Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach

Caibin Zhang; Zhibin Liang; Kam Chuen Yuen

doi:10.3934/jimo.2020156

Article Contents

2022, Volume 18, Issue 1: 341-366. Doi: 10.3934/jimo.2020156

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Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach

1.
School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China
2.
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
3.
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong

^* Corresponding author: Zhibin Liang
^* Corresponding author: Zhibin Liang

Received: January 31, 2019

Revised: May 31, 2020

Early access: November 2020

Published: January 2022

This research is supported by National Natural Science Foundation of China (Grant No.12071224 and Grant No.11771079), the Research Grants Council of the Hong Kong Special Administrative Region (Project No. HKU17306220), the Philosophy and Social Science Foundation for Colleges and Universities in Jiangsu Province (Grant No.2020SJA0261), and the MOE Project of Humanities and Social Sciences (Grant No.19YJCZH083)

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Abstract

In this paper, an optimal portfolio selection problem with mean-variance utility is considered for a financial market consisting of one risk-free asset and two risky assets, whose price processes are modulated by jump-diffusion model, the two jump number processes are correlated through a common shock, and the Brownian motions are supposed to be dependent. Moreover, it is assumed that not only the risk aversion coefficient but also the market parameters such as the appreciation and volatility rates as well as the jump amplitude depend on a Markov chain with finite states. In addition, short selling is supposed to be prohibited. Using the technique of stochastic control theory and the corresponding extended Hamilton-Jacobi-Bellman equation, the explicit expressions of the optimal strategies and value function are obtained within a game theoretic framework, and the existence and uniqueness of the solutions are proved as well. In the end, some numerical examples are presented to show the impact of the parameters on the optimal strategies, and some further discussions on the case of $ n\geq 3 $ risky assets are given to demonstrate the important effect of the correlation coefficient of the Brownian motions on the optimal results.

Keywords:

Mean-variance,
Portfolio selection,
Markov regime-switching,
Jump-diffusion,
Common shock,
No short selling,
Extended Hamilton-Jacobi-Bellman equation.

Mathematics Subject Classification: Primary: 62P05, 91G10; Secondary: 93E20.

Citation:

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Figure 1. The comparison of value functions with and without constraint

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Figure 2. The comparison of optimal strategies between regime 1 and regime 2

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Figure 3. The effect of parameter $ \lambda_{01} $ on optimal strategies with regime 1

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Figure 4. The effect of jump's parameters on optimal strategies with regime 1

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Figure 5. The effect of risk-free rate on optimal strategies with regime 1

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Table 1. The value of parameters

parameters	$ T $	$ x $	$ \gamma_{i} $	$ r_0 $	$ r_{1i} $	$ r_{2i} $	$ \rho $	$ \sigma_{1i} $	$ \sigma_{2i} $	$ \mu_{1i} $	$ \mu_{2i} $	$ \beta_{1i} $	$ \beta_{2i} $	$ \lambda_{1i} $	$ \lambda_{2i} $	$ \lambda_{0i} $
$ e_1 $(bullish)	5	2	0.5	0.3	0.60	0.30	0.2	0.3	0.8	0.06	0.08	0.02	0.055	2	0.5	1
$ e_2 $(bearish)	5	2	1	0.3	0.25	0.07	0.1	0.4	0.5	0.03	0.04	0.03	0.032	1	0.3	0.6

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Table 2. The parameters-set

parameters	$ T $	$ x $	$ \gamma_{i} $	$ r_0 $	$ r_{1i} $	$ r_{2i} $	$ \rho $	$ \sigma_{1i} $	$ \sigma_{2i} $	$ \mu_{1i} $	$ \mu_{2i} $	$ \beta_{1i} $	$ \beta_{2i} $	$ \lambda_{1i} $	$ \lambda_{2i} $	$ \lambda_{0i} $
$ e_1 $(bullish)	5	2	0.5	0.3	0.60	0.65	0.2	0.7	0.8	0.06	0.08	0.04	0.045	2	0.5	1
$ e_2 $(bearish)	5	2	1	0.3	0.35	0.37	0.1	0.4	0.5	0.03	0.04	0.03	0.032	1	0.3	0.6

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