This paper considers a portfolio optimization problem with delay. The finance market is consisted of one risk-free asset and one risk asset which price process is modeled by Cox-Ingersoll-Ross stochastic volatility model. In addition, considering the history information related to investment performance, the dynamic of wealth is modeled by stochastic delay differential equation. The investor’s objective is to maximize her expected utility for a linear combination of the terminal wealth and the average performance. By applying stochastic dynamic programming approach, we provide the corresponding Hamilton-Jacobin-Bellman equation and verification theorem, and the closed-form expressions of optimal strategy and optimal value function for CRRA utility are derived. Finally, a numerical example is provided to show our results.
Hull, J. and White, A. (1987) The Pricing of Options on Assets with Stochastic Volatilities. Journal of Finance, 42, 281-300. https://doi.org/10.1111/j.1540-6261.1987.tb02568.x
Scott, L.O. (1987) Option Pricing When the Variance Changes Randomly Theory Estimation, and an Application. Journal of Financial and Quantitative Analysis, 22, 419-438. https://doi.org/10.2307/2330793
Stein, E.M. and Stein, J.C. (1991) Stock Price Distributions with Stochastic Volatility: An Analytic Approach. Review of Financial Studies, 4, 727-752. https://doi.org/10.1093/rfs/4.4.727
Heston, S.L. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6, 327-343. https://doi.org/10.1093/rfs/6.2.327
Ball, C.A. and Roma, A. (1994) Stochastic Volatility Option Pricing. Journal of Financial and Quantitative Analysis, 29, 589-607. https://doi.org/10.2307/2331111
Chacko, G. and Viceira, L.M. (2005) Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets. Review of Financial Studies, 8, 1369-1402. https://doi.org/10.1093/rfs/hhi035
Fleming, W.H. and Hernandez-Hernandez, D. (2003) An Optimal Consumption Model with Stochastic Volatility. Finance and Stochastics, 7, 245-262. https://doi.org/10.1007/s007800200083
Liu, J. (2007) Portfolio Selection in Stochastic Environment. Review of Financial Studies, 20, 1-39. https://doi.org/10.1093/rfs/hhl001
Zariphopoulou, T. (1999) Optimal Investment and Consumption Models with Nonlinear Stock Dynamics. Mathematical Methods of Operations Research, 50, 271-296. https://doi.org/10.1007/s001860050098
Kraft, H. (2005) Optimal Portfolio and Heston’s Stochastic Volatility Model: An Explicit Solution for Power Utility. Quantitative Finance, 5, 303-313. https://doi.org/10.1080/14697680500149503
Taksar, M.I. and Zeng, X.D. (2009). A General Stochastic Volatility Model and Optimal Portfolio with Explicit Solutions. Working Paper. http://www.math.missouri.edu/zeng/pub/ageneral.pdf.
Ferland, R. and Watier, F. (2010) Mean-Variance Efficiency with Extended CIR Interest Rates. Applied Stochastic Models in Business and Industry, 26, 71-84. https://doi.org/10.1002/asmb.767
Li, J.Z. and Wu, R. (2009) Optimal Investment Problem with Stochastic Interest Rate and Stochastic Volatility: Maximizing a Power Utility. Applied Stochastic Models in Business and Industry, 25, 407-420. https://doi.org/10.1002/asmb.759
Noh, E.J. and Kim, J.H. (2010) An Optimal Portfolio Model with Stochastic Volatility and Stochastic Interest Rate. Journal of Mathematical Analysis and Applications, 375, 510-522. https://doi.org/10.1016/j.jmaa.2010.09.055
Li, Z.F., Zeng, Y. and Lai, Y.Z. (2012) Optimal Time-Consistent Investment and Reinsurance Strategies for Insurers under Heston’s SV Model. Insurance: Mathematics and Economics, 51, 191-203. https://doi.org/10.1016/j.insmatheco.2011.09.002
Chen, L., Peng, J., Zhang, B. and Rosyida, I. (2017) Diversified Models for Portfolio Selection Based on Uncertain Semivariance. International Journal of Systems Science, 48, 637-648. https://doi.org/10.1080/00207721.2016.1206985
Qin, Z.F., Kar, S. and Zheng, H.T. (2016) Uncertain Portfolio Adjusting Model Using Semiabsolute Deviation. Soft Computing, 20, 717-725. https://doi.org/10.1007/s00500-014-1535-y
Zhang, B., Peng, J. and Li, S.G. (2015) Uncertain Programming Models for Portfolio Selection with Uncertain Returns. International Journal of Systems Science, 46, 2510-2519. https://doi.org/10.1080/00207721.2013.871366
Akgiray, V. (1998) Conditional Heteroscedasticity in Time Series of Stock Returns. Journal of Business, 62, 55-80. https://doi.org/10.1086/296451
Dibeh, G. (2005) Speculative Dynamics in a Time-Delay Model of Asset Prices. Physica A, 355, 199-208. https://doi.org/10.1016/j.physa.2005.02.084
Sheinkman, J. and LeBaron, B. (1989) Nonlinear Dynamics and Stock Returns. Journal of Business, 62, 311-337. https://doi.org/10.1086/296465
Elsanousi, L. and Larssen, B. (2001) Optimal Consumption under Partial Observations for a Stochastic System with Delay. Preprint 9, University of Oslo, Oslo.
Chang, M.-H., Pang, T. and Yang, Y.P. (2011) A Stochastic Portfolio Optimization Model with Bounded Memory. Mathematics of Operations Research, 36, 604-619. https://doi.org/10.1287/moor.1110.0508
Lee, M.K., Kim, J.H. and Kim, J. (2011) A Delay Financial Model with Stochastic Volatility: Martingale Method. Physica A, 390, 2909-2919. https://doi.org/10.1016/j.physa.2011.03.032
A, C.X. and Li, Z.F. (2015) Optimal Investment and Excess-of-Loss Reinsurance Problem with Delay for an Insurer under Heston’s SV Model. Insurance: Mathematics and Economics, 61, 181-196. https://doi.org/10.1016/j.insmatheco.2015.01.005
Shen, Y. and Zeng, Y. (2014) Optimal Investment-Reinsurance with Delay for Mean-Variance Insurers: A Maximum Principle Approach. Insurance: Mathematics and Economics, 57, 1-12. https://doi.org/10.1016/j.insmatheco.2014.04.004
Liu, J. and Pan, J. (2003) Dynamic Derivative Strategies. Journal of Financial Economics, 69, 401-430. https://doi.org/10.1016/S0304-405X(03)00118-1