Abstract
This paper investigates optimal investment problems in the presence of stochastic interest rates and stochastic volatility under the expected utility maximization criterion. The financial market consists of three assets: a risk-free asset, a risky asset, and zero-coupon bonds (rolling bonds). The short interest rate is assumed to follow an affine diffusion process, which includes the Vasicek and the Cox–Ingersoll–Ross (CIR) models, as special cases. The risk premium of the risky asset depends on a square-root diffusion (CIR) process, while the return rate and volatility coefficient are unspecified and possibly given by non-Markovian processes. This framework embraces the family of the state-of-the-art 4/2 stochastic volatility models and some non-Markovian models, as exceptional examples. The investor aims to maximize the expected utility of the terminal wealth for two types of utility functions, power utility, and logarithmic utility. By adopting a backward stochastic differential equation (BSDE) approach to overcome the potentially non-Markovian framework and solving two BSDEs explicitly, we derive, in closed form, the optimal investment strategies and optimal value functions. Furthermore, explicit solutions to some special cases of our model are provided. Finally, numerical examples illustrate our results under one specific case, the hybrid Vasicek-4/2 model.
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References
Boulier, J., Huang, S., Taillard, G.: Optimal management under stochastic interest rates: the case of a protected defined contribution pension fund. Insur. Math. Econ. 28, 173–189 (2001)
Briand, P., Hu, Y.: Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Relat. Fields 141, 543–567 (2008)
Chacko, G., Viceira, L.M.: Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets. Rev. Financ. Stud. 18, 1369–1402 (2005)
Chang, H., Li, X.: Optimal consumption and portfolio decision with convertible bond in affine stochastic interest rate and Heston’s SV framework. Math. Probl. Eng. (2016). https://doi.org/10.1155/2016/4823451
Chang, H., Rong, X.: An investment and consumption problem with CIR interest rate and stochastic volatility. Abstr. Appl. Anal. (2013). https://doi.org/10.1155/2013/219397
Chang, H., Wang, C., Fang, Z., Ma, D.: Defined contribution pension planning with a stochastic interest rate and mean-reverting returns under the hyperbolic absolute risk aversion preference. IMA J. Manag. Math. 31, 167–189 (2020)
Cheng, Y., Escobar, M.: Optimal investment strategy in the family of 4/2 stochastic volatility models. Quant Financ. 21, 1723–1751 (2021)
Cheridito, P., Hu, Y.: Optimal consumption and investment in incomplete markets with general constraints. Stoch. Dyn. 11, 283–299 (2011)
Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)
Cox, J.C.: The constant elasticity of variance option pricing model. J. Portf. Manag. 22, 16–17 (1996)
Cui, Z., Kirkby, J.L., Nguyen, D.: Equity-linked annuity pricing with cliquet-style guarantees in regime-switching and stochastic volatility models with jumps. Insur. Math. Econ. 74, 46–62 (2017)
Deelstra, G., Grasselli, M., Koehl, P.F.: Optimal investment strategies in the presence of a minimum guarantee. Insur. Math. Econ. 33, 189–207 (2003)
Duffie, D., Kan, R.: A yield-factor model of interest rates. Math Financ. 6, 379–406 (1996)
El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math Financ. 7, 1–71 (1997)
Escobar, M., Neykova, D., Zagst, R.: HARA utility maximization in a Markov-switching bond-stock market. Quant Financ. 17, 1715–1733 (2017)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006)
Gnoatto, A., Grasselli, M., Platen, E.: Calibration to FX triangles of the 4/2 model under the benchmark approach. Decis. Econ. Financ. 45, 1–34 (2022)
Grasselli, M.: The 4/2 stochastic volatility model: a unified approach for the Heston and the 3/2 model. Math Financ. 27, 1013–1034 (2017)
Guan, G., Liang, Z.: Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework. Insur. Math. Econ. 57, 58–66 (2014)
Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993)
Hu, Y., Imkeller, P., Müller, M.: Utility maximization in incomplete markets. Ann. Appl. Probab. 15, 1691–1712 (2005)
Huang, Z., Wang, H., Wu, Z.: A kind of optimal investment problem under inflation and uncertain time horizon. Appl. Math. Comput. 375, 125084 (2020)
Kallsen, J., Muhle-Karbe, J.: Utility maximization in affine stochastic volatility models. Int. J. Theor. Appl. Financ. 13, 459–477 (2010)
Karatzas, I., Lehoczky, J., Shreve, S.: Optimal portfolio and consumption decisions for a small investor on a finite horizon. SIAM J. Control. Optim. 25, 1557–1586 (1987)
Karatzas, I., Lehoczky, J., Shreve, S., Xu, G.L.: Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control. Optim. 29, 702–730 (1991)
Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28, 558–602 (2000)
Korn, R., Kraft, H.: A stochastic control approach to portfolio problems with stochastic interest rates. SIAM J. Control. Optim. 40, 1250–1269 (2002)
Korn, R., Kraft, H.: On the stability of continuous-time portfolio problems with stochastic opportunity set. Math Financ. 14, 403–414 (2004)
Kraft, H.: Optimal portfolios and Heston’s stochastic volatility model: an explicit solution for power utility. Quant Financ. 5, 303–313 (2005)
Lewis, A.L.: Option Valuation under Stochastic Volatility. Finance Press, Newport Beach (2000)
Li, J., Wu, R.: Optimal investment problem with stochastic interest rate and stochastic volatility: maximizing a power utility. Appl. Stoch. Models. Bus Ind. 25, 407–420 (2009)
Lin, W., Li, S., Luo, X., Chern, S.: Consistent pricing of VIX and equity derivatives with the 4/2 stochastic volatility plus jumps model. J. Math. Anal. Appl. 447, 778–797 (2017)
Liu, J.: Portfolio selection in stochastic environments. Rev. Financ. Stud. 20, 1–39 (2007)
Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time cases. Rev. Econ. Stat. 51, 247–257 (1969)
Merton, R.C.: An intertemporal capital asset pricing model. Econometrica 41, 867–887 (1973)
Pan, J., Hu, S., Zhou, X.: Optimal investment strategy for asset-liability management under the Heston model. Optimization 68, 895–920 (2019)
Pham, H.: Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, Berlin (2009)
Platen, E., Heath, D.: A Benchmark Approach to Quantitative Finance. Springer, Berlin (2006)
Pliska, R.: A stochastic calculus model of continuous trading: optimal portfolios. Math. Oper. Res. 11, 371–382 (1986)
Shen, Y., Siu, T.K.: Asset allocation under stochastic interest rate with regime switching. Econ. Model. 29, 1126–1136 (2012)
Shen, Y., Zeng, Y.: Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process. Insur. Math. Econ. 62, 118–137 (2015)
Siu, T.K.: Functional Itô’s calculus and dynamic convex risk measures for derivative securities. Commun. Stoch. Anal. 6, 339–358 (2012)
Sun, Z., Zhang, X., Yuen, K.C.: Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option. Scan Actua J. 3, 218–244 (2020)
Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)
Zeng, X., Taksar, M.: A stochastic volatility model and optimal portfolio selection. Quant Financ. 13, 1547–1558 (2013)
Zhang, J.: Backward Stochastic Differential Equations: from Linear to Fully Nonlinear Theory. Springer, New York (2017)
Zhu, S., Wang, B.: Unified approach for the affine and non-affine models: an empirical analysis on the S &P 500 volatility dynamics. Comput. Econ. 53, 1421–1442 (2019)
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The authors are very grateful to Prof. Jesper Lund Pedersen, the editor, and two anonymous reviewers for their constructive comments and suggestions, which greatly improve the quality of this paper.
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Zhang, Y. Utility maximization in a stochastic affine interest rate and CIR risk premium framework: a BSDE approach. Decisions Econ Finan 46, 97–128 (2023). https://doi.org/10.1007/s10203-022-00374-x
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DOI: https://doi.org/10.1007/s10203-022-00374-x
Keywords
- Affine diffusion process
- CIR risk premium
- Power utility
- Logarithmic utility
- Backward stochastic differential equation