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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

Shuang Li 1, , Chuong Luong 1, , Francisca Angkola 1, and  Yonghong Wu 1,

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.
Keywords: Nash equilibrium theory, Hamilton-Jacobi-Bellman equation., mean-variance utility, stochastic volatility, Optimal portfolio.
Mathematics Subject Classification: Primary: 91B16, 91B70, 91G10; Secondary: 91G8.
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.

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