Optimal portfolio for a defined-contribution pension plan under a constant elasticity of variance model with exponential utility

Abstract

Based on the Lie symmetry method, we derive the explicit optimal invest strategy for an investor who seeks to maximize the expected exponential (CARA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model. We examine the point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem. The symmetries compatible with the terminal condition enable us to transform the (2+ 1)-dimensional HJB equation into a (1+ 1)-dimensional nonlinear equation which is linearized by its infinite-parameter Lie group of point transformations. Finally, the ansatz technique based on variables separation is applied to solve the linear equation and the optimal strategy is obtained. The algorithmic procedure of the Lie symmetry analysis method adopted here is quite general compared with conjectures used in the literature.

Key words： Lie symmetry; portfolio; defined-contribution (DC) pension plan; constant elasticity of variance (CEV) model; exponential utility

References

1 Ali M R, Ma W-X. New exact solutions of Bratu Gelfand model in two dimensions using Lie symmetry analysis. Chinese J Phys, 2020, 65: 198–206

2 Bakkaloglu A,Aziz T, Fatima A, Mahomed F M, Khalique C M.Invariant approach to optimal investment-consumption problem: the constant elasticity of variance (CEV) model. Math Methods Appl Sci, 2016, 40: 1382–1395

3 Bluman G W, Kumei S. Symmetries and Differential Equations. New York: Springer, 1989

4 Bozhkov Y, Dimas S. Group classification of a generalized Black-Scholes-Merton equation. Commun Nonlinear Sci Numer Simul, 2014, 19: 2200–2211

5 Bozhkov Y, Dimas S.Enhanced group analysis of a semi linear generalization of a general bond-pricing equation. Commun Nonlinear Sci Numer Simul, 2018, 54: 210–220

6 Cheviakov A F. GeM software package for computation of symmetries and conservation laws of differential equations. Comput Phys Commun, 2007, 176: 48–61

7 Cheviakov A F. Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models. Comput Phys Commun, 2017, 220: 56–73

8 Davis E P. Pension Funds: Retirement-Income Security and Capital Markets: An International Perspective. Oxford: Oxford Univ Press, 1995

9 Gao J W. Optimal portfolios for DC pension plans under a CEV model. Insurance Math Econom, 2009, 44: 479–490

10 Gao J W. Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model. Insurance Math Econom, 2009, 45: 9–18

11 Gazizov R K, Ibragimov N H. Lie symmetry analysis of differential equations in finance. Nonlinear Dynam, 1998, 17: 387–407

12 Goard J.New solutions to the bond-pricing equation via Lie's classical method. Math Comput Model, 2000, 32: 299–313

13 Goard J. Exact and approximate solutions for options with time-dependent stochastic volatility. Appl Math Model, 2014, 38: 2771–2780

14 Henderson V. Explicit solutions to an optimal portfolio choice problem with stochastic income . J Econom Dynam Control, 2005, 29: 1237–1266

15 Kumei S, Bluman G W.When nonlinear differential equations are equivalent to linear differential equations. SIAM J Appl Math, 1982, 42: 1157–1173

16 Leach P G L,O'Hara J G,Sinkala W. Symmetry-based solution of a model for a combination of a risky investment and a riskless investment. J Math Anal Appl, 2007, 334: 368–381

17 Lo C F, Hui C H. Valuation of financial derivatives with time-dependent parameters: Lie-algebraic approach. Quant Finance, 2001, 1: 73–78

18 Lo C F, Hui C H. Pricing multi{asset financial derivatives with time-dependent parametersLie algebraic approach. Int J Math Math Sci, 2002, 32: 401–410

19 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

20 Ma W-X, Zhang Y, Tang Y N. Symbolic computation of lump solutions to a combined equation involving three types of nonlinear terms. East Asian J Appl Math, 2020, 10: 732–745

21 Naicker V, Andriopoulos K, Leach P G L. Symmetry reductions of a Hamilton-Jacobi- Bellman equation arising in financial mathematics. J Nonlinear Math Phys, 2005, 12: 268–283

22 Olver P J. Applications of Lie Groups to Differential Equations. New York: Springer- Verlag, 1986

23 Ovsiannikov L V. Group Analysis of Differential Equations. New York: Academic Press, 1982

24 Qin C Y, Tian S F, Wang X B, Zou L, Zhang T T. Lie symmetry analysis, conservation laws and analytic solutions of the time fractional Kolmogorov-Petrovskii- Piskunov equation. Chinese J Phys, 2018, 56(4): 1734–1742

25 Sethi S P, Thompson G L. Optimal Control Theory: Applications to Management Science and Economics. 2nd ed. New York: Springer, 2000

26 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, 2001, 201: 95–107

27 Sinkala W, Leach P G L, O'Hara J G. Invariance properties of a general bond-pricing equation. J Differential Equations, 2008, 244: 2820–2835

28 Sophocleous C, O'Hara J G, Leach P G L. Algebraic solution of the Stein-Stein model for stochastic volatility. Commun Nonlinear Sci Numer Simul, 2009, 16: 1752–1759

29 Sophocleous C, O'Hara J G, Leach P G L. Symmetry analysis of a model of stochastic volatility with time-dependent parameters. J Comput Appl Math, 2011, 235: 4158–4164

30 Yan Z Y.Financial rogue waves. Commun Theor Phys (Beijing), 2010, 11: 947–949

31 Yan Z Y. Vector financial rogue waves. Phys Lett A, 2011, 375(48): 4274–4279

32 Yang J Y, Ma W-X, Khalique C H. Determining lump solutions for a combined soliton equation in (2+ 1)-dimensions. Eur Phys J Plus, 2020, 135(6): Art No 494

33 Yang S J, Xu T Z. Lie symmetry analysis for a parabolic Monge-Ampere equation in the optimal investment theory. J Comput Appl Math, 2019, 346: 483–489