IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i17p3695-d1226881.html
   My bibliography  Save this article

Optimal Consumption and Investment Problem under 4/2-CIR Stochastic Hybrid Model

Author

Listed:
  • Aiqin Ma

    (School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, China)

  • Cuiyun Zhang

    (School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, China)

  • Yubing Wang

    (School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, China)

Abstract

In this paper, we investigate the optimal consumption and investment problem under the expected utility maximization criterion. It is supposed that the financial market consists of a risky asset and a risk-free asset, and the risky asset prices follow the 4/2 Cox–Ingersoll–Ross (CIR) stochastic hybrid model. The investment objective is to obtain an optimal consumption–investment strategy by maximizing the objective function. The closed-form expression of the optimal consumption–investment strategy is obtained by using optimal control theory and the corresponding Hamilton–Jacobi–Bellman (HJB) equation under the power utility function. In addition, we present a numerical example to illustrate the influence of model parameters on the optimal consumption–investment strategy.

Suggested Citation

  • Aiqin Ma & Cuiyun Zhang & Yubing Wang, 2023. "Optimal Consumption and Investment Problem under 4/2-CIR Stochastic Hybrid Model," Mathematics, MDPI, vol. 11(17), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3695-:d:1226881
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/17/3695/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/17/3695/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. George Chacko & Luis M. Viceira, 2005. "Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets," The Review of Financial Studies, Society for Financial Studies, vol. 18(4), pages 1369-1402.
    2. Vila, Jean-Luc & Zariphopoulou, Thaleia, 1997. "Optimal Consumption and Portfolio Choice with Borrowing Constraints," Journal of Economic Theory, Elsevier, vol. 77(2), pages 402-431, December.
    3. Yumo Zhang, 2021. "Dynamic Optimal Mean-Variance Portfolio Selection with a 3/2 Stochastic Volatility," Risks, MDPI, vol. 9(4), pages 1-21, March.
    4. Aleš Černý & Jan Kallsen, 2008. "Mean–Variance Hedging And Optimal Investment In Heston'S Model With Correlation," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 473-492, July.
    5. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    6. Li, Danping & Shen, Yang & Zeng, Yan, 2018. "Dynamic derivative-based investment strategy for mean–variance asset–liability management with stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 72-86.
    7. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    8. Wei, Pengyu & Yang, Charles & Zhuang, Yi, 2023. "Robust consumption and portfolio choice with derivatives trading," European Journal of Operational Research, Elsevier, vol. 304(2), pages 832-850.
    9. Mou-Hsiung Chang & Tao Pang & Yipeng Yang, 2011. "A Stochastic Portfolio Optimization Model with Bounded Memory," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 604-619, November.
    10. Xudong Zeng & Michael Taksar, 2013. "A stochastic volatility model and optimal portfolio selection," Quantitative Finance, Taylor & Francis Journals, vol. 13(10), pages 1547-1558, October.
    11. Ben G. Fitzpatrick & Wendell H. Fleming, 1991. "Numerical Methods for an Optimal Investment-Consumption Model," Mathematics of Operations Research, INFORMS, vol. 16(4), pages 823-841, November.
    12. Wendell H. Fleming & Thaleia Zariphopoulou, 1991. "An Optimal Investment/Consumption Model with Borrowing," Mathematics of Operations Research, INFORMS, vol. 16(4), pages 802-822, November.
    13. Egloff, Daniel & Leippold, Markus & Wu, Liuren, 2010. "The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 45(5), pages 1279-1310, October.
    14. Escobar, Marcos & Ferrando, Sebastian & Rubtsov, Alexey, 2018. "Dynamic derivative strategies with stochastic interest rates and model uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 86(C), pages 49-71.
    15. Wenyuan Wang & Dmitry Muravey & Yang Shen & Yan Zeng, 2023. "Optimal investment and reinsurance strategies under 4/2 stochastic volatility model," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2023(5), pages 413-449, May.
    16. Jun Liu, 2007. "Portfolio Selection in Stochastic Environments," The Review of Financial Studies, Society for Financial Studies, vol. 20(1), pages 1-39, January.
    17. Martino Grasselli, 2017. "The 4/2 Stochastic Volatility Model: A Unified Approach For The Heston And The 3/2 Model," Mathematical Finance, Wiley Blackwell, vol. 27(4), pages 1013-1034, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Cheng, Yuyang & Escobar-Anel, Marcos, 2023. "A class of portfolio optimization solvable problems," Finance Research Letters, Elsevier, vol. 52(C).
    2. Wang, Ning & Zhang, Yumo, 2024. "Robust asset-liability management games for n players under multivariate stochastic covariance models," Insurance: Mathematics and Economics, Elsevier, vol. 117(C), pages 67-98.
    3. Nicole Branger & Matthias Muck & Stefan Weisheit, 2019. "Correlation risk and international portfolio choice," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(1), pages 128-146, January.
    4. Chenxu Li & Olivier Scaillet & Yiwen Shen, 2020. "Wealth Effect on Portfolio Allocation in Incomplete Markets," Papers 2004.10096, arXiv.org, revised Aug 2021.
    5. Yang Shen, 2020. "Effect of Variance Swap in Hedging Volatility Risk," Risks, MDPI, vol. 8(3), pages 1-34, July.
    6. Yumo Zhang, 2023. "Utility maximization in a stochastic affine interest rate and CIR risk premium framework: a BSDE approach," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 97-128, June.
    7. Chenxu Li & O. Scaillet & Yiwen Shen, 2020. "Decomposition of Optimal Dynamic Portfolio Choice with Wealth-Dependent Utilities in Incomplete Markets," Swiss Finance Institute Research Paper Series 20-22, Swiss Finance Institute.
    8. Wang, Ning & Zhang, Yumo, 2023. "Robust optimal asset-liability management with mispricing and stochastic factor market dynamics," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 251-273.
    9. Yumo Zhang, 2022. "Dynamic optimal mean-variance portfolio selection with stochastic volatility and stochastic interest rate," Annals of Finance, Springer, vol. 18(4), pages 511-544, December.
    10. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    11. Zheng, Xiaoxiao & Zhou, Jieming & Sun, Zhongyang, 2016. "Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 77-87.
    12. Johannes Muhle-Karbe & Max Reppen & H. Mete Soner, 2016. "A Primer on Portfolio Choice with Small Transaction Costs," Papers 1612.01302, arXiv.org, revised May 2017.
    13. An Chen & Thai Nguyen & Manuel Rach, 2021. "A collective investment problem in a stochastic volatility environment: The impact of sharing rules," Annals of Operations Research, Springer, vol. 302(1), pages 85-109, July.
    14. Rytchkov, Oleg, 2016. "Time-Varying Margin Requirements and Optimal Portfolio Choice," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 51(2), pages 655-683, April.
    15. Escobar, Marcos & Ferrando, Sebastian & Rubtsov, Alexey, 2018. "Dynamic derivative strategies with stochastic interest rates and model uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 86(C), pages 49-71.
    16. Escobar, Marcos & Ferrando, Sebastian & Rubtsov, Alexey, 2016. "Portfolio choice with stochastic interest rates and learning about stock return predictability," International Review of Economics & Finance, Elsevier, vol. 41(C), pages 347-370.
    17. Marcos Escobar-Anel & Ben Spies & Rudi Zagst, 2024. "Optimal consumption and investment in general affine GARCH models," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 46(3), pages 987-1026, September.
    18. Legendre, François & Togola, Djibril, 2016. "Explicit solutions to dynamic portfolio choice problems: A continuous-time detour," Economic Modelling, Elsevier, vol. 58(C), pages 627-641.
    19. Oliva, I. & Renò, R., 2018. "Optimal portfolio allocation with volatility and co-jump risk that Markowitz would like," Journal of Economic Dynamics and Control, Elsevier, vol. 94(C), pages 242-256.
    20. Wang, Hang & Hu, Zhijun, 2020. "Optimal consumption and portfolio decision with stochastic covariance in incomplete markets," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3695-:d:1226881. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.