Coherent Risk Measures
Alexander Cherny
Department of Probability Theory
Faculty of Mechanics and Mathematics
Moscow State University
Tuesday, March 6, 2007
4:30 - 5:30 PM
Terman Engineering Center, Room 217
Abstract:
The classical measures of risk are variance and V@R. However, both have certain
drawbacks. In 1997, Artzner, Delbaen, Eber, and Heath [1] introduced the notion of
a coherent risk measure as a new way of quantifying risk. Being that, it provides new
approaches to various problems of finance, including pricing, hedging, and portfolio choice.
The theory of coherent risks is a very rapidly growing branch of financial mathematics.
As an example, the paper of Artzner, Delbaen, Eber, and Heath has the third download
rank at www.gloriamundi.org and there are about 350 papers on that
site citing it.
In the talk I will first give a basic introduction to coherent risks, providing a comparison
with the classical risk measures. Then I will speak about the best representatives of
this class. In a sense, the most convenient and important subclass of coherent risks is
Weighted V@R. I will describe basic properties of this class, including the description
of law invariant risk measures [6], strict diversification property [2], and the description
of factor monotone risk measures [3], which is closely connected to the notion of factor
risks introduced in [4] (these are aimed at measuring the risk of a portfolio driven by a
particular risk factor). Finally, I will introduce some new nice representatives of the class
Weighted V@R proposed in [5] and compare between different representatives of this class
using the notion of a coherent state-price density [5].
The talk will be completely self-contained.
References
[1] P. Artzner, F. Delbaen, J.-M. Eber, D. Heath. Thinking coherently. Risk, 10 (1997),
No. 11, p. 68-71.
[2] A.S. Cherny. Weighted V@R and its properties. Finance and Stochastics, 10 (2006),
p. 367-393.
[3] A.S. Cherny, P.G. Grigoriev. Dilatation monotone risk measures are law invariant.
Finance and Stochastics, 11 (2007), No. 2, 8 p.
[4] A.S. Cherny, D.B. Madan. Coherent measurement of factor risks. Preprint,
www.ssrn.com.
[5] A.S. Cherny, D.B. Madan. On measuring the degree of market effciency. Preprint,
www.ssrn.com.
[6] S. Kusuoka. On law invariant coherent risk measures. Advances in Mathematical
Economics, 3 (2001), p. 83-95.
Operations Research Colloquia: http://or.stanford.edu/oras_seminars.html