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Optimal portfolios for DC pension plans under a CEV model. (English) Zbl 1162.91411
Summary: This paper studies the portfolio optimization problem for an investor who seeks to maximize the expected utility of the terminal wealth in a DC pension plan. We focus on a constant elasticity of variance (CEV) model to describe the stock price dynamics, which is an extension of geometric Brownian motion. By applying stochastic optimal control, power transform and variable change technique, we derive the explicit solutions for the CRRA and CARA utility functions, respectively. Each solution consists of a moving Merton strategy and a correction factor. The moving Merton strategy is similar to the result of {\it P. Devolder, M. Bosch Princep} and {\it I. Dominguez Fabian} [Insur. Math. Econ. 33, No. 2, 227--238 (2003; Zbl 1103.91346)], whereas it has an updated instantaneous volatility at the current time. The correction factor denotes a supplement term to hedge the volatility risk. In order to have a better understanding of the impact of the correction factor on the optimal strategy, we analyze the property of the correction factor. Finally, we present a numerical simulation to illustrate the properties and sensitivities of the correction factor and the optimal strategy.

##### MSC:
 91B30 Risk theory, insurance 91B28 Finance etc. (MSC2000) 93E99 Stochastic systems and stochastic control
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##### References:
 [1] Barndorff-Nielsen, O.; Shephard, N.: Non-Gaussian Ornstein--Uhlenbeck-based models and some of their uses in financial economics (with discussion), Journal of the royal statistical society. Series B 63, 167-241 (2001) · Zbl 0983.60028 · doi:10.1111/1467-9868.00282 [2] Bates, D.: Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options, Review of financial studies 9, 69-107 (1996) [3] Beckers, S.: The constant elasticity of variance model and its implications for option pricing, Journal of finance 35, 661-673 (1980) [4] Black, F.; Scholes, M.: The pricing of options and corporate liabilities, Journal of political economy 81, 637-654 (1973) · Zbl 1092.91524 [5] Blomvall, J.; Lindberg, P. O.: Back-testing the performance of an actively managed option portfolio at the swedish stock market, 1990--1999, Journal of economic dynamics control 27, 1099-1112 (2003) · Zbl 1178.91174 [6] Boulier, J. F.; Huang, S.; Taillard, G.: Optimal management under stochastic interest rates: the case of a protected defined contribution pension fund, Insurance: mathematics economics 28, 173-189 (2001) · Zbl 0976.91034 · doi:10.1016/S0167-6687(00)00073-1 [7] Cox, J. C.; Huang, C. F.: Optimal consumption and portfolio policies when asset prices follow a diffusion process, Journal of economic theory 49, 33-83 (1989) · Zbl 0678.90011 · doi:10.1016/0022-0531(89)90067-7 [8] Cox, J. C.; Ross, S. A.: The valuation of options for alternative stochastic processes, Journal of financial economics 4, 145-166 (1976) [9] Cox, J. C.: The constant elasticity of variance option pricing model, The journal of portfolio management 22, 16-17 (1996) [10] Davydov, D.; Linetsky, V.: The valuation and hedging of barrier and lookback option under the CEV process, Management science 47, 949-965 (2001) · Zbl 1232.91659 [11] Deelstra, G.; Grasselli, M.; Koehl, P. F.: Optimal design of the guarantee for defined contribution funds, Journal of economic dynamics control 28, 2239-2260 (2004) · Zbl 1202.91124 · doi:10.1016/j.jedc.2003.10.003 [12] Detemple, J.; Tian, W. D.: The valuation of American options for a class of diffusion processes, Management science 48, 917-937 (2002) · Zbl 1232.91660 · doi:10.1287/mnsc.48.7.917.2815 [13] Devolder, P.; Bosch, P. M.; Dominguez, F. I.: Stochastic optimal control of annuity contracts, Insurance: mathematics economics 33, 227-238 (2003) · Zbl 1103.91346 [14] Emanuel, D. C.; Macbeth, J. D.: Further results on the constant elasticity of variance call option pricing model, Journal of financial and quantitative analysis 17, 533-554 (1982) [15] Gerrard, R.; Haberman, S.; Vigna, E.: Optimal investment choices post-retirement in a defined contribution pension scheme, Insurance: mathematics economics 35, 321-342 (2004) · Zbl 1093.91027 · doi:10.1016/j.insmatheco.2004.06.002 [16] Haberman, S.; Vigna, E.: Optimal investment strategies and risk measures in defined contribution pension schemes, Insurance: mathematics economics 31, 35-69 (2002) · Zbl 1039.91025 · doi:10.1016/S0167-6687(02)00128-2 [17] Henderson, V.: Explicit solutions to an optimal portfolio choice problem with stochastic income, Journal of economic dynamics control 29, 1237-1266 (2005) · Zbl 1198.91188 · doi:10.1016/j.jedc.2004.07.004 [18] Heston, S.: A closed-form solution for options with stochastic volatility with applications to Bond and currency options, Review of financial studies 6, 327-343 (1993) [19] Jones, C.: The dynamics of the stochastic volatility: evidence from underlying and options markets, Journal of econometrics 116, 181-224 (2003) · Zbl 1016.62122 · doi:10.1016/S0304-4076(03)00107-6 [20] Lo, C. F.; Yuen, P. H.; Hui, C. H.: Constant elasticity of variance option pricing model with time- dependent parameters, International journal of theoretical and applied finance 3, 661-674 (2000) · Zbl 1006.91050 · doi:10.1142/S0219024900000814 [21] Macbeth, J. D.; Merville, L. J.: Tests of the black--Scholes and Cox call option valuation models, Journal of finance 35, 285-300 (1980) [22] Merton, R.: Lifetime portfolio selection under uncertainty: the continuous-time case, Review of economics and statistics 51, 247-257 (1969) [23] Merton, R.: Optimum consumption and portfolio rules in a continuous-time model, Journal of economic theory 31, 373-413 (1971) · Zbl 1011.91502 [24] Munk, C.; Sørensen, C.; Vinther, N. T.: Dynamic asset allocation under mean-reverting returns, stochastic interest rates, and inflation uncertainty: are popular recommendations consistent with rational behavior?, International review of economics and finance 13, 141-166 (2004) [25] Widdicks, M.; Duck, P.; Andricopoulos, A.; Newton, D. P.: The black--Scholes equation revisited: symptotic expansions and singular perturbations, Mathematical finance 15, 373-391 (2005) · Zbl 1124.91342 · doi:10.1111/j.0960-1627.2005.00224.x [26] Xiao, J.; Zhai, H.; Qin, C.: The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance: mathematics economics 40, 302-310 (2007) · Zbl 1141.91473 · doi:10.1016/j.insmatheco.2006.04.007 [27] Yuen, K. C.; Yang, H.; Chu, K. L.: Estimation in the constant elasticity of variance model, British actuarial journal 7, 275-292 (2001)