IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v19y2016i02ns0219024916500096.html
   My bibliography  Save this article

Worst-Case Portfolio Optimization In A Market With Bubbles

Author

Listed:
  • CHRISTOPH BELAK

    (Department IV – Mathematics, University of Trier, Universitätsring 19, 54296 Trier, Germany)

  • SÖREN CHRISTENSEN

    (Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden)

  • OLAF MENKENS

    (School of Mathematical Sciences, Dublin City University, Collins Avenue, Dublin 9, Ireland)

Abstract

We investigate a utility maximization problem in the presence of asset price bubbles. At random times, the investor receives warnings that a bubble has formed in the market which may lead to a crash in the risky asset. We propose a regime-switching model for the warnings and we make no assumptions about the distribution of the timing and the size of the crashes. Instead, we assume that the investor takes a worst-case perspective towards their impacts, i.e. the investor maximizes her expected utility under the worst-case crash scenario. We characterize the value function by a system of Hamilton–Jacobi–Bellman equations and derive a coupled system of ordinary differential equations for the optimal strategies. Numerical examples are provided.

Suggested Citation

  • Christoph Belak & Sören Christensen & Olaf Menkens, 2016. "Worst-Case Portfolio Optimization In A Market With Bubbles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(02), pages 1-36, March.
    • Handle: RePEc:wsi:ijtafx:v:19:y:2016:i:02:n:s0219024916500096
    DOI: 10.1142/S0219024916500096
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024916500096
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219024916500096?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. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
    2. Marcos Escobar & Daniela Neykova & Rudi Zagst, 2015. "Portfolio Optimization In Affine Models With Markov Switching," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-46.
    3. Dilip Abreu & Markus K. Brunnermeier, 2003. "Bubbles and Crashes," Econometrica, Econometric Society, vol. 71(1), pages 173-204, January.
    4. Luz Rocío Sotomayor & Abel Cadenillas, 2009. "Explicit Solutions Of Consumption‐Investment Problems In Financial Markets With Regime Switching," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 251-279, April.
    5. Tirole, Jean, 1982. "On the Possibility of Speculation under Rational Expectations," Econometrica, Econometric Society, vol. 50(5), pages 1163-1181, September.
    6. Agostino Capponi & José Figueroa-López & Andrea Pascucci, 2015. "Dynamic credit investment in partially observed markets," Finance and Stochastics, Springer, vol. 19(4), pages 891-939, October.
    7. Francesca Biagini & Hans Föllmer & Sorin Nedelcu, 2014. "Shifting martingale measures and the birth of a bubble as a submartingale," Finance and Stochastics, Springer, vol. 18(2), pages 297-326, April.
    8. Belak, Christoph & Christensen, Sören & Menkens, Olaf, 2014. "Worst-case optimal investment with a random number of crashes," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 140-148.
    9. Sascha Desmettre & Ralf Korn & Frank Thomas Seifried, 2015. "Lifetime Consumption And Investment For Worst-Case Crash Scenarios," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(01), pages 1-30.
    10. Robert Jarrow & Philip Protter, 2011. "Foreign currency bubbles," Review of Derivatives Research, Springer, vol. 14(1), pages 67-83, April.
    11. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    12. Ralf Korn & Olaf Menkens, 2005. "Worst-Case Scenario Portfolio Optimization: a New Stochastic Control Approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(1), pages 123-140, September.
    13. Jose A. Scheinkman & Wei Xiong, 2003. "Overconfidence and Speculative Bubbles," Journal of Political Economy, University of Chicago Press, vol. 111(6), pages 1183-1219, December.
    14. Frank Thomas Seifried, 2010. "Optimal Investment for Worst-Case Crash Scenarios: A Martingale Approach," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 559-579, August.
    15. Ralf Korn & Paul Wilmott, 2002. "Optimal Portfolios Under The Threat Of A Crash," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 171-187.
    16. Robert A. Jarrow & Philip Protter, 2009. "Forward And Futures Prices With Bubbles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(07), pages 901-924.
    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. Robert A. Jarrow, 2015. "Asset Price Bubbles," Annual Review of Financial Economics, Annual Reviews, vol. 7(1), pages 201-218, December.
    2. Francesca Biagini & Hans Föllmer & Sorin Nedelcu, 2014. "Shifting martingale measures and the birth of a bubble as a submartingale," Finance and Stochastics, Springer, vol. 18(2), pages 297-326, April.
    3. Francesca Biagini & Andrea Mazzon & Thilo Meyer-Brandis, 2016. "Liquidity induced asset bubbles via flows of ELMMs," Papers 1611.01440, arXiv.org, revised Nov 2016.
    4. Francesca Biagini & Thomas Reitsam, 2019. "Asset Price Bubbles in market models with proportional transaction costs," Papers 1911.10149, arXiv.org, revised Dec 2020.
    5. Belak, Christoph & Christensen, Sören & Menkens, Olaf, 2014. "Worst-case optimal investment with a random number of crashes," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 140-148.
    6. Martin Herdegen & Sebastian Herrmann, 2017. "Strict Local Martingales and Optimal Investment in a Black-Scholes Model with a Bubble," Papers 1711.06679, arXiv.org.
    7. Martin Herdegen & Martin Schweizer, 2016. "Strong Bubbles And Strict Local Martingales," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-44, June.
    8. Johannes Muhle-Karbe & Marcel Nutz, 2018. "A risk-neutral equilibrium leading to uncertain volatility pricing," Finance and Stochastics, Springer, vol. 22(2), pages 281-295, April.
    9. Sascha Desmettre & Sebastian Merkel & Annalena Mickel & Alexander Steinicke, 2023. "Worst-Case Optimal Investment in Incomplete Markets," Papers 2311.10021, arXiv.org.
    10. Martin HERDEGEN & Martin SCHWEIZER, 2015. "Economics-Based Financial Bubbles (and Why They Imply Strict Local Martingales)," Swiss Finance Institute Research Paper Series 15-05, Swiss Finance Institute.
    11. Ralf Korn & Elisabeth Leoff, 2019. "Multi-Asset Worst-Case Optimal Portfolios," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    12. Francesca Biagini & Andrea Mazzon & Thilo Meyer-Brandis & Katharina Oberpriller, 2022. "Liquidity based modeling of asset price bubbles via random matching," Papers 2210.13804, arXiv.org, revised Nov 2022.
    13. Peter Carr & Travis Fisher & Johannes Ruf, 2014. "On the hedging of options on exploding exchange rates," Finance and Stochastics, Springer, vol. 18(1), pages 115-144, January.
    14. Franklin Allen & Elena Carletti, 2013. "Systemic risk from real estate and macro-prudential regulation," International Journal of Banking, Accounting and Finance, Inderscience Enterprises Ltd, vol. 5(1/2), pages 28-48.
    15. Matsushima, Hitoshi, 2013. "Behavioral aspects of arbitrageurs in timing games of bubbles and crashes," Journal of Economic Theory, Elsevier, vol. 148(2), pages 858-870.
    16. John Fender, 2020. "Beyond the efficient markets hypothesis: Towards a new paradigm," Bulletin of Economic Research, Wiley Blackwell, vol. 72(3), pages 333-351, July.
    17. Bidian, Florin, 2015. "Portfolio constraints, differences in beliefs and bubbles," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 317-326.
    18. Alessandro Fiori Maccioni, 2011. "Endogenous Bubbles in Derivatives Markets: The Risk Neutral Valuation Paradox," Papers 1106.5274, arXiv.org, revised Sep 2011.
    19. Allen, Franklin & Carletti, Elena, 2013. "New theories to underpin financial reform," Journal of Financial Stability, Elsevier, vol. 9(2), pages 242-249.
    20. Hirota, Shinichi & Huber, Juergen & Stöckl, Thomas & Sunder, Shyam, 2022. "Speculation, money supply and price indeterminacy in financial markets: An experimental study," Journal of Economic Behavior & Organization, Elsevier, vol. 200(C), pages 1275-1296.

    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:wsi:ijtafx:v:19:y:2016:i:02:n:s0219024916500096. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/ijtaf/ijtaf.shtml .

    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.