Measuring Risk with Multiple Eligible Assets
Mathematics and Financial Economics, 9 (2015)
28 Pages Posted: 25 Jan 2012 Last revised: 5 Nov 2015
Date Written: March 3, 2014
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: Suggested Citation