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Cash Sub-additive Risk Measures and Interest Rate Ambiguity

34 Pages Posted: 24 Jan 2007 Last revised: 8 Jul 2008

Nicole El Karoui

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees

Claudia Ravanelli

Center for Finance and Insurance

Date Written: April 4, 2008

Abstract

A new class of risk measures called cash sub-additive risk measures is introduced to assess the risk of future financial, nonfinancial and insurance positions. The debated cash additive axiom is relaxed into the cash sub-additive axiom to preserve the original difference between the numeraire of the current reserve amounts and future positions. Consequently, cash sub-additive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented and in such contexts cash additive risk measures cannot be used. Several representations of the cash sub-additive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sub-linear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash sub-additive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash sub-additive risk measure.

Keywords: Risk measures, Fenchel-Legendre transform, model uncertainty, inf-convolution, backward stochastic differential equations

JEL Classification: D81, G10, G13

Suggested Citation

El Karoui, Nicole and Ravanelli, Claudia, Cash Sub-additive Risk Measures and Interest Rate Ambiguity (April 4, 2008). Swiss Finance Institute Research Paper No. 08-09. Available at SSRN: https://ssrn.com/abstract=959092 or http://dx.doi.org/10.2139/ssrn.959092

Nicole El Karoui

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees ( email )

Palaiseau Cedex, 91128
France
(33) 1 69 33 41 48 (Phone)
(33) 1 69 33 70 31 (Fax)

Claudia Ravanelli (Contact Author)

Center for Finance and Insurance ( email )

Plattenstrasse 14
Zürich, 8032
Switzerland
+41 44 634 29 81 (Phone)

HOME PAGE: http://www.bf.uzh.ch/cms/en/ravanelli.claudia.html

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