Journal of Applied Probability

Distribution-invariant risk measures, entropy, and large deviations

Stefan Weber

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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.

Article information

J. Appl. Probab. Volume 44, Number 1 (2007), 16-40.

First available in Project Euclid: 30 March 2007

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Zentralblatt MATH identifier

Primary: 91B30: Risk theory, insurance 49Q20: Variational problems in a geometric measure-theoretic setting 62B10: Information-theoretic topics [See also 94A17] 62D05: Sampling theory, sample surveys 91B28

Risk measure average value at risk shortfall risk Monte Carlo large deviations principle Sanov's theorem relative entropy


Weber, Stefan. Distribution-invariant risk measures, entropy, and large deviations. J. Appl. Probab. 44 (2007), no. 1, 16--40. doi:10.1239/jap/1175267161.

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