IDEAS home Printed from https://ideas.repec.org/p/ehl/lserod/67689.html
   My bibliography  Save this paper

Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion

Author

Listed:
  • Czichowsky, Christoph
  • Schachermayer, Walter

Abstract

While absence of arbitrage in frictionlessfinancial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an It^o process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an It^o process in a one-sided manner without reflection.

Suggested Citation

  • Czichowsky, Christoph & Schachermayer, Walter, 2017. "Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion," LSE Research Online Documents on Economics 67689, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:67689
    as

    Download full text from publisher

    File URL: http://eprints.lse.ac.uk/67689/
    File Function: Open access version.
    Download Restriction: no

    References listed on IDEAS

    as
    1. Erhan Bayraktar & Hasanjan Sayit, 2010. "On the stickiness property," Quantitative Finance, Taylor & Francis Journals, vol. 10(10), pages 1109-1112.
    2. Cvitanic, Jaksa & Wang, Hui, 2001. "On optimal terminal wealth under transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 223-231, April.
    3. Luciano Campi & Walter Schachermayer, 2006. "A super-replication theorem in Kabanov’s model of transaction costs," Finance and Stochastics, Springer, vol. 10(4), pages 579-596, December.
    4. Benoit Mandelbrot, 1967. "The Variation of Some Other Speculative Prices," The Journal of Business, University of Chicago Press, vol. 40, pages 393-393.
    5. He, Hua & Pearson, Neil D., 1991. "Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case," Journal of Economic Theory, Elsevier, vol. 54(2), pages 259-304, August.
    6. Jakša Cvitanić & Ioannis Karatzas, 1996. "Hedging And Portfolio Optimization Under Transaction Costs: A Martingale Approach12," Mathematical Finance, Wiley Blackwell, vol. 6(2), pages 133-165, April.
    7. repec:dau:papers:123456789/7701 is not listed on IDEAS
    8. Attila Herczegh & Vilmos Prokaj & Mikl'os R'asonyi, 2013. "Diversity and no arbitrage," Papers 1301.4173, arXiv.org, revised Aug 2014.
    9. Griselda Deelstra & Huyên Pham & Nizar Touzi, 2001. "Dual formulation of the utility maximisation problem under transaction costs," ULB Institutional Repository 2013/7596, ULB -- Universite Libre de Bruxelles.
    10. repec:spr:compst:v:73:y:2011:i:2:p:251-262 is not listed on IDEAS
    11. repec:dau:papers:123456789/5455 is not listed on IDEAS
    12. M. H. A. Davis & A. R. Norman, 1990. "Portfolio Selection with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 676-713, November.
    13. Paolo Guasoni & Mikl'os R'asonyi & Walter Schachermayer, 2008. "Consistent price systems and face-lifting pricing under transaction costs," Papers 0803.4416, arXiv.org.
    14. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the Existence of Shadow Prices," Papers 1111.6633, arXiv.org, revised Jan 2013.
    15. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    16. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2013. "On the existence of shadow prices," Finance and Stochastics, Springer, vol. 17(4), pages 801-818, October.
    17. Michael Taksar & Michael J. Klass & David Assaf, 1988. "A Diffusion Model for Optimal Portfolio Selection in the Presence of Brokerage Fees," Mathematics of Operations Research, INFORMS, vol. 13(2), pages 277-294, May.
    18. Freddy Delbaen & Peter Grandits & Thorsten Rheinländer & Dominick Samperi & Martin Schweizer & Christophe Stricker, 2002. "Exponential Hedging and Entropic Penalties," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 99-123, April.
    19. Jan Kallsen & Johannes Muhle-Karbe, 2011. "Existence of shadow prices in finite probability spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 251-262, April.
    20. Hua He & Neil D. Pearson, 1991. "Consumption and Portfolio Policies With Incomplete Markets and Short‐Sale Constraints: the Finite‐Dimensional Case1," Mathematical Finance, Wiley Blackwell, vol. 1(3), pages 1-10, July.
    21. Kardaras, Constantinos & Platen, Eckhard, 2011. "On the semimartingale property of discounted asset-price processes," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2678-2691, November.
    22. Walter Schachermayer, 2013. "Admissible Trading Strategies under Transaction Costs," Papers 1308.1492, arXiv.org, revised May 2014.
    23. Giuseppe Benedetti & Luciano Campi, 2011. "Multivariate utility maximization with proportional transaction costs and random endowment," Working Papers hal-00586377, HAL.
    24. Luciano Campi & Mark Owen, 2011. "Multivariate utility maximization with proportional transaction costs," Finance and Stochastics, Springer, vol. 15(3), pages 461-499, September.
    25. repec:dau:papers:123456789/2318 is not listed on IDEAS
    26. Richard Rouge & Nicole El Karoui, 2000. "Pricing Via Utility Maximization and Entropy," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 259-276, April.
    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. Dániel Ágoston Bálint & Martin Schweizer, 2019. "Properly Discounted Asset Prices Are Semimartingales," Swiss Finance Institute Research Paper Series 19-53, Swiss Finance Institute.
    2. Huy N. Chau & Miklós Rásonyi, 2019. "Robust utility maximisation in markets with transaction costs," Finance and Stochastics, Springer, vol. 23(3), pages 677-696, July.
    3. Christoph Belak & Jörn Sass, 2019. "Finite-horizon optimal investment with transaction costs: construction of the optimal strategies," Finance and Stochastics, Springer, vol. 23(4), pages 861-888, October.

    More about this item

    Keywords

    portfolio choice; non-semimartingale price processes; fractional Brownian motion; proportional transaction costs; utilities on the whole real line; exponential utility; shadow price; convex duality; stickiness; optimal trading strategies;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:ehl:lserod:67689. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (LSERO Manager). General contact details of provider: http://edirc.repec.org/data/lsepsuk.html .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.