Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria

Hao Chang; Jiaao Li; Hui Zhao

doi:10.3934/jimo.2021025

Article Contents

2022, Volume 18, Issue 2: 1393-1423. Doi: 10.3934/jimo.2021025

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Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria

Hao Chang^1, ,,
Jiaao Li^1, and
Hui Zhao^2, ,

1.
School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
2.
School of Mathematical Sciences, Tianjin University, Tianjin 300072, China

^* Corresponding authors: Hao Chang and Hui Zhao
^* Corresponding authors: Hao Chang and Hui Zhao

Received: March 2020

Revised: December 2020

Early access: February 2021

Published: March 2022

This research is supported by the National Natural Science Foundation of China (Nos.71671122 and 11771329)

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Abstract

This paper studies a robust optimal investment problem under the mean-variance criterion for a defined contribution (DC) pension plan with an ambiguity-averse member (AAM), who worries about model misspecification and aims to find robust optimal strategy. The member has access to a risk-free asset (i.e., cash or bank account) and a risky asset (i.e., the stock) in a financial market. In order to get closer to the actual environment, we assume that both the income level and stock price are driven by Heston's stochastic volatility model. A continuous-time mean-variance model with ambiguity aversion for a DC pension plan is established. By using the Lagrangian multiplier method and stochastic optimal control theory, the closed-form expressions for robust efficient strategy and efficient frontier are derived. In addition, some special cases are derived in detail. Finally, a numerical example is presented to illustrate the effects of model parameters on the robust efficient strategy and the efficient frontier, and some economic implications have been revealed.

Keywords:

Mathematics Subject Classification: Primary: 60H30, 91B28; Secondary: 93E20.

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Figure 1. The effects of volatility parameters $ \alpha $, $ \sigma _0 $, $ \lambda _S $ and $ \rho $ on $ \pi ^{\ast }\left( t \right) $

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Figure 2. The effects of income parameters $ \mu $, $ \sigma _1 $ and $ k $ on $ \pi ^ {\ast }\left( t \right) $

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Figure 3. The effect of ambiguity-aversion coefficient $ \beta $ on $ \pi ^ {\ast }\left( t \right) $

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Figure 4. The effects of volatility parameters $ \alpha $ and $ \sigma _0 $ on $ \sigma \left( {X\left( T \right)} \right) $

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Figure 5. The effects of income parameters $ \mu $ and $ k $ on $ \sigma \left( {X\left( T \right)} \right) $

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Figure 6. The effect of ambiguity-aversion coefficient $ \beta $ on $ \sigma \left( {X\left( T \right)} \right) $

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Figure 7. Comparisons of the efficient strategies and the efficient frontiers

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Figure 8. When $ \rho = -0.65 $ and $ \lambda _S = 1.5 $, we have $ \Delta>0 $; when $ \rho = -0.87 $ and $ \lambda _S = 12.56 $, we get $ \Delta = 0 $; when $ \rho = -0.9 $ and $ \lambda _S = 10 $, we have $ \Delta<0 $. The effects of different symbols of $ \Delta $ on $ \pi ^ {\ast }(t) $ and $ \sigma (X(T)) $

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