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SET-VALUED CASH SUB-ADDITIVE RISK MEASURES

Published online by Cambridge University Press:  11 April 2018

Fei Sun  and
Yijun Hu
Fei Sun
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, People's Republic of China E-mails: sunfei@whu.edu.cn; yjhu.math@whu.edu.cn
Yijun Hu
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, People's Republic of China E-mails: sunfei@whu.edu.cn; yjhu.math@whu.edu.cn
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Abstract

In this paper, we introduce a new class of set-valued risk measures which satisfies cash sub-additivity. Dual representation for them is provided. Moreover, we also investigate dynamic set-valued cash sub-additive risk measures and discuss the corresponding multi-portfolio time consistency. The equivalent characterization of the multi-portfolio time consistency is given. Finally, an example is also given to illustrate the introduction of set-valued cash sub-additive risk measures. The present paper can be considered as a set-valued extension of scalar cash sub-additive risk measures introduced by El Karouii and Ravanelli [8].

Keywords

cash additivecash sub-additivedynamicrisk measureset-valued
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 2 , April 2019 , pp. 241 - 257
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Ararat, C., Hamel, A.H. & Rudloff, B (2017). Set-valued shortfall and divergence risk measures. International Journal of Theoretical and Applied Finance 20(5): 1–48.Google Scholar
2.Artzner, P., Dellbaen, F., Eber, J.M. & Heath, D (1997). Thinking coherently. Risk 10: 68–71.Google Scholar
3.Artzner, P., Dellbaen, F., Eber, J.M. & Heath, D (1999). Coherent measures of risk. Mathematical Finance 9(3): 203–228.Google Scholar
4.Cascos, I. & Molchanov, I (2007). Multivariate risks and depth-trimmed regions. Finance and Stochastics 11(3): 373–397.Google Scholar
5.Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M. & Montrucchio, L (2011). Risk measures: rationality and diversification. Mathematical Finance 21(4): 743–774.Google Scholar
6.Cheridito, P. & Kupper, M (2011). Composition of time-consistent dynamic monetary risk measures in discrete time. International Journal of Theoretical and Applied Finance 14: 137–162.Google Scholar
7.Cont, R., Deguest, R. & He, X.D (2013). Loss-based risk measures. Statistics and Risk Modeling with Applications in Finance and Insurance 30(2): 133–167.Google Scholar
8.EL Karouii, N., Ravanelli, C (2009). Cash subadditive risk measures and Interest rate ambiguity. Mathematical Finance 19: 561–590.Google Scholar
9.Farkas, W., Koch-Medina, P. & Munari, C (2015). Measuring risk with multiple eligible assets. Mathematics and Financial Economics 9(1): 3–27.Google Scholar
10.Feinstein, Z. & Rudloff, B (2013). Time consistency of dynamic risk measures in markets with transaction costs. Quantitative Finance 13(9): 1473–1489.Google Scholar
11.Feinstein, Z. & Rudloff, B (2015a). Multi-portfolio time consistency for set-valued convex and coherent risk measures. Finance and Stochastics 19: 67–107.Google Scholar
12.Feinstein, Z. & Rudloff, B. (2015b). A comparison of techniques for dynamic multivariate risk measures. In Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B. & Schrage, C. (eds) Set Optimization and Applications in Finance. The State of the Art, Springer PROMS series, Vol. 151, 3–41. ISBN: 978-3-662-48668-9.Google Scholar
13.Föllmer, H. & Schied, A (2002). Convex measures of risk and trading constrains. Finance and Stochastics 6: 429–447.Google Scholar
14.Frittelli, M., Rosazza, , Gianin, E (2002). Putting order in risk measures. Journal of Banking and Finance 26: 1473–1486.Google Scholar
15.Hamel, A.H (2009). A duality theory for set-valued functions I: Fenchel conjugation theory. Set-valued and Variational Analysis 17(2): 153–182.Google Scholar
16.Hamel, A.H. & Heyde, F (2010). Duality for set-valued measures of risk. SIAM Journal on Finance Mathematics 1(1): 66–95.Google Scholar
17.Hamel, A.H., Heyde, F. & Rudloff, B (2011). Set-valued risk measures for conical market models. Mathematics and Financial Economics 5(1): 1–28.Google Scholar
18.Hamel, A.H., Rudloff, B. & Yankova, M (2013). Set-valued average value at risk and its computation. Mathematics and Financial Economics 7(2): 229–246.Google Scholar
19.Jouini, E., Meddeb, M. & Touzi, N (2004). Vector-valued coherent risk measures. Finance and Stochastics 8(4): 531–552.Google Scholar
20.Labuschagne, C.C.A., Offwood-Le Roux, T.M (2014). Representations of set-valued risk measures definded on the l-tensor product of Banach lattices. Positivity 18(3): 619–639.Google Scholar
21.Lepinette, E. & Molchanov, I. (2016). Risk arbitrage and hedging to acceptability, arXiv: 1605.07884v2 [q-fin.MF] 15 Jun.Google Scholar
22.Mastrogiacomo, E. & Rosazza Gianin, E. (2015). Time-consistency of cash-subadditive risk measures, arXiv: 1512.03641v1 [q-fin.RM] 11 Dec.Google Scholar
23.Molchanov, I. & Cascos, I (2016). Multivariate risk measures: a constructive approach based on selections. Mathematical Finance 26(4): 867–900.Google Scholar
24.Ng, K.W., Yang, H. & Zhang, L (2004). Ruin probability under compound poisson models with random discount factor. PROBAB ENG INFORM SC. 18: 55–70.Google Scholar
25.Sun, F., Chen, Y.H. & Hu, Y.J (2018). Set-valued loss-based risk measures. Positivity. https://doi.org/10.1007/s11117-017-0550-5.Google Scholar
26.Tahar, I. & Lépinette, E (2014). Vector valued coherent risk measure processes. International Journal of Theoretical and Applied Finance 17, 1450011.Google Scholar