• Previous Article
    Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays
  • JIMO Home
  • This Issue
  • Next Article
    On linear convergence of projected gradient method for a class of affine rank minimization problems
October  2016, 12(4): 1521-1533. doi: 10.3934/jimo.2016.12.1521

Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion

1. 

Department of Mathematics and Statistics, Curtin University, Kent Street, Bentley, Perth, Western Australia 6102, Australia, Australia, Australia, Australia

Received  December 2014 Revised  August 2015 Published  January 2016

This paper studies the portfolio optimization of mean-variance utility with state-dependent risk aversion, where the stock asset is driven by a stochastic process. The sub-game perfect Nash equilibrium strategies and the extended Hamilton-Jacobi-Bellman equations have been used to derive the system of non-linear partial differential equations. From the economic point of view, we demonstrate the numerical evaluation of the suggested solution for a special case where the risk aversion rate is proportional to the wealth value. Our results show that the asset driven by the stochastic volatility process is more general and reasonable than the process with a constant volatility.
Citation: Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521
References:
[1]

I. Bajeux-Besnainou and R. Portait, Dynamic asset allocation in a mean-variance framework, Management Science, 44 (1998), 79-95. doi: 10.1287/mnsc.44.11.S79.

[2]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016.

[3]

T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244. doi: 10.1111/j.0960-1627.2005.00218.x.

[4]

T. Börk, A. Murgoci and X. Y. Zhou, Mean-variance Portfolio Optimization with State-Dependent Risk Aversion, Mathematical Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.

[5]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, preprint.

[6]

J. C. Cox, J. E. Innersole and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242.

[7]

M. Dai, Z. Xu and X. Y. Zhou, Continuous-Time Markowitz's model with transaction costs, SIAM Journal on Financial Mathematics, 1 (2010), 96-125. doi: 10.1137/080742889.

[8]

J. B. Detemple and F. Zapatero, Asset prices in an exchange economy with habit formation, Econometrica, 59 (1991), 1633-1657. doi: 10.2307/2938283.

[9]

S. M. Goldman, Consistent plans, The Review of Economic Studies, 47 (1980), 533-537. doi: 10.2307/2297304.

[10]

P. Krusell and A. A. Smith, Consumption-savings decisions with quasi-geometric discounting, Econometrica, 71 (2003), 366-375. doi: 10.1111/1468-0262.00400.

[11]

F. E. Kydland and E. C. Prescott, Rules rather than discretion: The inconsistency of optimal plans, The Journal of Political Economy, 85 (1997), 473-492.

[12]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod Mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100.

[13]

A. E. B. Lim and X. Y. Zhou, Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120. doi: 10.1287/moor.27.1.101.337.

[14]

A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161. doi: 10.1287/moor.1030.0065.

[15]

H. Markowitz, Portfolio selection, The journal of finance, 7 (1952), 77-98.

[16]

R. A. Pollak, Consistent planning, The Review of Economic Studies, 35 (1968), 201-208. doi: 10.2307/2296548.

[17]

B. Peleg and M. E. Yaar, On the existence of a consistent course of action when tastes are changing, The Review of Economic Studies, 40 (1973), 391-401. doi: 10.2307/2296458.

[18]

H. R. Richardson, A minimum variance result in continuous trading portfolio optimization, Management Science, 35 (1989), 1045-1055. doi: 10.1287/mnsc.35.9.1045.

[19]

J. Xia, Mean-variance portfolio choice: Quadratic partial hedging, Mathematical Finance, 15 (2005), 533-538. doi: 10.1111/j.1467-9965.2005.00231.x.

[20]

J. E. Zhang, H. Zhao and E. C. Chang, Equilibrium asset and option pricing under jump diffusion, Mathematical Finance, 22 (2012), 538-568. doi: 10.1111/j.1467-9965.2010.00468.x.

show all references

References:
[1]

I. Bajeux-Besnainou and R. Portait, Dynamic asset allocation in a mean-variance framework, Management Science, 44 (1998), 79-95. doi: 10.1287/mnsc.44.11.S79.

[2]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016.

[3]

T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244. doi: 10.1111/j.0960-1627.2005.00218.x.

[4]

T. Börk, A. Murgoci and X. Y. Zhou, Mean-variance Portfolio Optimization with State-Dependent Risk Aversion, Mathematical Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.

[5]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, preprint.

[6]

J. C. Cox, J. E. Innersole and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242.

[7]

M. Dai, Z. Xu and X. Y. Zhou, Continuous-Time Markowitz's model with transaction costs, SIAM Journal on Financial Mathematics, 1 (2010), 96-125. doi: 10.1137/080742889.

[8]

J. B. Detemple and F. Zapatero, Asset prices in an exchange economy with habit formation, Econometrica, 59 (1991), 1633-1657. doi: 10.2307/2938283.

[9]

S. M. Goldman, Consistent plans, The Review of Economic Studies, 47 (1980), 533-537. doi: 10.2307/2297304.

[10]

P. Krusell and A. A. Smith, Consumption-savings decisions with quasi-geometric discounting, Econometrica, 71 (2003), 366-375. doi: 10.1111/1468-0262.00400.

[11]

F. E. Kydland and E. C. Prescott, Rules rather than discretion: The inconsistency of optimal plans, The Journal of Political Economy, 85 (1997), 473-492.

[12]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod Mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100.

[13]

A. E. B. Lim and X. Y. Zhou, Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120. doi: 10.1287/moor.27.1.101.337.

[14]

A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161. doi: 10.1287/moor.1030.0065.

[15]

H. Markowitz, Portfolio selection, The journal of finance, 7 (1952), 77-98.

[16]

R. A. Pollak, Consistent planning, The Review of Economic Studies, 35 (1968), 201-208. doi: 10.2307/2296548.

[17]

B. Peleg and M. E. Yaar, On the existence of a consistent course of action when tastes are changing, The Review of Economic Studies, 40 (1973), 391-401. doi: 10.2307/2296458.

[18]

H. R. Richardson, A minimum variance result in continuous trading portfolio optimization, Management Science, 35 (1989), 1045-1055. doi: 10.1287/mnsc.35.9.1045.

[19]

J. Xia, Mean-variance portfolio choice: Quadratic partial hedging, Mathematical Finance, 15 (2005), 533-538. doi: 10.1111/j.1467-9965.2005.00231.x.

[20]

J. E. Zhang, H. Zhao and E. C. Chang, Equilibrium asset and option pricing under jump diffusion, Mathematical Finance, 22 (2012), 538-568. doi: 10.1111/j.1467-9965.2010.00468.x.

[1]

Xuhui Wang, Lei Hu. A new method to solve the Hamilton-Jacobi-Bellman equation for a stochastic portfolio optimization model with boundary memory. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021137

[2]

Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369

[3]

Bian-Xia Yang, Shanshan Gu, Guowei Dai. Existence and multiplicity for Hamilton-Jacobi-Bellman equation. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3767-3793. doi: 10.3934/cpaa.2021130

[4]

Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial and Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161

[5]

Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, 2021, 29 (5) : 3429-3447. doi: 10.3934/era.2021046

[6]

Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251

[7]

Hao Chang, Jiaao Li, Hui Zhao. Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1393-1423. doi: 10.3934/jimo.2021025

[8]

Yan Zeng, Zhongfei Li, Jingjun Liu. Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers. Journal of Industrial and Management Optimization, 2010, 6 (3) : 483-496. doi: 10.3934/jimo.2010.6.483

[9]

Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1887-1912. doi: 10.3934/jimo.2020051

[10]

Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial and Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133

[11]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933

[12]

Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022048

[13]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295

[14]

Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial and Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166

[15]

Zhen Wang, Sanyang Liu. Multi-period mean-variance portfolio selection with fixed and proportional transaction costs. Journal of Industrial and Management Optimization, 2013, 9 (3) : 643-657. doi: 10.3934/jimo.2013.9.643

[16]

Ning Zhang. A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems. Journal of Industrial and Management Optimization, 2020, 16 (2) : 991-1008. doi: 10.3934/jimo.2018189

[17]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

[18]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[19]

Liming Zhang, Rongming Wang, Jiaqin Wei. Open-loop equilibrium mean-variance reinsurance, new business and investment strategies with constraints. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021140

[20]

Xianping Wu, Xun Li, Zhongfei Li. A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints. Journal of Industrial and Management Optimization, 2018, 14 (1) : 249-265. doi: 10.3934/jimo.2017045

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (176)
  • HTML views (0)
  • Cited by (2)

[Back to Top]