IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2410.05524.html
   My bibliography  Save this paper

Deep Learning Methods for S Shaped Utility Maximisation with a Random Reference Point

Author

Listed:
  • Ashley Davey
  • Harry Zheng

Abstract

We consider the portfolio optimisation problem where the terminal function is an S-shaped utility applied at the difference between the wealth and a random benchmark process. We develop several numerical methods for solving the problem using deep learning and duality methods. We use deep learning methods to solve the associated Hamilton-Jacobi-Bellman equation for both the primal and dual problems, and the adjoint equation arising from the stochastic maximum principle. We compare the solution of this non-concave problem to that of concavified utility, a random function depending on the benchmark, in both complete and incomplete markets. We give some numerical results for power and log utilities to show the accuracy of the suggested algorithms.

Suggested Citation

  • Ashley Davey & Harry Zheng, 2024. "Deep Learning Methods for S Shaped Utility Maximisation with a Random Reference Point," Papers 2410.05524, arXiv.org.
    • Handle: RePEc:arx:papers:2410.05524
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2410.05524
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jennifer N. Carpenter, 2000. "Does Option Compensation Increase Managerial Risk Appetite?," Journal of Finance, American Finance Association, vol. 55(5), pages 2311-2331, October.
    2. Ashley Davey & Harry Zheng, 2022. "Deep Learning for Constrained Utility Maximisation," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 661-692, June.
    3. Dong, Yinghui & Zheng, Harry, 2020. "Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to defined contribution pension plan," European Journal of Operational Research, Elsevier, vol. 281(2), pages 341-356.
    4. Alex S. L. Tse & Harry Zheng, 2023. "Speculative trading, prospect theory and transaction costs," Finance and Stochastics, Springer, vol. 27(1), pages 49-96, January.
    5. (**), Hui Wang & Jaksa Cvitanic & (*), Walter Schachermayer, 2001. "Utility maximization in incomplete markets with random endowment," Finance and Stochastics, Springer, vol. 5(2), pages 259-272.
    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. Marcos Escobar-Anel & Michel Kschonnek & Rudi Zagst, 2022. "Portfolio optimization: not necessarily concave utility and constraints on wealth and allocation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 101-140, February.
    2. Christian Dehm & Thai Nguyen & Mitja Stadje, 2020. "Non-concave expected utility optimization with uncertain time horizon," Papers 2005.13831, arXiv.org, revised Oct 2021.
    3. Fangyuan Zhang, 2023. "Non-concave portfolio optimization with average value-at-risk," Mathematics and Financial Economics, Springer, volume 17, number 3, October.
    4. Zongxia Liang & Yang Liu & Litian Zhang, 2021. "A Framework of State-dependent Utility Optimization with General Benchmarks," Papers 2101.06675, arXiv.org, revised Dec 2023.
    5. Chen, An & Stadje, Mitja & Zhang, Fangyuan, 2024. "On the equivalence between Value-at-Risk- and Expected Shortfall-based risk measures in non-concave optimization," Insurance: Mathematics and Economics, Elsevier, vol. 117(C), pages 114-129.
    6. Anne MacKay & Adriana Ocejo, 2022. "Portfolio Optimization With a Guaranteed Minimum Maturity Benefit and Risk-Adjusted Fees," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1021-1049, June.
    7. John Armstrong & Damiano Brigo & Alex S. L. Tse, 2020. "The importance of dynamic risk constraints for limited liability operators," Papers 2011.03314, arXiv.org.
    8. Pietro Siorpaes, 2015. "Optimal investment and price dependence in a semi-static market," Finance and Stochastics, Springer, vol. 19(1), pages 161-187, January.
    9. Basak, Suleyman & Makarov, Dmitry, 2012. "Difference in interim performance and risk taking with short-sale constraints," Journal of Financial Economics, Elsevier, vol. 103(2), pages 377-392.
    10. Chai, Naijie & Zhou, Wenliang & Hu, Xinlei, 2022. "Safety evaluation of urban rail transit operation considering uncertainty and risk preference: A case study in China," Transport Policy, Elsevier, vol. 125(C), pages 267-288.
    11. Bin Zou, 2017. "Optimal Investment In Hedge Funds Under Loss Aversion," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-32, May.
    12. Kim Weston, 2016. "Stability of utility maximization in nonequivalent markets," Finance and Stochastics, Springer, vol. 20(2), pages 511-541, April.
    13. Armstrong, Christopher & Nicoletti, Allison & Zhou, Frank S., 2022. "Executive stock options and systemic risk," Journal of Financial Economics, Elsevier, vol. 146(1), pages 256-276.
    14. Dong, Yinghui & Zheng, Harry, 2019. "Optimal investment of DC pension plan under short-selling constraints and portfolio insurance," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 47-59.
    15. Horst, Ulrich & Hu, Ying & Imkeller, Peter & Réveillac, Anthony & Zhang, Jianing, 2014. "Forward–backward systems for expected utility maximization," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1813-1848.
    16. Raphaëlle Bellando & Sébastien Ringuedé, 2007. "Compétition entre fonds et prise de risque excessive : une application empirique au cas des OPCVM actions de droit français," Post-Print halshs-00226341, HAL.
    17. Francis, Bill & Hasan, Iftekhar & Sharma, Zenu, 2011. "Leverage and growth: Effect of stock options," Journal of Economics and Business, Elsevier, vol. 63(6), pages 558-581.
    18. E. Nasakkala & J. Keppo, 2008. "Hydropower with Financial Information," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(5-6), pages 503-529.
    19. Jakša Cvitanić & Julien Hugonnier, 2022. "Optimal fund menus," Mathematical Finance, Wiley Blackwell, vol. 32(2), pages 455-516, April.
    20. Cuthbertson, Keith & Nitzsche, Dirk & O'Sullivan, Niall, 2016. "A review of behavioural and management effects in mutual fund performance," International Review of Financial Analysis, Elsevier, vol. 44(C), pages 162-176.

    More about this item

    Statistics

    Access and download statistics

    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:arx:papers:2410.05524. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.