Original Paper

Finance and Stochastics

, Volume 6, Issue 4, pp 429-447

First online:

Convex measures of risk and trading constraints

  • Hans FöllmerAffiliated withInstitut für Mathematik, Humboldt-Universität, Unter den Linden 6, 10099 Berlin, Germany (e-mail: {foellmer,schied}@mathematik.hu-berlin.de)
  • , Alexander SchiedAffiliated withInstitut für Mathematik, Humboldt-Universität, Unter den Linden 6, 10099 Berlin, Germany (e-mail: {foellmer,schied}@mathematik.hu-berlin.de)

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Abstract.

We introduce the notion of a convex measure of risk, an extension of the concept of a coherent risk measure defined in Artzner et al. (1999), and we prove a corresponding extension of the representation theorem in terms of probability measures on the underlying space of scenarios. As a case study, we consider convex measures of risk defined in terms of a robust notion of bounded shortfall risk. In the context of a financial market model, it turns out that the representation theorem is closely related to the superhedging duality under convex constraints.

Key words: Risk measure, convex measure of risk, shortfall, trading constraints, efficient hedging
JEL Classification: G18, G11, C60, C61
Mathematics Subject Classification (2000): 46N10, 91B16, 91B28, 91B30