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Multivariate risk measures in the non-convex setting. (English) Zbl 1458.91229

When evaluating the risk of a random vector \(X\), describing positions in several assets, e.g., in several currencies, it is indispensable to take into account eventual transfers between its components. If transfers are forbidden, then the risk of \(X\) is the vector of risk measures calculated for its components. A different situation arises when transfers are allowed, and then one should normally take into account transaction costs associated with them. In this case, it is natural to evaluate risks of all admissible positions which can be attained from \(X\) by exchanges and call \(X\) acceptable if there exists a transfer (a selection of the set of attainable positions) that converts \(X\) to a vector with all individually acceptable components.
The attainable positions form a random closed set, which is convex for most transaction costs models. The paper under review deals with the non-convex case, which arises, e.g., if fixed transaction costs are imposed. For this, it follows the approach based on considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex sets of attainable positions are presented. A particular attention is devoted to the special case of fixed transaction costs.

MSC:

91G70 Statistical methods; risk measures
60D05 Geometric probability and stochastic geometry
91G99 Actuarial science and mathematical finance
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References:

[1] F. Delbaen, Monetary Utility Functions, Osaka University Press, Osaka, 2012. · Zbl 1187.91095
[2] H. Föllmer and A. Schied, Stochastic Finance. An Introduction in Discrete Time, 2nd ed., De Gruyter, Berlin, 2004. · Zbl 1126.91028
[3] A. Haier, I. Molchanov and M. Schmutz, Intragroup transfers, intragroup diversification and their risk assessment, Ann. Finance 12 (2016), 363-392. · Zbl 1398.91330
[4] A. H. Hamel and F. Heyde, Duality for set-valued measures of risk, SIAM J. Financial Math. 1 (2010), 66-95. · Zbl 1197.91112
[5] A. H. Hamel, F. Heyde and B. Rudloff, Set-valued risk measures for conical market models, Math. Financ. Economics 5 (2011), 1-28. · Zbl 1275.91077
[6] E. Jouini, W. Schachermayer and N. Touzi, Law invariant risk measures have the Fatou property, Adv. Math. Econ. 9 (2006), 49-71. · Zbl 1198.46028
[7] Y. M. Kabanov and M. Safarian, Markets with Transaction Costs. Mathematical Theory, Springer, Berlin, 2009. · Zbl 1186.91006
[8] M. Kaina and L. Rüschendorf, On convex risk measures on \(L^p\)-spaces, Math. Methods Oper. Res. 69 (2009), 475-495. · Zbl 1168.91019
[9] J. Leitner, Balayage monotonous risk measures, Int. J. Theor. Appl. Finance 7 (2004), 887-900. · Zbl 1090.91053
[10] E. Lépinette and I. Molchanov, Conditional cores and conditional convex hulls of random sets, J. Math. Anal. Appl. (2019), 10.1016/j.jmaa.2019.05.010. · Zbl 1479.60019
[11] I. Molchanov, Theory of Random Sets, 2nd ed., Springer, London, 2017. · Zbl 1406.60006
[12] I. Molchanov and I. Cascos, Multivariate risk measures: A constructive approach based on selections, Math. Finance 26 (2016), 867-900. · Zbl 1368.91183
[13] N. Sagara, A Lyapunov-type theorem for nonadditive vector measures, Modeling Decisions for Artificial Intelligence—MDAI 2009, Springer, Berlin (2009), 72-80. · Zbl 1273.28010
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