, in: CRC Handbook of Lie Group Analysis of Differential Equations, vol. III, CRC Press, Boca Raton, 1996.
[28] G.W.Bluman, A.F. Cheviakov, S.C. Anco, Application of Symmetry Methods to Partial Differential Equations, Springer,2010
[29] J.F. Ganghoffer, I. Mladenov, Similarity and Similarity Methods, Springer, 2014.
[30] G.W. Bluman, Symmetry and integration Methods for Differential Equations, Springer, New York, 2002
[31] Sinkala, W., Leach, P.G.L., O’Hara, J.G.: An optimal system and group-invariant solutions of the Cox-Ingersoll-Ross pricing equation. Appl. Math. Comput. 201, 95– 107 (2008)
[32] Kumei, S., Bluman,G.W.: When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173 (1982)
[33] Ovsiannikov, L.V. Group Analysis of Differential Equations. Academic Press, New York (1982)
Published
2023-12-09
How to Cite
Ijioma, O. (2023). On The Optimization of DC-Pension Fund Investment by Lie Symmetry Analysis. GPH - International Journal of Mathematics, 6(11), 17-27. https://doi.org/10.5281/zenodo.10318502
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Copyright (c) 2023 Onwukwe Ijioma
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