Top

Mathematics and Financial Economics

Published in:

01-03-2015

Pareto optimal allocations and optimal risk sharing for quasiconvex risk measures

Authors: Elisa Mastrogiacomo, Emanuela Rosazza Gianin

Published in: Mathematics and Financial Economics | Issue 2/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The main goal of this paper is to generalize the characterization of Pareto optimal allocations known for convex risk measures (see, among others, Jouini et al., in Math Financ 18(2):269–292, 2008 and Filipovic and Kupper, in Int J Theor Appl Financ, 11:325–343, 2008) to the wider class of quasiconvex risk measures. Following the approach of Jouini et al., in Math Financ 18(2):269–292, 2008 for convex risk measures, in the quasiconvex case we provide sufficient conditions for allocations to be (weakly) Pareto optimal in terms of exactness of the so-called quasiconvex inf-convolution as well as an existence result for weakly Pareto optimal allocations. Moreover, we give a necessary condition for weakly optimal risk sharing that is also sufficient under cash-additivity of at least one between the risk measures.

previous article Optimal acquisition of a partially hedgeable house

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

\(\pi \) is said to be Lipschitz on the strict lower level set \(\{\pi < \pi (\xi ) \}\) if there exists \(c>0\) such that \(\vert \pi (X)-\pi (Y) \vert \le c \Vert X-Y \Vert _{\infty }\) for any \(X,Y \in L^{\infty }\) such that \(\pi (X), \pi (Y) < \pi (\xi )\). Roughly speaking, this means that \(\pi \) is \(\Vert \cdot \Vert _{\infty }\)-continuous on the strict lower level set.

Acciaio, B.: Optimal risk sharing with non-monotone monetary functionals. Financ. Stoch. 11, 267–289 (2007)CrossRefMATHMathSciNet

Barrieu, P., El Karoui, N.: Inf-convolution of risk measures and optimal risk transfer. Financ. Stoch. 9, 269–298 (2005)CrossRefMATH

Barrieu, P., El Karoui, N.: Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Carmona, R. (ed.) Indifference Pricing, pp. 77–144. Princeton University Press, Princeton (2005)

Borch, Z.: Equilibrium in a reinsurance market. Econometrica 30, 424–444 (1962)CrossRefMATH

Bühlmann, H., Jewell, S.: Optimal risk exchanges. Astin Bull. 10, 243–262 (1979)

Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Risk measures: rationality and diversification. Math. Financ. 21(4), 743–774 (2011)MATHMathSciNet

Chateauneuf, A., Dana, R.-A., Tallon, J.-M.: Optimal risk-sharing rules and equilibria with Choquet-expected-utility. J. Math. Econ. 34, 191–214 (2000)CrossRefMATHMathSciNet

Delbaen, F.: Coherent risk measures on general probability spaces. In: Sandmann, K., Schönbucher, P.J. (eds.) Advances in Finance and Stochastics, pp. 1–37. Springer-Verlag, New York (2002)CrossRef

Deprez, O., Gerber, H.U.: On convex principles of premium calculation. Insurance 4, 179–189 (1985)MATHMathSciNet

10.

Drapeau, S., Kupper, M.: Risk preferences and their robust representation. Math. Oper. Res. 38, 28–62 (2013)CrossRefMATHMathSciNet

11.

Elqortobi, A.: Inf-convolution quasi-convexe des fonctionnelles positives. Oper. Res. 26(4), 301–311 (1992)MATHMathSciNet

12.

Filipovic, D., Kupper, M.: Equilibrium prices for monetary utility functions. Int. J. Theor. Appl. Financ. 11, 325–343 (2008)CrossRefMATHMathSciNet

13.

Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Financ. Stoch. 6, 429–447 (2002a)CrossRefMATH

14.

Frittelli, M., Maggis, M.: Dual representation of quasiconvex conditional maps. SIAM J. Financ. Math. 2, 357–382 (2011)CrossRefMATHMathSciNet

15.

Frittelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Bank. Financ. 26, 1473–1486 (2002)CrossRef

16.

Jouini, E., Schachermayer, W., Touzi, N.: Optimal risk sharing for law invariant monetary utility functions. Math. Financ. 18(2), 269–292 (2008)CrossRefMATHMathSciNet

17.

Klöppel, S., Schweizer, M.: Dynamic utility indifference valuation via convex risk measures. Math. Financ. 17(4), 599–627 (2007)CrossRefMATH

18.

Luc, D.T., Volle, M.: Level sets, infimal convolution and level addition. J. Optim. Theory Appl. 94(3), 695–714 (1997)CrossRefMATHMathSciNet

19.

Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Et Appl. 49, 109–154 (1970)MATHMathSciNet

20.

Penot, J.-P.: Characterization of solution sets of quasiconvex programs. J. Optim. Theory Appl. 117(3), 627–636 (2003)CrossRefMATHMathSciNet

21.

Penot, J.-P.: Are generalized derivatives useful for generalized convex functions? In: Crouzeix, J.-P., et al. (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results, pp. 3–59. Kluwer, Dordrecht (1998)CrossRef

22.

Penot, J.-P., Volle, M.: On quasiconvex duality. Math. Oper. Res. 15, 597–625 (1990)CrossRefMATHMathSciNet

23.

Penot, J.-P., Zalinescu, C.: Elements of quasiconvex subdifferential calculus. J. Convex Anal. 7(2), 243–269 (2000)MATHMathSciNet

24.

Ravanelli, C., Svindland, G.: Comonotone Pareto optimal allocations for law invariant robust utilities on \(L^1\). Financ. Stoch. 18, 249–269 (2014)CrossRefMATHMathSciNet

25.

Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATH

26.

Seeger, A., Volle, M.: On a convolution operation obtained by adding level sets: classical and new results. Oper. Res. 29(2), 131–154 (1995)MATHMathSciNet

27.

Volle, M.: Duality for the level sum of quasiconvex functions and applications. Control Optim. Calc. Var. 3, 329–343 (1998)CrossRefMATHMathSciNet

Title: Pareto optimal allocations and optimal risk sharing for quasiconvex risk measures
Authors: Elisa Mastrogiacomo
Emanuela Rosazza Gianin
Publication date: 01-03-2015
Publisher: Springer Berlin Heidelberg
Published in: Mathematics and Financial Economics / Issue 2/2015
Print ISSN: 1862-9679
Electronic ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-014-0139-8