Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes

By Patrick Cheridito, Freddy Delbaen, Michael Kupper. 2004

370 citations recorded by the source corpus.

We study dynamic monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a dynamic risk measure time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time. We show that this condition translates into a decomposition property for the corresponding acceptance sets, and we demonstrate how time-consistent dynamic monetary risk measures can be constructed by pasting together one-period risk measures. For conditional coherent and convex monetary risk measures, we provide dual representations of Legendre--Fenchel type based on linear functionals induced by adapted increasing processes of integrable variation. Then we give dual characterizations of time-consistency for dynamic coherent and convex monetary risk measures. To this end, we introduce a concatenation operation for adapted increasing processes of integrable variation, which generalizes the pasting of probability measures. In the coherent case, time-consistency corresponds to stability under concatenation in the dual. For dynamic convex monetary risk measures, the dual characterization of time-consistency generalizes to a condition on the family of convex conjugates of the conditional risk measures at different times. The theoretical results are applied by discussing the time-consistency of various specific examples of dynamic monetary risk measures that depend on bounded discrete-time processes.

Subfields: Economics and Econometrics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/c1a2c5287a6fb3137fdd24c47ea9e1af33fec8d8

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Citations in the corpus

Dual characterization of properties of risk measures on Orlicz hearts (2008, background): “For the convex monetary case, see Lemma 3.22 and Theorem 3.23 in Cheridito et al. (2006) .”
Lectures on Stochastic Programming: Modeling and Theory (2009, background): “In the finance literature time consistency is usually defined via the composition property of dynamic risk measures; see, Artzner, Delbaen, Eber, Heath and Ku [9], BionNadal [20, 21], Cheridito, Delbaen and Kupper [38], Cheridito and Kupper [39], Föllmer and Penner [79], Frittelli and Scandolo [81],”
Convex risk measures and the dynamics of their penalty functions (2006, methodology): “3 of [6] shows that monotonicity and cash invariance imply this local property.”
Conditional and dynamic convex risk measures (2005, background): “Key words: Conditional convex risk measure, robust representation, entropic risk measures, dynamic convex risk measures, time-consistency JEL Classification: D81 MS Classification (2000): 91B16, 91B70, 91B30, 46A20 This version: February 28, 2005”
Time consistent dynamic risk processes (2009, background): “[7] by a condition on the acceptance set and also a “concatenation condition”.”
Pricing with Coherent Risk (2006, background): “, [20], [31], [32], [38], [40], [53] 1 and the reviews [21], [33; Ch. 4], [47]. Currently, one of the major tasks is the problem of proper risk measurement in the dynamic setting; see, in particular, [12], [25], [36], [45], and [46]. The theory of coherent risk measures is important not only for ris”
Risk-averse dynamic programming for Markov decision processes (2010, background): “Particularly relevant for us are the duality theory [1,10,15–17,34,43,45,46], and the theory of conditional and dynamic risk measures [2,8,11,14,18,19,28,38,48,40,45,47].”
Separation and duality in locally L 0 -convex modules (2009, background): “The restriction ρt to bounded risks, that is L∞(F ), has been analyzed in [2,5,7,9].”
Risk measuring under model uncertainty (2010, background): “Dynamic risk measures have then been studied in many papers, among them [13], [11] [24] [7] [8] [30].”
Optimal Stopping for Dynamic Convex Risk Measures (2009, background): “5] and Cheridito et al. [2006, Sections 5.2 and 5.3]. Delbaen [2006] and Karatzas and Zamfirescu [2006], on the other hand, considered continuous-time optimal stopping problems in which the essential infimum over the stopping times in (1.1) is replaced by an essential supremum. The controller-and-st”
Bid-Ask Dynamic Pricing in Financial Markets with Transaction Costs and Liquidity Risk (2007, methodology): “The close notion of monetary utility functions for processes in a discrete time setting has been studied by Cheridito, Delbaen and Kupper [18].”
Recent progress in random metric theory and its applications to conditional risk measures (2010, background): “Based on this kind of conditional convex risk measure, the corresponding dynamic risk measures of L−type were developed in [11, 18], see [6, 7] for L−type of dynamic risk measures of bounded stochastic processes.”
Dynamic risk measures (2010, background): “As its static counterpart, a conditional convex risk measure can be represented as the worst conditional expected loss over a class of suitably penalized probability measures; see [31, 30, 17, 6, 26, 12].”
Time-Inconsistency of VaR and Time-Consistent Alternatives (2007, neutral): “Similar or equivalent conditions have been studied for dynamic preferences; see for instance, Koopmans (1960), Kreps and Porteus (1978), Wang (2003), Epstein and Schneider (2003), Maccheroni et al (2006). In the context of risk measurement time-consistency has been investigated among others, by Ried”