×
Ma, Hui-qiang

Continuous-time mean-variance portfolio selection under the CEV process. (English) Zbl 1406.91419

Abstr. Appl. Anal. 2014, Article ID 363046, 14 p. (2014).
Summary: We consider a continuous-time mean-variance portfolio selection model when stock price follows the constant elasticity of variance (CEV) process. The aim of this paper is to derive an optimal portfolio strategy and the efficient frontier. The mean-variance portfolio selection problem is formulated as a linearly constrained convex program problem. By employing the Lagrange multiplier method and stochastic optimal control theory, we obtain the optimal portfolio strategy and mean-variance efficient frontier analytically. The results show that the mean-variance efficient frontier is still a parabola in the mean-variance plane, and the optimal strategies depend not only on the total wealth but also on the stock price. Moreover, some numerical examples are given to analyze the sensitivity of the efficient frontier with respect to the elasticity parameter and to illustrate the results presented in this paper. The numerical results show that the price of risk decreases as the elasticity coefficient increases.

MSC:

91G10 Portfolio theory
90C25 Convex programming
93E20 Optimal stochastic control

Keywords:

mean-variance portfolio selection; convex program problem; stochastic optimal control
Cite Review PDF
Full Text: DOI
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] Markowitz, H., Portfolio selection, The Journal of Finance, 7, 77-91 (1952)
[2] Grauer, R. R.; Hakansson, N. H., On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: a comparison of returns and investment policies, Management Science, 39, 856-871 (1993)
[3] Hakansson, N. H., Multi-period mean-variance analysis: toward a general theory of portfolio choice, Journal of Finance, 26, 857-884 (1971)
[4] Pliska, S. R., Introduction to Mathematical Finance (1997), Blackwell, UK: Malden, Blackwell, UK
[5] Samuelson, P. A., Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51, 239-246 (1969)
[6] Li, D.; Ng, W. L., Optimal dynamic portfolio selection: multiperiod mean-variance formulation, Mathematical Finance, 10, 3, 387-406 (2000) · Zbl 0997.91027 · doi:10.1111/1467-9965.00100
[7] Zhou, X. Y.; Li, D., Continuous-time mean-variance portfolio selection: a stochastic {LQ} framework, Applied Mathematics and Optimization, 42, 1, 19-33 (2000) · Zbl 0998.91023 · doi:10.1007/s002450010003
[8] Chiu, M. C.; Li, D., Asset and liability management under a continuous-time mean-variance optimization framework, Insurance: Mathematics and Economics, 39, 3, 330-355 (2006) · Zbl 1151.91493 · doi:10.1016/j.insmatheco.2006.03.006
[9] Li, X.; Zhou, X. Y.; Lim, A. E. B., Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40, 5, 1540-1555 (2002) · Zbl 1027.91040 · doi:10.1137/S0363012900378504
[10] Lim, A. E. B.; Zhou, X. Y., Mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27, 1, 101-120 (2002) · Zbl 1082.91521 · doi:10.1287/moor.27.1.101.337
[11] Lim, A. E. B., Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29, 1, 132-161 (2004) · Zbl 1082.91050 · doi:10.1287/moor.1030.0065
[12] Xie, S.; Li, Z.; Wang, S., Continuous-time portfolio selection with liability: mean-variance model and stochastic LQ approach, Insurance: Mathematics & Economics, 42, 3, 943-953 (2008) · Zbl 1141.91474 · doi:10.1016/j.insmatheco.2007.10.014
[13] Xie, S. X., Continuous-time mean-variance portfolio selection with liability and regime switching, Insurance: Mathematics & Economics, 45, 1, 148-155 (2009) · Zbl 1231.91417 · doi:10.1016/j.insmatheco.2009.05.005
[14] Zhou, X. Y.; Yin, G., Markowitz’s mean-variance portfolio selection with regime switching: a continuous-time model, SIAM Journal on Control and Optimization, 42, 4, 1466-1482 (2003) · Zbl 1175.91169 · doi:10.1137/S0363012902405583
[15] Beckers, S., The constant elasticity of variance model and its implications for option pricing, The Journal of Finance, 35, 661-673 (1980)
[16] Campbell, J. Y., Stock returns and the term structure, Journal of Financial Economics, 18, 2, 373-399 (1987) · doi:10.1016/0304-405X(87)90045-6
[17] Yuen, K. C.; Yang, H.; Chu, K. L., Estimation in the constant elasticity of variance model, British Actuarial Journal, 7, 275-292 (2001)
[18] Cox, J. C., Notes on option pricing I: constant elasticity of diffusions, Working Paper (1975), Stanford, Calif, USA: Stanford University, Stanford, Calif, USA
[19] Davydov, D.; Linetsky, V., Pricing and hedging path-dependent options under the CEV process, Management Science, 47, 7, 949-965 (2001) · Zbl 1232.91659 · doi:10.1287/mnsc.47.7.949.9804
[20] Xiao, J.; Hong, Z.; Qin, C., The constant elasticity of variance ({CEV}) model and the Legendre transform-dual solution for annuity contracts, Insurance: Mathematics and Economics, 40, 2, 302-310 (2007) · Zbl 1141.91473 · doi:10.1016/j.insmatheco.2006.04.007
[21] Gao, J., Optimal portfolios for DC pension plans under a CEV model, Insurance: Mathematics & Economics, 44, 3, 479-490 (2009) · Zbl 1162.91411 · doi:10.1016/j.insmatheco.2009.01.005
[22] Gu, M.; Yang, Y.; Li, S.; Zhang, J., Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance: Mathematics and Economics, 46, 3, 580-587 (2010) · Zbl 1231.91193 · doi:10.1016/j.insmatheco.2010.03.001
[23] Cox, J. C.; Ross, S. A., The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3, 1-2, 145-166 (1976) · doi:10.1016/0304-405X(76)90023-4
[24] Bäuerle, N., Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62, 1, 159-165 (2005) · Zbl 1101.93081 · doi:10.1007/s00186-005-0446-1
[25] Zeng, Y.; Li, Z.; Liu, J., Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers, Journal of Industrial and Management Optimization, 6, 3, 483-496 (2010) · Zbl 1269.90085 · doi:10.3934/jimo.2010.6.483
[26] Luenberger, D. G., Optimization by Vector Space Methods (1968), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0184.44502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.