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On Approximating Law-Invariant Comonotonic Coherent Risk Measures

Published online by Cambridge University Press:  09 August 2013

Yumiharu Nakano
Yumiharu Nakano*
Affiliation:
Graduate School of Innovation Management, Tokyo Institute of Technology, 2-12-1 W9-117 Ookayama 152-8552 Tokyo, Japan
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Abstract

The optimal quantization theory is applied for approximating law-invariant comonotonic coherent risk measures. Simple Lp-norm estimates for the risk measures provide the rate of convergence of that approximation as the number of quantization points goes to infinity.

Keywords

Coherent risk measuresoptimal quantizationaverage value-at-riskcomonotonicity
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 1 , May 2012 , pp. 343 - 353
Copyright
Copyright © International Actuarial Association 2012

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References

Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. J. Banking Finance 26, 1487–1503.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measurement of risk. Math. Finance 9, 203–228.CrossRefGoogle Scholar
Delbaen, F. (2002) Coherent measures of risk on general probability spaces. In Advances in Finance and Stochastics, Essays in Honor of Dieter Sondermann (eds. Sandmann, K. and Schönbucher, P. J.), pp. 1–37, Springer, Berlin.Google Scholar
Feuerverger, A. and Wong, A.C. (2000) Computation of Value at Risk for nonlinear portfolios. J. Risk 3, 37–55.CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2004) Stochastic Finance: An Introduction in Discrete Time. 2nd ed., Walter de Gruyter, Berlin.CrossRefGoogle Scholar
Glasserman, P. (2004) Monte Carlo methods in financial engineering. Springer, New York.Google Scholar
Graf, S. and Luschgy, H. (2000) Foundation of quantization for probability distributions. Springer, Berlin.CrossRefGoogle Scholar
Inoue, A. (2003) On the worst conditional expectation. J. Math. Anal. Appl. 286, 237–247.CrossRefGoogle Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001) Modern actuarial risk theory. Kluwer, Boston.Google Scholar
Kaina, M. and Rüschendorf, L. (2009) On convex risk measures on L p -spaces. Math. Methods Oper. Res. 69, 475–495.CrossRefGoogle Scholar
Kusuoka, S. (2001) On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95.CrossRefGoogle Scholar
Landsman, Z.M. and Valdez, E.A. (2003) Tail conditional expectations for elliptical distributions. N. Amer. Actuarial J. 7, 55–71.CrossRefGoogle Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative risk management: concepts, techniques and tools. Princeton University Press, Princeton.Google Scholar
Pagès, G. (1997) A space quantization method for numerical integration. J. Comput. Appl. Math. 89, 1–38.CrossRefGoogle Scholar
Pagès, G., Pham, H. and Printems, J. (2004) An optimal Markovian quantization algorithm for multidimensional stochastic control problems. Stoch. Dyn. 4, 501–545.CrossRefGoogle Scholar
Pagès, G. and Printems, J. (2003) Optimal quadratic quantization for numerics: the Gaussian case. Monte Carlo Methods Appl. 9, 135–168.CrossRefGoogle Scholar
Yamai, Y. and Yoshiba, T. (2002) Comparative analyses of expected shortfall and value-at-risk: their estimation error, decomposition, and optimization. Monet. Econ. Stud. 20, 87–121.Google Scholar
Young, V.R. (2004) Premium principles. In Encyclopedia of actuarial sciences. Wiley, New York.Google Scholar