# On dynamic measures of risk

## Author Info

• Ioannis Karatzas

()

(Departments of Mathematics and Statistics, Columbia University, New York, NY 10027, USA Manuscript)

• Jaksa Cvitanic

(Department of Statistics, Columbia University, New York, NY 10027, USA)

## Abstract

In the context of complete financial markets, we study dynamic measures of the form $\rho(x;C):=\sup_{\nu\in\D} \inf_{\pi(\cdot)\in\A(x)}{\bf E}_\nu\left(\frac{C-X^{x, \pi}(T)}{S_0(T)}\right)^+,$ for the risk associated with hedging a given liability C at time t = T. Here x is the initial capital available at time t = 0, ${\cal A}(x)$ the class of admissible portfolio strategies, $S_0(\cdot)$ the price of the risk-free instrument in the market, ${\cal P}=\{{\bf P}_\nu\}_{\nu\in{\cal D}}$ a suitable family of probability measures, and [0,T] the temporal horizon during which all economic activity takes place. The classes ${\cal A}(x)$ and ${\cal D}$ are general enough to incorporate capital requirements, and uncertainty about the actual values of stock-appreciation rates, respectively. For this latter purpose we discuss, in addition to the above "max-min" approach, a related measure of risk in a "Bayesian" framework. Risk-measures of this type were introduced by Artzner, Delbaen, Eber and Heath in a static setting, and were shown to possess certain desirable "coherence" properties.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

## Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 3 (1999)
Issue (Month): 4 ()
Pages: 451-482

as
in new window

 Handle: RePEc:spr:finsto:v:3:y:1999:i:4:p:451-482 Note: received: February 1998; final version received: February 1999 Contact details of provider: Web page: http://www.springer.com Order Information: Web: http://www.springer.com/mathematics/quantitative+finance/journal/780/PS2

## References

No references listed on IDEAS
You can help add them by filling out this form.

## Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

## Corrections

When requesting a correction, please mention this item's handle: RePEc:spr:finsto:v:3:y:1999:i:4:p:451-482. 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: (Sonal Shukla)

or (Rebekah McClure)

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 references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link 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 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.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.