The paper explores non-convex risk measures and their characterizations.
problem Characterizing non-convex risk measures without convexity or weak convexity.
method Characterizes monetary risk measures as lower envelopes of families of convex or coherent risk measures, considering law-invariance and SSD-consistency.
result Unified representation theorems for law-invariant risk measures, including VaR.
Paper studies convex risk measures linked to optimization.
problem Risk assessment in finance and insurance.
method Investigates a wide class of risk measures on Orlicz spaces.
result Characterizes the dual of risk measures and provides complementary representations.
Study cash-subadditive risk measures without quasi-convexity.
problem Cash subadditivity without quasi-convexity.
method Represent cash-subadditive risk measures as lower envelopes of quasi-convex measures and introduce quasi-star-shapedness.
result General cash-subadditive risk measures can be represented as lower envelopes of quasi-convex measures.
Submodularity is studied for convex risk measures, including Expected Shortfall.
problem Characterizing submodularity in convex risk measures.
method Analyzing submodularity properties of law-invariant coherent risk measures, including Expected Shortfall and Value-at-Risk.
result AES is submodular only when it reduces to ES, and empirical analysis shows AES violations are less frequent than VaR and ES violations.
Paper introduces quasi-logconvex risk measures and their properties.
problem Characterizing and understanding new risk measures.
method Characterization through dual representation and properties of acceptance sets.
result Established dual representation and taxonomy of quasi-logconvex risk measures.
Introduces GG-convex risk measures and derives their dual representations.
problem Defining and studying GG-convex risk measures.
method Introduces GG-convex conjugate, derives dual representations, and studies Orlicz risk measures.
result Derives a general dual representation for GG-convex risk measures.
Unified framework for robust risk measures beyond convexity.
problem Developing risk measures for uncertainty beyond classical convexity.
method Constructing robust quasi-convex measures through uncertainty sets.
result Unified framework for robust quasi-convex risk measures.
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
In the present contribution we characterize law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (2006), we show that these risk measures can be identified with a class of generalized shortfall risk measures. As a direct consequence, we are …
Optimal hedging framework with variational preferences under convex risk measures.
problem Optimal hedging with variational preferences under convex risk measures.
method Theoretical hedging optimization framework with dual representation of risk measures and utilities.
result Derivation of optimality and indifference pricing conditions.
Paper characterizes star-shaped risk measures and their properties.
problem Characterizing risk measures in the presence of liquidity risk and competitive delegation.
method Characterization of star-shaped risk measures, study of their properties.
result Star-shaped risk measures include all practically used risk measures.
Constructs new elicitable risk measures with multiplicative scoring functions.
problem Defining new elicitable risk measures with specific properties.
method Constructs new elicitable risk measures using a multiplicative scoring function.
result Encompasses and allows construction of novel elicitable risk measures.
Optimal risk sharing without convex preferences using aggregate convexity.
problem Risk sharing among non-convex preferences.
method Aggregate convexity principles and Lyapunov convexity, combined with approximation arguments for law invariant risk measures.
result Derivation of a computationally tractable formula for the conjugate of the value function.
To provide a solid analytic foundation for the module approach to conditional risk measures, this paper establishes a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the (ε,λ)--topology and the locally L0-- convex topolo…
In this work we study the Lebesgue property for convex risk measures on the space of bounded càdlàg random processes (R∞). Lebesgue property has been defined for one period convex risk measures in \cite{Jo} and earlier had been studied in \cite{De} for coherent risk measures. We introduce and study th…
The paper refines and generalizes worst-case law invariant convex risk measures.
problem Developing robust convex risk measures under uncertainty sets.
method Generalizing closed forms for worst-case law invariant convex risk measures with uncertainty sets based on norms and moment constraints.
result Explicit closed forms for convex risk measures are developed and assessed through numerical simulations.
Novel convex risk measures aggregate multiple uncertain sources for insurance firms.
problem Managing risk from multiple uncertain sources in insurance.
method Proposes convex risk measures based on Fréchet mean.
result Allows for robust risk characterization and closed-form expressions.
The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measu…
Paper characterizes monotonic mean-deviation risk measures.
problem Developing consistent risk measures from mean-deviation models.
method Applying a risk-weighting function to the deviation part of a mean-deviation model.
result Characterizes monotonic mean-deviation measures as consistent risk measures.
The paper characterizes dynamic return and star-shaped risk measures via BSDEs.
problem Characterizing dynamic return and star-shaped risk measures.
method Characterization of star-shaped functionals and BSDEs.
result Existence of convex BSDEs with non-empty set of supersolutions.
Paper discusses natural quasiconvexity and its relation to decomposable sums in risk measures.
problem Understanding natural quasiconvexity and its implications in risk measures.
method Relates natural quasiconvexity to decomposable sums, proposes a general treatment of convexity index, and proves equivalence for certain spaces.
result Natural quasiconvexity and convexity are equivalent for conditional risk measures on Lp spaces under mild conditions. We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized …
Dual representations for robust risk measures and uncertainty sets.
problem Characterizing continuity of robust risk measures and their uncertainty sets.
method Develop dual representations for robust risk measures and uncertainty sets based on distinct geometric assumptions.
result Two dual frameworks for consolidated uncertainty sets are complementary, not interchangeable.
This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty …
Sharp bounds found for various risk measures using generalized FGM copulas.
problem Finding sharp bounds for risk measures in high dimensions.
method Proved that generalized FGM copulas form a convex polytope, used this structure to find bounds for risk measures.
result Sharp analytical bounds for convex risk measures in the class of generalized FGM copulas.
The paper analyzes elicitability of return risk measures and their scoring functions.
problem Elicitability of return risk measures and their scoring functions.
method Dual representation results for convex and geometrically convex return risk measures, axiomatic characterizations of Orlicz premia, and construction of strictly consistent scoring functions.
result Orlicz premia are the only elicitable return risk measures under different sets of conditions.
The paper develops a new approach to conditional risk measures using modular convex analysis.
problem Developing a new method for conditional risk measures.
method Random modular approach to conditional certainty equivalents and niveloids in the conditional L∞-space. result Retrieves a conditional variational formula for optimized certainty equivalents and applies it to the conditional entropic risk measure.
This paper deals with multidimensional dynamic risk measures induced by conditional g-expectations. A notion of multidimensional g-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and…
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.
Geometrically convex return risk measures on AM-algebras
problem Quantifying risk in time series analysis
method Extending return risk measures to general ordered vector spaces
result Establishing results on finiteness, continuity, separability, and dual and aggregation-based representations
Extends return risk measures to multiple assets, proving properties and comparing different risk models.
problem Evaluating risk in financial markets with multiple assets.
method Develops multi-asset return risk measures (MARRMs), analyzes their properties, and compares them with other risk models.
result Proves that a positively homogeneous MARRM is quasi-convex if and only if it is convex, and provides conditions to avoid inconsistent risk evaluations.
The paper studies dynamic star-shaped risk measures and their representation.
problem Representing dynamic star-shaped risk measures and their properties.
method Representation theorems for dynamic monetary and star-shaped risk measures.
result Dynamic star-shaped risk measures can be represented as the lower envelope of a family of dynamic convex risk measures.
New multivariate risk measures improve on univariate OCE methods.
problem Improving risk assessment in multivariate settings.
method Inspired by univariate OCE, introduces convex, monotonic, cash-invariant measures.
result Numerical algorithms provide error estimates for computations.
The paper analyzes risk assessment for cash flows in continuous time using the notion of convex risk measures for processes. By combining a decomposition result for optional measures, and a dual representation of a convex risk measure for bounded \cd processes, we show that this framework provides a systematic approach…
Simple conditions for comonotonic additive risk measures from acceptance sets.
problem Conditions for comonotonic additive risk measures from acceptance sets.
method Conditions on acceptance sets for induced comonotonic additive risk measures.
result Acceptance sets induce comonotonic additive risk measures if and only if the acceptance sets and their complements are stable under convex combinations of comonotonic random variables.
Paper proposes equal risk pricing for financial derivatives using convex risk measures.
problem Equal risk pricing and hedging in financial derivatives with convex risk measures.
method Established that the problem reduces to solving independently hedging problems for writer and buyer with zero initial capital. Provided dynamic programming equations for European and American options under Markovian decompositions of convex risk measures.
result Equal risk pricing leads to more similar and smaller risks for both writer and buyer compared to other pricing methods.
Develops a new method for risk diversification using dynamic risk measures.
problem Dynamic risk diversification in investment portfolios.
method Introduces dynamic risk contributions and a recursive optimization approach for coherent dynamic distortion risk measures.
result Dynamic risk budgeting strategies can be solved using deep learning.
The paper bounds payoffs and option prices in discrete models.
problem Measuring risk in discrete models and incomplete markets.
method Analytical and simulated bounds for payoff functions and option prices.
result Analytical and simulated bounds for European and American options.
Study dynamic risk measures with distributional uncertainty using optimal transport.
problem Risk robustification under distributional uncertainty in Markovian models.
method Characterize risk measures via convex monotone semigroups and optimal transport costs.
result Identify generator and correction terms for dynamic risk measures under different scaling regimes.
The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g. in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks…
Extends inf-convolution to countable risk measures for risk sharing.
problem Limited inf-convolution theory to finite sets of risk measures.
method Extends inf-convolution to countable sets, investigates properties and results.
result Generalizes known properties and results to countable case.
Set risk measures extend traditional risk measures to handle sets of positions.
problem Handling sets of positions with a single capital requirement.
method Developed an axiomatic framework for set risk measures, dual representation through topology and measures.
result Characterized worst-case set risk measures and provided examples.
Monitoring means to observe a system for any changes which may occur over time, using a monitor or measuring device of some sort. In this paper we formulate a problem of monitoring dates of maximal risk of a financial position. Thus, the systems we are going to observe arise from situations in finance. The measuring de…
The left tail of the implied volatility skew, coming from quotes on out-of-the-money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In …
Risk measures applied to dynamic Markov processes with varying risk aversion.
problem Investigating dynamic risk measures in Markov decision processes with varying risk aversion.
method Distributional viewpoint on law-invariant convex risk measures, applied to Markov decision processes with latent costs and random actions.
result Existence of optimal policies in finite and infinite time horizons under mild assumptions.
It is well known that Expected Shortfall (also called Average Value-at-Risk) is a convex risk measure, i. e. Expected Shortfall of a convex linear combination of arbitrary risk positions is not greater than a convex linear combination with the same weights of Expected Shortfalls of the same risk positions. In this shor…
This paper approaches the definition and properties of dynamic convex risk measures through the notion of a family of concave valuation operators satisfying certain simple and credible axioms. Exploring these in the simplest context of a finite time set and finite sample space, we find natural risk-transfer and time-co…
Develops risk measures for markets with constraints and costs.
problem Risk measures in markets with portfolio constraints and transaction costs.
method Embeds portfolio constraints and transaction costs into securities market; provides comprehensive analysis of risk measures properties.
result Establishes dual representations for convex and quasiconvex risk measures.