IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v87y2018i3d10.1007_s00186-017-0609-x.html
   My bibliography  Save this article

Non-linear filtering and optimal investment under partial information for stochastic volatility models

Author

Listed:
  • Dalia Ibrahim

    (CentraleSupélec)

  • Frédéric Abergel

    (CentraleSupélec)

Abstract

This paper studies the question of filtering and maximizing terminal wealth from expected utility in partial information stochastic volatility models. The special feature is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named “Kushner–Stratonovich equations”. Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that, for power and logarithmic utility, the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.

Suggested Citation

  • Dalia Ibrahim & Frédéric Abergel, 2018. "Non-linear filtering and optimal investment under partial information for stochastic volatility models," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(3), pages 311-346, June.
  • Handle: RePEc:spr:mathme:v:87:y:2018:i:3:d:10.1007_s00186-017-0609-x
    DOI: 10.1007/s00186-017-0609-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-017-0609-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s00186-017-0609-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Lakner, Peter, 1998. "Optimal trading strategy for an investor: the case of partial information," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 77-97, August.
    2. Dothan, Michael U & Feldman, David, 1986. "Equilibrium Interest Rates and Multiperiod Bonds in a Partially Observable Economy," Journal of Finance, American Finance Association, vol. 41(2), pages 369-382, June.
    3. Yoichi Kuwana, 1995. "Certainty Equivalence And Logarithmic Utilities In Consumption/Investment Problems," Mathematical Finance, Wiley Blackwell, vol. 5(4), pages 297-309, October.
    4. Lakner, Peter, 1995. "Utility maximization with partial information," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 247-273, April.
    5. Detemple, Jerome B, 1986. "Asset Pricing in a Production Economy with Incomplete Information," Journal of Finance, American Finance Association, vol. 41(2), pages 383-391, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Olga V. Klimova, 2016. "«A Monologue About Foreign Ships» by Sugita Genpaku," HSE Working papers WP BRP 135/HUM/2016, National Research University Higher School of Economics.

    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. Michele Longo & Alessandra Mainini, 2015. "Learning and Portfolio Decisions for HARA Investors," Papers 1502.02968, arXiv.org.
    2. Michele Longo & Alessandra Mainini, 2016. "Learning And Portfolio Decisions For Crra Investors," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-21, May.
    3. Nikolai Dokuchaev, 2015. "Optimal portfolio with unobservable market parameters and certainty equivalence principle," Papers 1502.02352, arXiv.org.
    4. Thomas Lim & Marie-Claire Quenez, 2010. "Portfolio optimization in a default model under full/partial information," Papers 1003.6002, arXiv.org, revised Nov 2013.
    5. David Feldman, 2007. "Incomplete information equilibria: Separation theorems and other myths," Annals of Operations Research, Springer, vol. 151(1), pages 119-149, April.
    6. Dalia Ibrahim & Frédéric Abergel, 2018. "Non-linear filtering and optimal investment under partial information for stochastic volatility models," Post-Print hal-01018869, HAL.
    7. 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.
    8. Zongxia Liang & Qi Ye, 2024. "Optimal information acquisition for eliminating estimation risk," Papers 2405.09339, arXiv.org.
    9. Kristoffer Lindensjö, 2016. "Optimal investment and consumption under partial information," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 87-107, February.
    10. Tomas Björk & Mark Davis & Camilla Landén, 2010. "Optimal investment under partial information," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(2), pages 371-399, April.
    11. Erik Ekstrom & Juozas Vaicenavicius, 2015. "Optimal liquidation of an asset under drift uncertainty," Papers 1509.00686, arXiv.org.
    12. Kristoffer Lindensjö, 2016. "Optimal investment and consumption under partial information," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(1), pages 87-107, February.
    13. Hening Liu, 2011. "Dynamic portfolio choice under ambiguity and regime switching mean returns," Post-Print hal-00781344, HAL.
    14. Liu, Hening, 2011. "Dynamic portfolio choice under ambiguity and regime switching mean returns," Journal of Economic Dynamics and Control, Elsevier, vol. 35(4), pages 623-640, April.
    15. Michal Pakos & Hui Chen, 2008. "Asset Pricing with Uncertainty About the Long Run," 2008 Meeting Papers 295, Society for Economic Dynamics.
    16. Andrew Papanicolaou, 2018. "Backward SDEs for Control with Partial Information," Papers 1807.08222, arXiv.org.
    17. Wang, Xiao-Tian & Li, Zhe & Zhuang, Le, 2017. "European option pricing under the Student’s t noise with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 469(C), pages 848-858.
    18. Carmine De Franco & Johann Nicolle & Huyên Pham, 2019. "Dealing with Drift Uncertainty: A Bayesian Learning Approach," Risks, MDPI, vol. 7(1), pages 1-18, January.
    19. Guidolin, Massimo & Timmermann, Allan, 2007. "Properties of equilibrium asset prices under alternative learning schemes," Journal of Economic Dynamics and Control, Elsevier, vol. 31(1), pages 161-217, January.
    20. Bidarkota, Prasad V. & Dupoyet, Brice V. & McCulloch, J. Huston, 2009. "Asset pricing with incomplete information and fat tails," Journal of Economic Dynamics and Control, Elsevier, vol. 33(6), pages 1314-1331, June.

    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:spr:mathme:v:87:y:2018:i:3:d:10.1007_s00186-017-0609-x. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.