Optimal investment strategy under the CEV model with stochastic interest rate. (English) Zbl 07346351
Summary: Interest rate is an important macrofactor that affects asset prices in the financial market. As the interest rate in the real market has the property of fluctuation, it might lead to a great bias in asset allocation if we only view the interest rate as a constant in portfolio management. In this paper, we mainly study an optimal investment strategy problem by employing a constant elasticity of variance (CEV) process and stochastic interest rate. The assets of investment for individuals are supposed to be composed of one risk-free asset and one risky asset. The interest rate for risk-free asset is assumed to follow the Cox-Ingersoll-Ross (CIR) process, and the price of risky asset follows the CEV process. The objective is to maximize the expected utility of terminal wealth. By applying the dual method, Legendre transformation, and asymptotic expansion approach, we successfully obtain an asymptotic solution for the optimal investment strategy under constant absolute risk aversion (CARA) utility function. In the end, some numerical examples are provided to support our theoretical results and to illustrate the effect of stochastic interest rates and some other model parameters on the optimal investment strategy.
References:
| [1] | Merton, R. C., Lifetime portfolio selection under uncertainty: the continuous-time case, The Review of Economics and Statistics, 51, 3, 247-257 (1969) · doi:10.2307/1926560 |
| [2] | Merton, R. C., Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3, 4, 373-413 (1971) · Zbl 1011.91502 · doi:10.1016/0022-0531(71)90038-x |
| [3] | Fleming, W. H.; Zariphopoulou, T., An optimal investment/consumption model with borrowing, Mathematics of Operations Research, 16, 4, 802-822 (1991) · Zbl 0744.90004 · doi:10.1287/moor.16.4.802 |
| [4] | Vila, J.-L.; Zariphopoulou, T., Optimal consumption and portfolio choice with borrowing constraints, Journal of Economic Theory, 77, 2, 402-431 (1997) · Zbl 0897.90078 · doi:10.1006/jeth.1997.2285 |
| [5] | Munk, C.; Sørensen, C., Optimal consumption and investment strategies with stochastic interest rates, Journal of Banking & Finance, 28, 8, 1987-2013 (2004) · doi:10.1016/j.jbankfin.2003.07.002 |
| [6] | Fleming, W. H.; Pang, T., An application of stochastic control theory to financial economics, SIAM Journal on Control and Optimization, 43, 2, 502-531 (2004) · Zbl 1101.93085 · doi:10.1137/s0363012902419060 |
| [7] | Yao, R.; Zhang, H. H., Optimal consumption and portfolio choices with risky housing and borrowing constraints, Review of Financial Studies, 18, 1, 197-239 (2004) · doi:10.1093/rfs/hhh007 |
| [8] | Dumas, B.; Luciano, E., An exact solution to a dynamic portfolio choice problem under transactions costs, The Journal of Finance, 46, 2, 577-595 (1991) · doi:10.1111/j.1540-6261.1991.tb02675.x |
| [9] | Shreve, S. E.; Soner, H. M., Optimal investment and consumption with transaction costs, The Annals of Applied Probability, 4, 3, 609-692 (1994) · Zbl 0813.60051 · doi:10.1214/aoap/1177004966 |
| [10] | Dai, M.; Jiang, L.; Li, P.; Yi, F., Finite horizon optimal investment and consumption with transaction costs, SIAM Journal on Control and Optimization, 48, 2, 1134-1154 (2009) · Zbl 1189.35166 · doi:10.1137/070703685 |
| [11] | Zhang, C. B.; Rong, X. M., Optimal investment strategies for DC pension with stochastic salary under affine interest rate model, Discrete Dynamics in Nature and Society, 10 (2013) · Zbl 1264.91120 · doi:10.1155/2013/297875 |
| [12] | Zhang, C. B., Mean-Variance portfolio selection for defined-contribution pension funds with stochastic salary, Discrete The Scientific World Journal, 10 (2014) · doi:10.1155/2014/826125 |
| [13] | 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 |
| [14] | Hsu, Y. L.; Lin, T. I.; Lee, C. F., Constant elasticity of variance (cev) option pricing model: integration and detailed derivation, Mathematics and Computers in Simulation, 79, 1, 60-71 (2008) · Zbl 1144.91325 · doi:10.1016/j.matcom.2007.09.012 |
| [15] | Chen, R. R.; Lee, C. F., A constant elasticity of variance (CEV) family of stock price distributions in option pricing, review, and integration, Handbook of Quantitative Finance and Risk Management, 10 (2010) |
| [16] | Chen, R.-R.; Lee, C.-F.; Lee, H.-H., Empirical performance of the constant elasticity variance option pricing model, Review of Pacific Basin Financial Markets and Policies, 12, 2, 177-217 (2009) · doi:10.1142/s0219091509001605 |
| [17] | Nicholls, D. P.; Sward, A., A discontinuous galerkin method for pricing american options under the constant elasticity of variance model, Communications in Computational Physics, 17, 3, 761-778 (2015) · Zbl 1375.91243 · doi:10.4208/cicp.190513.131114a |
| [18] | Gao, J., Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance: Mathematics and Economics, 45, 1, 9-18 (2009) · Zbl 1231.91402 · doi:10.1016/j.insmatheco.2009.02.006 |
| [19] | Gao, J., An extended CEV model and the Legendre transform-dual-asymptotic solutions for annuity contracts, Insurance: Mathematics and Economics, 46, 3, 511-530 (2010) · Zbl 1231.91432 · doi:10.1016/j.insmatheco.2010.01.009 |
| [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] | Lin, X.; Li, Y., Optimal reinsurance and investment for a jump diffusion risk process under the CEV model, North American Actuarial Journal, 15, 3, 417-431 (2011) · Zbl 1291.91121 · doi:10.1080/10920277.2011.10597628 |
| [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] | Gu, A.; Guo, X.; Li, Z.; Zeng, Y., Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance: Mathematics and Economics, 51, 3, 674-684 (2012) · Zbl 1285.91057 · doi:10.1016/j.insmatheco.2012.09.003 |
| [24] | Zhao, H.; Rong, X., Portfolio selection problem with multiple risky assets under the constant elasticity of variance model, Insurance: Mathematics and Economics, 50, 1, 179-190 (2012) · Zbl 1235.91159 · doi:10.1016/j.insmatheco.2011.10.013 |
| [25] | Jung, E. J.; Kim, J. H., Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance: Mathematics and Economics, 51, 1, 667-673 (2012) · Zbl 1285.91119 · doi:10.1016/j.insmatheco.2012.09.009 |
| [26] | Jonsson, M.; Sircar, R., Optimal investment problems and volatility homogenization approximations, NATO Science Series II, 51 (2002) · Zbl 1104.91302 |
| [27] | Takahashi, A., An asymptotic expansion approach to pricing contingent claims, Asia-Pacific Financial Markets, 6, 1, 115-151 (1999) · Zbl 1153.91568 · doi:10.1023/a:1010080610650 |
| [28] | Widdicks, M.; Duck, P.; Andricopoulos, A.; Newton, D. P., The Black Scholes equation revisited: asymptotic expansions and singular perturbations, Mathematical Finance, 15, 3, 373-391 (2005) · Zbl 1124.91342 · doi:10.1111/j.0960-1627.2005.00224.x |
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