IDEAS home Printed from
   My bibliography  Save this paper

Improved Fr\'echet$-$Hoeffding bounds on $d$-copulas and applications in model-free finance


  • Thibaut Lux
  • Antonis Papapantoleon


We derive upper and lower bounds on the expectation of $f(\mathbf{S})$ under dependence uncertainty, i.e. when the marginal distributions of the random vector $\mathbf{S}=(S_1,\dots,S_d)$ are known but their dependence structure is partially unknown. We solve the problem by providing improved \FH bounds on the copula of $\mathbf{S}$ that account for additional information. In particular, we derive bounds when the values of the copula are given on a compact subset of $[0,1]^d$, the value of a functional of the copula is prescribed or different types of information are available on the lower dimensional marginals of the copula. We then show that, in contrast to the two-dimensional case, the bounds are quasi-copulas but fail to be copulas if $d>2$. Thus, in order to translate the improved \FH bounds into bounds on the expectation of $f(\mathbf{S})$, we develop an alternative representation of multivariate integrals with respect to copulas that admits also quasi-copulas as integrators. By means of this representation, we provide an integral characterization of orthant orders on the set of quasi-copulas which relates the improved \FH bounds to bounds on the expectation of $f(\mathbf{S})$. Finally, we apply these results to compute model-free bounds on the prices of multi-asset options that take partial information on the dependence structure into account, such as correlations or market prices of other traded derivatives. The numerical results show that the additional information leads to a significant improvement of the option price bounds compared to the situation where only the marginal distributions are known.

Suggested Citation

  • Thibaut Lux & Antonis Papapantoleon, 2016. "Improved Fr\'echet$-$Hoeffding bounds on $d$-copulas and applications in model-free finance," Papers 1602.08894,, revised Jun 2017.
  • Handle: RePEc:arx:papers:1602.08894

    Download full text from publisher

    File URL:
    File Function: Latest version
    Download Restriction: no

    References listed on IDEAS

    1. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    2. Genest, C. & Quesada Molina, J. J. & Rodriguez Lallena, J. A. & Sempi, C., 1999. "A Characterization of Quasi-copulas," Journal of Multivariate Analysis, Elsevier, vol. 69(2), pages 193-205, May.
    3. Peter Tankov, 2010. "Improved Frechet bounds and model-free pricing of multi-asset options," Papers 1004.4153,, revised Mar 2011.
    4. M. Taylor, 2007. "Multivariate measures of concordance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(4), pages 789-806, December.
    Full references (including those not matched with items on IDEAS)


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

    Cited by:

    1. Thibaut Lux & Antonis Papapantoleon, 2016. "Model-free bounds on Value-at-Risk using extreme value information and statistical distances," Papers 1610.09734,, revised Nov 2018.
    2. Stephan Eckstein & Michael Kupper, 2018. "Computation of optimal transport and related hedging problems via penalization and neural networks," Papers 1802.08539,

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    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:1602.08894. 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: (arXiv administrators). General contact details of provider: .

    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.