Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model. (English) Zbl 1231.91402

Summary: This paper focuses on the constant elasticity of variance (CEV) model for studying the optimal investment strategy before and after retirement in a defined contribution pension plan where benefits are paid under the form of annuities; annuities are supposed to be guaranteed during a certain fixed period of time. Using Legendre transform, dual theory and variable change technique, we derive the explicit solutions for the power and exponential utility functions in two different periods (before and after retirement). Each solution contains a modified factor which reflects an investor’s decision to hedge the volatility risk. In order to investigate the influence of the modified factor on the optimal strategy, we analyze the property of the modified factor. The results show that the dynamic behavior of the modified factor for the power utility mainly depends on the time and the investor’s risk aversion coefficient, whereas it only depends on the time in the exponential case.


91G10 Portfolio theory
91G80 Financial applications of other theories
93E20 Optimal stochastic control
Full Text: DOI


[1] Albrecht, P.; Maurer, R., Self-annuitization, consumption shortfall in retirement and asset allocation: The annuity benchmark, Journal of Pension Economics and Finance, 1, 269-288 (2002)
[2] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654 (1973) · Zbl 1092.91524
[3] Blake, D.; Cairns, A. J.G.; Dowd, K., Pensionmetrics: Stochastic pension plan design and Valute-at-risk during the accumulation phase, Insurance: Mathematics & Economics, 29, 187-215 (2001) · Zbl 0989.62057
[4] Blake, D.; Cairns, A. J.G.; Dowd, K., Pensionmetrics 2: Stochastic pension plan design during the distribution phase, Insurance: Mathematics & Economics, 33, 29-47 (2003) · Zbl 1043.62086
[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 Dynamic & Control, 27, 1099-1112 (2003) · Zbl 1178.91174
[6] Booth, P.; Yakoubov, Y., Investment policy for defined contribution scheme members close to retirement: An analysis of the lifestyle concept, North American Actuarial Journal, 4, 1-19 (2000) · Zbl 1083.91527
[7] 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
[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 and Control, 28, 2239-2260 (2004) · Zbl 1202.91124
[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
[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] 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
[15] 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
[16] Hsu, Y. L.; Lin, T. I.; Lee, C. F., Constant elasticity of variance (CEV) option pricing model: Integration and detailed derivation, Mathematics and Computer in Simulation, 79, 60-71 (2008) · Zbl 1144.91325
[17] Jones, C., The dynamics of the stochastic volatility: Evidence from underlying and options markets, Journal of Econometrics, 116, 181-224 (2003) · Zbl 1016.62122
[18] Jonsson, M.; Sircar, R., Optimal investment problems and volatility homogenization approximations, (Modern Methods in Scientific Computing and Applications NATO Science Series II, vol. 75 (2002), Springer: Springer Germany), 255-281 · Zbl 1104.91302
[19] MacBeth, J. D.; Merville, L. J., Tests of the Black-Scholes and Cox call option valuation models, Journal of Finance, 35, 285-300 (1980)
[20] 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)
[21] Schroder, M., Computing the constant elasticity of variance option pricing formula, Journal of Finance, 44, 211-219 (1989)
[22] 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.