Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T07:11:46.948Z Has data issue: false hasContentIssue false
  • English
  • Français

Distribution-Invariant Risk Measures, Entropy, and Large Deviations

Part of:Sampling theory, sample surveysSufficiency and informationManifoldsMathematical economics

Published online by Cambridge University Press:  14 July 2016

Stefan Weber
Stefan Weber*
Affiliation:
Cornell University
*
∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, 279 Rhodes Hall, Ithaca, NY 14853, USA. Email address: sweber@orie.cornell.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.

Keywords

Risk measureaverage value at riskshortfall riskMonte Carlolarge deviations principleSanov's theoremrelative entropy

MSC classification

Primary:91B30: Risk theory, insurance49Q20: Variational problems in a geometric measure-theoretic setting62B10: Information-theoretic topics62D05: Sampling theory, sample surveys

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007, pp. 16 - 40
Copyright
Copyright © Applied Probability Trust 2007 

References

Aliprantis, C. D. and Border, K. C. (1999). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer, Berlin.Google Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9, 203–228.Google Scholar
Csiszár, I. (1975). I-divergence geometry of probability distributions and minimization problems. Ann. Prob. 3, 146–158.CrossRefGoogle Scholar
Delbaen, F. (2002). Coherent risk measures on general probability spaces. In Advances in Finance and Stochastics, eds Sandmann, K. and Schönbucher, P. J., Springer, Berlin, pp. 1–38.Google Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications. Springer, New York.Google Scholar
Dunkel, J. and Weber, S. (2005). Efficient Monte Carlo methods for risk measures. Working paper, Cornell University. Available at http://people.orie.cornell.edu/∼sweber/research.htm.Google Scholar
Föllmer, H. and Schied, A. (2002). Robust preferences and convex meaures of risk. In Advances in Finance and Stochastics, eds Sandmann, K. and Schönbucher, P. J., Springer, Berlin, pp. 39–56.Google Scholar
Föllmer, H. and Schied, A. (2004). Stochastic Finance. An Introduction in Discrete Time, 2nd edn. De Gruyter, Berlin.Google Scholar
Frittelli, M. and Rosazza, G. E. (2002). Putting order in risk measures. J. Banking Finance 26, 1473–1486.CrossRefGoogle Scholar
Fu, M. C., Jin, X. and Xiong, X. (2003). Probabilistic error bounds for simulation quantile estimators. Manag. Sci. 49, 230–246.Google Scholar
Giesecke, K., Schmidt, T. and Weber, S. (2005). Measuring the risk of extreme events. Working paper, Cornell University. Available at http://people.orie.cornell.edu/∼sweber/research.htm.Google Scholar
Jouini, E., Schachermayer, W. and Touzi, N. (2006). Law invariant risk measures have the Fatou property. Adv. Math. Econom. 9, 49–71.CrossRefGoogle Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Kusuoka, S. (2001). On law invariant coherent risk measures. Adv. Math. Econom. 3, 83–95.Google Scholar
Schmeidler, D. (1986). Integral representation without additivity. Proc. Amer. Math. Soc. 97, 255–261.Google Scholar
Weber, S. (2004). Measures and models of financial risk. Doctoral Thesis, Humboldt-Universität zu Berlin.Google Scholar
Weber, S. (2006). Distribution-invariant risk measures, information, and dynamic consistency. Math. Finance 16, 419–442.Google Scholar