Measuring Risk with Multiple Eligible Assets

Mathematics and Financial Economics, 9 (2015)

28 Pages Posted: 25 Jan 2012 Last revised: 5 Nov 2015

Walter Farkas

University of Zurich - Department of Banking and Finance; Swiss Finance Institute; ETH Zurich

Pablo Koch-Medina

University of Zurich - Department of Banking and Finance; Swiss Finance Institute

Cosimo Munari

University of Zurich - Department of Banking and Finance; Swiss Finance Institute

Date Written: March 3, 2014

Abstract

The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.

Keywords: risk measures, multiple eligible assets, acceptance sets, dual representations, set-valued risk measures, superhedging with shortfall risk, optimal risk sharing

JEL Classification: C60, G11, G22

Suggested Citation

Farkas, Walter and Koch-Medina, Pablo and Munari, Cosimo, Measuring Risk with Multiple Eligible Assets (March 3, 2014). Mathematics and Financial Economics, 9 (2015). Available at SSRN: https://ssrn.com/abstract=1989077 or http://dx.doi.org/10.2139/ssrn.1989077

Walter Farkas (Contact Author)

University of Zurich - Department of Banking and Finance ( email )

Schönberggasse 1
Zürich, 8001
Switzerland
+41-44-634 3953 (Phone)
+41-44-634 4345 (Fax)

HOME PAGE: http://https://people.math.ethz.ch/~farkas/

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

ETH Zurich ( email )

Rämistrasse 101
ZUE F7
Zürich, 8092
Switzerland

Pablo Koch-Medina

University of Zurich - Department of Banking and Finance ( email )

Plattenstrasse 14
Zürich, 8032
Switzerland

Swiss Finance Institute ( email )

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

Cosimo Munari

University of Zurich - Department of Banking and Finance ( email )

Schönberggasse 1
Zürich, 8001
Switzerland

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

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