Asymptotic solution for the optimal consumption, life insurance, and investment problem under the 4/2 stochastic volatility model with habit formation

Abstract

This paper explores the optimal consumption, life insurance, and investment strategies of an individual under the influence of habit formation. We assume that the individual can invest in a risk-free asset, a stock, and an index bond in the financial market, where the stock price follows the 4/2 stochastic volatility model. The primary aim of this paper is to maximize the expected utility of consumption, total bequests, and terminal wealth before retirement or death; the utility of consumption is derived from actual consumption exceeding the established habitual consumption level. By applying the dynamic programming method, we derive the Hamilton-Jacobi-Bellman (HJB) equation that the value function satisfies, obtain the asymptotic solutions for the optimal consumption, life insurance, and investment strategies using the asymptotic expansion method, and prove the corresponding verification theorem. Furthermore, we provide numerical examples to analyze the influence of consumption habit patterns and model parameters on the individual’s optimal strategies.

Keywords

Habit formation / 4/2 stochastic volatility model / Hamilton-Jacobi-Bellman equation / Asymptotic expansion method / Optimal strategies

Acknowledgements

We would like to thank the two anonymous referees for their constructive and helpful comments. This work was supported by the National Key R&D Program of China (Grant No. 2023YFA1009204).

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