Abstract
In this article, explicit representation of solution for the Hamilton–Jacobi–Bellman (HJB) equation associated with the portfolio optimization problem for an investor who seeks to maximize the expected power (CRRA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model is derived based on the application of the Lie symmetry method to the partial differential equation and its associated terminal condition. Compared with the ingenious ansatz techniques used before, here we present a group theoretical analysis of the terminal value problem for the solution following the algorithmic procedure of the Lie symmetry analysis. It shows that the interesting properties of the group structures of the original HJB equation and its successive similarity reduced equations lead to an elegant resolution of the problem. Moreover, we identify the meaningful range of risk aversion coefficient which is ignored in the previous work. At last, the properties and sensitivity analysis of the derived optimal strategy are demonstrated by numerical simulations and several figures. The method used here is quite general and can be applied to other equations obtained in financial mathematics.
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Acknowledgements
This work is supported by the 13th Five Year National Key Research and Development Program of China with Grant No.2016YFC0401406, Beijing Natural Science Foundation with Grant No. Z200001 and the Fundamental Research Funds of the Central Universities with Grant Nos.2019MS050, 2020MS043.
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Yong, X., Sun, X. & Gao, J. Symmetry-based optimal portfolio for a DC pension plan under a CEV model with power utility. Nonlinear Dyn 103, 1775–1783 (2021). https://doi.org/10.1007/s11071-021-06203-x
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DOI: https://doi.org/10.1007/s11071-021-06203-x