IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i10p1503-d1392711.html
   My bibliography  Save this article

The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets

Author

Listed:
  • Shuhui Liu

    (Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China)

Abstract

The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket.

Suggested Citation

  • Shuhui Liu, 2024. "The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets," Mathematics, MDPI, vol. 12(10), pages 1-18, May.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:10:p:1503-:d:1392711
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/10/1503/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/10/1503/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2021. "Consistent investment of sophisticated rank‐dependent utility agents in continuous time," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 1056-1095, July.
    2. Jakša Cvitanić & Jin Ma & Jianfeng Zhang, 2003. "Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 135-151, January.
    3. repec:eme:mfppss:v:35:y:2009:i:5:p:427-438 is not listed on IDEAS
    4. Lars Peter Hansen & Thomas J Sargent, 2014. "Robust Control and Model Uncertainty," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 5, pages 145-154, World Scientific Publishing Co. Pte. Ltd..
    5. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    6. Chen, Zengjing & Epstein, Larry G., 2022. "A central limit theorem for sets of probability measures," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 424-451.
    7. Nicholas Barberis, 2000. "Investing for the Long Run when Returns Are Predictable," Journal of Finance, American Finance Association, vol. 55(1), pages 225-264, February.
    8. Tomasz R. Bielecki & Hanqing Jin & Stanley R. Pliska & Xun Yu Zhou, 2005. "Continuous‐Time Mean‐Variance Portfolio Selection With Bankruptcy Prohibition," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 213-244, April.
    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. Jianjun Miao, 2009. "Ambiguity, Risk and Portfolio Choice under Incomplete Information," Annals of Economics and Finance, Society for AEF, vol. 10(2), pages 257-279, November.
    2. Cvitanic, Jaksa & Lazrak, Ali & Wang, Tan, 2008. "Implications of the Sharpe ratio as a performance measure in multi-period settings," Journal of Economic Dynamics and Control, Elsevier, vol. 32(5), pages 1622-1649, May.
    3. Junna Bi & Qingbin Meng & Yongji Zhang, 2014. "Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer," Annals of Operations Research, Springer, vol. 212(1), pages 43-59, January.
    4. Agarwal, Vikas & Arisoy, Y. Eser & Naik, Narayan Y., 2017. "Volatility of aggregate volatility and hedge fund returns," Journal of Financial Economics, Elsevier, vol. 125(3), pages 491-510.
    5. Stadje, M.A. & Pelsser, A., 2014. "Time-Consistent and Market-Consistent Evaluations (Revised version of 2012-086)," Discussion Paper 2014-002, Tilburg University, Center for Economic Research.
    6. Suleyman Basak & Georgy Chabakauri, 2010. "Dynamic Mean-Variance Asset Allocation," The Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 2970-3016, August.
    7. Antoon Pelsser & Mitja Stadje, 2014. "Time-Consistent And Market-Consistent Evaluations," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 25-65, January.
    8. Bi, Junna & Jin, Hanqing & Meng, Qingbin, 2018. "Behavioral mean-variance portfolio selection," European Journal of Operational Research, Elsevier, vol. 271(2), pages 644-663.
    9. Kim, Dong-Hyuk, 2013. "Optimal choice of a reserve price under uncertainty," International Journal of Industrial Organization, Elsevier, vol. 31(5), pages 587-602.
    10. Roger J. A. Laeven & Mitja Stadje, 2014. "Robust Portfolio Choice and Indifference Valuation," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1109-1141, November.
    11. Massimo Guidolin & Francesca Rinaldi, 2013. "Ambiguity in asset pricing and portfolio choice: a review of the literature," Theory and Decision, Springer, vol. 74(2), pages 183-217, February.
    12. Dejian Tian, 2022. "Pricing principle via Tsallis relative entropy in incomplete market," Papers 2201.05316, arXiv.org, revised Oct 2022.
    13. Akihiko Takahashi & Toshihiro Yamada, 2016. "An Asymptotic Expansion for Forward–Backward SDEs: A Malliavin Calculus Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 23(4), pages 337-373, December.
    14. Philipp K. Illeditsch & Jayant V. Ganguli & Scott Condie, 2021. "Information Inertia," Journal of Finance, American Finance Association, vol. 76(1), pages 443-479, February.
    15. repec:esx:essedp:770 is not listed on IDEAS
    16. Hui Chen & Nengjiu Ju & Jianjun Miao, 2014. "Dynamic Asset Allocation with Ambiguous Return Predictability," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 17(4), pages 799-823, October.
    17. Dirk Becherer & Klebert Kentia, 2017. "Hedging under generalized good-deal bounds and model uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 171-214, August.
    18. Holger Kraft & Thomas Seiferling & Frank Thomas Seifried, 2017. "Optimal consumption and investment with Epstein–Zin recursive utility," Finance and Stochastics, Springer, vol. 21(1), pages 187-226, January.
    19. Zhang, Jinqing & Jin, Zeyu & An, Yunbi, 2017. "Dynamic portfolio optimization with ambiguity aversion," Journal of Banking & Finance, Elsevier, vol. 79(C), pages 95-109.
    20. Bi, Junna & Liang, Zhibin & Xu, Fangjun, 2016. "Optimal mean–variance investment and reinsurance problems for the risk model with common shock dependence," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 245-258.
    21. Kraft, Holger & Seiferling, Thomas & Seifried, Frank Thomas, 2016. "Optimal consumption and investment with Epstein-Zin recursive utility," SAFE Working Paper Series 52, Leibniz Institute for Financial Research SAFE, revised 2016.

    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:12:y:2024:i:10:p:1503-:d:1392711. 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.