Hostname: page-component-6766d58669-zlvph Total loading time: 0 Render date: 2026-05-25T02:12:22.083Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Information

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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