Scientific Research

An Academic Publisher

Optimal Investment Problem with Multiple Risky Assets under the Constant Elasticity of Variance (CEV) Model ()

This paper studies the optimal investment problem for utility maximization with multiple risky assets under the constant elasticity of variance (CEV) model. By applying stochastic optimal control approach and variable change technique, we derive explicit optimal strategy for an investor with logarithmic utility function. Finally, we analyze the properties of the optimal strategy and present a numerical example.

Share and Cite:

H. Zhao, X. Rong, W. Ma and B. Gao, "Optimal Investment Problem with Multiple Risky Assets under the Constant Elasticity of Variance (CEV) Model,"

*Modern Economy*, Vol. 3 No. 6, 2012, pp. 718-725. doi: 10.4236/me.2012.36092.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | R. C. Merton, “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case,” The Review of Economics and Statistics, Vol. 51, No. 3, 1969, pp. 247-257. doi:10.2307/1926560 |

[2] | R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of economic theory, Vol. 3, No. 4, 1971, pp. 373-413. doi:10.1016/0022-0531(71)90038-X |

[3] | S. Browne, “Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin,” Mathematics of Operations Research, Vol. 20, No. 4, 1995, pp. 937-958. doi:10.1287/moor.20.4.937 |

[4] | H. Yang and L. Zhang, “Optimal Investment for Insurer with Jump-Diffusion Risk Process,” Insurance: Mathematics and Economics, Vol. 37, No. 3, 2005, pp. 615-634. doi:10.1016/j.insmatheco.2005.06.009 |

[5] | S. Pliska, “A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios,” Mathematics of Operations Research, Vol. 11, No. 2, 1986, pp. 371-382. doi:10.1287/moor.11.2.371 |

[6] | I. Karatzas, J. P. Lehoczky and S. E. Shreve, “Optimal Portfolio and Consumption Decisions for a ‘Small Investor’ on a Finite Horizon,” SIAM Journal on Control and Optimization, Vol. 25, No. 6, 1987, pp. 1557-1586. doi:10.1137/0325086 |

[7] | J. C. Cox and C. F. Huang, “Optimal Consumption and Portfolio Policies when Asset Prices Follow a Diffusion Process,” Journal of Economic Theory, Vol. 49, No. 1, 1989, pp. 33-83. doi:10.1016/0022-0531(89)90067-7 |

[8] | I. Karatzas, “Optimization Problems in the Theory of Continuous Trading,” SIAM Journal on Control and Optimization, Vol. 27, No. 6, 1989, pp. 1221-1259. doi:10.1137/0327063 |

[9] | I. Karatzas and S. E. Shreve, “Brownian Motion and Stochastic Calculus,” Springer Verlag, New York, 1991. doi:10.1007/978-1-4612-0949-2 |

[10] | I. Karatzas, J. P. Lehoczky, S. E. Shreve and G. L. Xu, “Martingale and Duality Methods for Utility Maximization in an Incomplete Market,” SIAM Journal on Control and Optimization, Vol. 29, No. 3, 1991, pp. 702-730. doi:10.1137/0329039 |

[11] | A. Zhang, “A Secret to Create a Complete Market from an Incomplete Market,” Applied Mathematics and Computation, Vol. 191, No. 1, 2007, pp. 253-262. doi:10.1016/j.amc.2007.02.086 |

[12] | Z. Wang, J. Xia and L. Zhang, “Optimal Investment for an Insurer: The Martingale Approach,” Insurance: Mathematics and Economics, Vol. 40, No. 2, 2007, pp. 322-334. doi:10.1016/j.insmatheco.2006.05.003 |

[13] | Q. Zhou, “Optimal Investment for an Insurer in the Lévy Market: The Martingale Approach,” Statistics & Probability Letters, Vol. 79, No. 14, 2009, pp. 1602-1607. doi:10.1016/j.spl.2009.03.027 |

[14] | D. G. Hobson and L. C. G. Rogers, “Complete Models with Stochastic Volatility,” Mathematical Finance, Vol. 8, No. 1, 1998, pp. 27-48. doi:10.1111/1467-9965.00043 |

[15] | J. C. Cox and S. A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 145-166. doi:10.1016/0304-405X(76)90023-4 |

[16] | J. C. Cox, “The Constant Elasticity of Variance Option Pricing Model,” The Journal of Portfolio Management, Vol. 22, 1996, pp. 15-17. |

[17] | C. F. Lo, P. H. Yuen and C. H. Hui, “Constant Elasticity of Variance Option Pricing Model with Time-Dependent Parameters,” International Journal of Theoretical and Applied Finance, Vol. 3, No. 4, 2000, pp. 661-674. doi:10.1142/S0219024900000814 |

[18] | D. Davydov and V. Linetsky, “Pricing and Hedging Path- Dependent Options under the CEV Process,” Management Science, Vol. 47, No. 7, 2001, pp. 949-965. doi:10.1287/mnsc.47.7.949.9804 |

[19] | Y. L. Hsu, T. I. Lin and C. F. Lee, “Constant Elasticity of Variance (CEV) Option Pricing Model: Integration and Detailed Derivation,” Mathematics and Computers in Simulation, Vol. 79, No. 1, 2008, pp. 60-71. doi:10.1016/j.matcom.2007.09.012 |

[20] | J. Xiao, H. Zhai and C. Qin, “The Constant Elasticity of Variance (CEV) Model and the Legendre Transform-dual Solution for Annuity Contracts,” Insurance: Mathematics and Economics, Vol. 40, No. 2, 2007, 302-310. doi:10.1016/j.insmatheco.2006.04.007 |

[21] | J. Gao, “Optimal Portfolios for DC Pension Plans under a CEV Model,” Insurance: Mathematics and Economics, Vol. 44, No. 3, 2009, 479-490. doi:10.1016/j.insmatheco.2009.01.005 |

[22] | J. Gao, “Optimal Investment Strategy for Annuity Contracts under the Constant Elasticity of Variance (CEV) Model,” Insurance: Mathematics and Economics, Vol. 45, No. 1, 2009, 9-18. doi:10.1016/j.insmatheco.2009.02.006 |

[23] | M. Gu, Y. Yang, S. Li and J. Zhang, “Constant Elasticity of Variance Model for Proportional Reinsurance and Investment Strategies,” Insurance: Mathematics and Economics, Vol. 46, No. 3, 2010, 580-587. doi:10.1016/j.insmatheco.2010.03.001 |

[24] | H. Zhao and X. Rong, “Portfolio Selection Problem with Multiple Risky Assets under the Constant Elasticity of Variance Model,” Insurance: Mathematics and Economics, Vol. 50, No. 1, 2012, 179-190. doi:10.1016/j.insmatheco.2011.10.013 |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.