Introduction to Stochastic Finance pp 263-305 | Cite as

# Static Risk Measures

## Abstract

The financial market faces risks arising from many types of uncertain losses, including market risk, credit risk, liquidity risk, operational risk, etc. In 1988, the Basel Committee on Banking Supervision proposed measures to control credit risk in banking. A risk measure called the *value-at-risk*, acronym VaR, became, in the 1990s, an important tool of risk assessment and management for banks, securities companies, investment funds, and other financial institutions in asset allocation and performance evaluation. The VaR associated with a given confidence level for a venture capital is the upper limit of possible losses in the next certain period of time. In 1996 the Basel Committee on Banking Supervision endorsed the VaR as one of the acceptable methods for the bank’s internal risk measure. However, due to the defects of VaR, a variety of new risk measures came into being. This chapter focuses on the representation theorems for static risk measures. For an overview of the subject we refer to Song and Yan (2009b).

## References

- Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Thinking coherently. Risk
**10**, 68–71 (1997)Google Scholar - Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Financ.
**9**(3), 203–228 (1999)MathSciNetCrossRefGoogle Scholar - Choquet, G.: Theory of capacities. Ann. Inst. Fourier Grenoble. 5, 131–295 (1953–54)MathSciNetCrossRefGoogle Scholar
- Dana, R.-A.: A representation result for concave Schur concave functions. Math. Financ.
**15**, 615–634 (2005)MathSciNetCrossRefGoogle Scholar - Denneberg, D.: Non-additive Measure and Integral. Kluwer Academic, Boston (1994)CrossRefGoogle Scholar
- Dhaene, J., Vanduffel, S., Goovaerts, M.J., Kaas, R., Tang, Q., Vyncke, D.: Risk measures and comotononicity: a review. Stoch. Model.
**22**, 573–606 (2006)CrossRefGoogle Scholar - Embrechts, P., McNeil, A.J., Straumann, D.: Correlation and dependence in risk management: properties and pitfalls. In: Dempster, M., Moffatt, H.K. (eds.) Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge (2000)Google Scholar
- Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Financ. Stoch.
**6**(4), 429–447 (2002)MathSciNetCrossRefGoogle Scholar - Föllmer, H., Schied, A.: Stochastic Finance, an Introduction in Discrete Time, 2nd revised and extended edition. Welter de Gruyter, Berlin, New York (2004)Google Scholar
- Frittelli, M., Gianin, E.R.: Putting order in risk measures. J. Bank. Financ.
**26**(7), 1473–1486 (2002)CrossRefGoogle Scholar - Heyde, C.C., Kou, S.G., Peng, X.H.: What is a good risk measure: bridging the gaps between data, coherent risk measure, and insurance risk measure. Preprint (2006)Google Scholar
- Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ.
**3**, 83–95 (2001)MathSciNetCrossRefGoogle Scholar - Song, Y., Yan, J.A.: The representations of two types of functionals on \(L^\infty (\varOmega , {\cal F})\) and \(L^\infty (\varOmega , {\cal F}, P)\). Sci. China, Ser. A Math.
**49**(10), 1376–1382 (2006)Google Scholar - Song, Y., Yan, J.A.: Risk measures with co-monotonic subadditivity or convexity and respecting stochastic orders. Insur. Math. Econ.
**45**, 459–465 (2009a)CrossRefGoogle Scholar - Yan, J.A.: A short presentation of Choquet integral. In: Duan, J. et al., (eds.) Recent Development in Stochastic Dynamics and Stochastic Analysis. Interdisciplinary Mathematical Science, vol. 8, pp. 269–291. Wold Scientific, New Jersey (2010)CrossRefGoogle Scholar