The authors thank two anonymous referees and an associate Editor who helped improve the paper. The authors are grateful to Guillaume Carlier, Rose-Anne Dana, Nicole El Karoui, Jean-Charles Rochet, Ludger Rüschendorf, and Nizar Touzi, as well as participants to the workshop “Multivariate and Dynamic Risk Measures” (IHP, Paris, October 2008), the SAS seminar at Princeton University, and the Statistics Department Risk seminar at Columbia University for helpful discussions and comments. Galichon and Ekeland gratefully acknowledge support from Chaire EDF-Calyon “Finance and Développement Durable” and FiME, Laboratoire de Finance des Marchés de l’Energie (http://www.fime-lab.org), and Galichon that of Chaire Société Générale “Risques Financiers” and Chaire Axa “Assurance et Risques Majeurs.”
COMONOTONIC MEASURES OF MULTIVARIATE RISKS
Article first published online: 19 OCT 2010
© 2010 Wiley Periodicals, Inc.
Volume 22, Issue 1, pages 109–132, January 2012
How to Cite
Ekeland, I., Galichon, A. and Henry, M. (2012), COMONOTONIC MEASURES OF MULTIVARIATE RISKS. Mathematical Finance, 22: 109–132. doi: 10.1111/j.1467-9965.2010.00453.x
- Issue published online: 19 JAN 2012
- Article first published online: 19 OCT 2010
- Manuscript received October 2007; final revision received November 2009.
- regular risk measures;
- coherent risk measures;
- maximal correlation;
- optimal transportation;
- strongly coherent risk measures
We propose a multivariate extension of a well-known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals. This involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions. Moreover, we propose to replace the current law invariance, subadditivity, and comonotonicity axioms by an equivalent property we call strong coherence and that we argue has more natural economic interpretation. Finally, we reformulate the computation of regular and coherent risk measures as an optimal transportation problem, for which we provide an algorithm and implementation.