RISK MEASURES ON ORLICZ HEARTS

Coherent, convex, and monetary risk measures were introduced in a setup where uncertain outcomes are modeled by bounded random variables. In this paper, we study such risk measures on Orlicz hearts. This includes coherent, convex, and monetary risk measures on Lp‐spaces for 1 ≤p < ∞ and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. We prove that every coherent or convex monetary risk measure on an Orlicz heart which is real‐valued on a set with non‐empty algebraic interior is real‐valued on the whole space and admits a robust representation as maximal penalized expectation with respect to different probability measures. We also show that penalty functions of such risk measures have to satisfy a certain growth condition and that our risk measures are Luxemburg‐norm Lipschitz‐continuous in the coherent case and locally Luxemburg‐norm Lipschitz‐continuous in the convex monetary case. In the second part of the paper we investigate cash‐additive hulls of transformed Luxemburg‐norms and expected transformed losses. They provide two general classes of coherent and convex monetary risk measures that include many of the currently known examples as special cases. Explicit formulas for their robust representations and the maximizing probability measures are given.

[1]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[2]  M. Teboulle,et al.  Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming , 1986 .

[3]  Marc Teboulle,et al.  Penalty Functions and Duality in Stochastic Programming Via ϕ-Divergence Functionals , 1987, Math. Oper. Res..

[4]  Gerald A. Edgar,et al.  Stopping times and directed processes , 1992 .

[5]  D. Burkholder.  Stopping Times and Directed Processes, Encyclopedia of Mathematics and its Applications; Volume 47 - G. A. Edgar; L. Sucheston. , 1994 .

[6]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[7]  F. Delbaen.  Coherent risk measures , 2000 .

[8]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[9]  Hélyette Geman,et al.  Pricing and hedging in incomplete markets , 2001 .

[10]  S. Kusuoka.  On law invariant coherent risk measures , 2001 .

[11]  R. Rockafellar,et al.  Conditional Value-at-Risk for General Loss Distributions , 2001 .

[12]  Uwe Küchler,et al.  Coherent risk measures and good-deal bounds , 2001, Finance Stochastics.

[13]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[14]  Alexander Schied,et al.  Robust Preferences and Convex Measures of Risk , 2002 .

[15]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[16]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[17]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[18]  M. Frittelli,et al.  Putting order in risk measures , 2002 .

[19]  Marco Frittelli,et al.  Dynamic convex risk measures , 2004 .

[20]  G. P. Szegö,et al.  Risk measures for the 21st century , 2004 .

[21]  Terry J. Lyons,et al.  Stochastic finance. an introduction in discrete time , 2004 .

[22]  Jeremy Staum,et al.  Fundamental Theorems of Asset Pricing for Good Deal Bounds , 2004 .

[23]  F. Delbaen,et al.  Coherent and convex monetary risk measures for bounded càdlàg processes , 2004 .

[24]  Massimo Marinacci,et al.  PORTFOLIO SELECTION WITH MONOTONE MEAN‐VARIANCE PREFERENCES , 2009 .

[25]  Volker Krätschmer.  Robust representation of convex risk measures by probability measures , 2005, Finance Stochastics.

[26]  M. Frittelli,et al.  Law invariant convex risk measures , 2005 .

[27]  E. Jouini,et al.  Law Invariant Risk Measures Have the Fatou Property , 2005 .

[28]  Stan Uryasev,et al.  Generalized deviations in risk analysis , 2004, Finance Stochastics.

[29]  Patrick Cheridito,et al.  Coherent and convex monetary risk measures for unbounded càdlàg processes , 2004, Finance Stochastics.

[30]  Pavlo A. Krokhmal,et al.  Modeling and Implementation of Risk-Averse Preferences in Stochastic Programs Using Risk Measures , 2006 .

[31]  Alexander Shapiro,et al.  Optimization of Convex Risk Functions , 2006, Math. Oper. Res..

[32]  A. S. Cherny,et al.  Weighted V@R and its Properties , 2006, Finance Stochastics.

[33]  Damir Filipović,et al.  Monotone and cash-invariant convex functions and hulls , 2007 .

[34]  Alexander Schied,et al.  Optimal investments for risk- and ambiguity-averse preferences: a duality approach , 2006, Finance Stochastics.

[35]  RISK MEASURES FOR NON‐INTEGRABLE RANDOM VARIABLES , 2009 .