Time consistency of dynamic risk measures in markets with transaction costs

Abstract

Set-valued dynamic risk measures are defined on with and with an image space in the power set of . Primal and dual representations of dynamic risk measures are deduced. Definitions of different time consistency properties in the set-valued framework are given. It is shown that the recursive form for multivariate risk measures as well as an additive property for the acceptance sets is equivalent to a stronger time consistency property called multi-portfolio time consistency.

Keywords:

• Risk measures
• Multivariate contingent claims
• Dynamic models
• Transaction costs

Classification:

• D8
• D81
• G1

Acknowledgments

We would like to thank Patrick Cheridito, Andreas Hamel, Frank Heyde, and Frank Riedel for helpful comments and discussions, and two anonymous referees, who helped to significantly improve the paper. Birgit Rudloff’s research was supported by NSF award DMS-1007938 and Zachary Feinstein was supported by the NSF RTG grant 0739195.