arXiv subjects

Felix-Benedikt Liebrich

Publications and source records attributed to Felix-Benedikt Liebrich.

11 recordsLinked to original sources

arXiv2026

Meyer risk measures

Risk measures summarize the risk profile of financial positions in a single metric, which allows their comparison and supports investment decisions. When they respect common stochastic orders such as second-order stochastic dominance (SSD), they rest on sound conceptual grounds. Since SSD can be overly conservative and rigid, we study Meyer risk measures. Those are monetary risk measures that are consistent with stochastic orders induced by a threshold utility function $v$, where the defining test utilities are at least as risk averse as $v$. The flexibility in choosing $v$ allows to encompass a wide range of stochastic orders, including SSD. Our contribution is threefold. First, we show that Meyer risk measures whose threshold utility $v$ is CARA admit a lower envelope representation via adjusted risk measures, and analyse how they extend and compare to classical risk measures. Second, we show that monotone additive statistics, recently introduced in decision theory, are Meyer risk measures under mild conditions. Third, impossibility results for the situation beyond the CARA class establish that monetary risk measures compatible to the associated stochastic orders either fail to exist or cannot satisfy key properties simultaneously. Throughout, we highlight practical implications, for example to portfolio selection and risk management.

q-fin.MF↗

arXiv2026

Lambda-quantiles under the microscope

We study Lambda-quantiles, a generalisation of classical quantiles in which the constant probability level $λ\in [0,1]$ is replaced by a functional parameter $Λ\colon \mathbb{R} \to [0,1]$. We consider the general case of non-monotone $Λ$, which arises naturally if closure properties of the class of corresponding Lambda-quantiles with respect to inf-aggregation or with respect to mixtures are required. As preliminary results, we characterise finiteness, constancy, and what we call the attainment property known from classical quantiles. We then consider the problem of reconstructing $Λ$ from the values of $Λ$-quantiles on a suitable family of simple distributions, showing its identifiability under mild assumptions. Next, we substantially refine several results obtained in the literature on weak upper and lower semicontinuity and on the property of convexity of the level sets with respect to mixtures, obtaining in both cases almost complete characterisations without any monotonicity assumption. We then move to the case in which $Λ$ has bounded variation, which enables us to prove a mixture representation result: any such $Λ$-quantile can be rewritten as a Lambda-quantile with an increasing functional parameter, evaluated at a mixture of the original distribution with a fixed reference distribution at a fixed weight, thus reducing the complexity of the parameter from bounded variation to monotone. Finally, we introduce and study the notion of the ordinal covariance group of a risk measure, showing that in the case of a $Λ$-quantile it coincides with the compositional invariance group of $Λ$ and with a certain group of measure-preserving transformations of the signed measure associated with $Λ$.

math.ST↗

arXiv2026

Eliciting reference measures of law-invariant functionals

Law-invariant functionals are central to risk management and assign identical values to random prospects sharing the same distribution under an atomless reference probability measure. This measure is typically assumed fixed. Here, we adopt the reverse perspective: given only observed functional values, we aim to either recover the reference measure or identify a candidate measure to test for law invariance when that property is not {\em a priori} satisfied. Our approach is based on a key observation about law-invariant functionals defined on law-invariant domains. These functionals define lower (upper) supporting sets in dual spaces of signed measures, and the suprema (infima) of these supporting sets---if they exist---are scalar multiples of the reference measure. In specific cases, this observation can be formulated as a sandwich theorem. We illustrate the methodology through a detailed analysis of prominent examples: the entropic risk measure, Expected Shortfall, and Value-at-Risk. For the latter, our elicitation procedure initially fails due to the triviality of supporting set extrema. We therefore develop a suitable modification.

q-fin.RM↗

arXiv2026

A unifying perspective on the collapse to the mean for law-invariant functionals

We revisit the ``collapse to the mean'' phenomenon, which refers to mild structural conditions, such as local linearity, that force a law-invariant functional $\ph$ defined on finite-mean random variables to depend solely on the expectation of its argument $X$, and not on any other distributional feature. Starting from a concise characterisation of the convex order, our simplified approach unifies and extends existing results without assuming the functional to be convex or monotone in the almost-sure order, and clarifies the conceptual foundations of the ``collapse to the mean" phenomenon. In addition, we establish a new ``dual collapse'' result for quasi-star-shaped functionals.

q-fin.RM↗

arXiv2025

Multi-asset return risk measures

We revisit the recently introduced concept of return risk measures (RRMs) and extend it by incorporating risk management via multiple so-called eligible assets. The resulting new class of risk measures, termed multi-asset return risk measures (MARRMs), introduces a novel economic model for multiplicative risk sharing. We analyze properties of these risk measures. In particular, we prove that a positively homogeneous MARRM is quasi-convex if and only if it is convex. Furthermore, we provide conditions to avoid inconsistent risk evaluations. Then, we point out the connection between MARRMs and the well-known concept of multi-asset risk measures (MARMs). This is used to obtain various dual representations of MARRMs. Moreover, we conduct a series of case studies, in which we use typical continuous-time financial markets and different notions of acceptability of losses to compare RRMs, MARMs, and MARRMs and draw conclusions about the cost of risk mitigation.

q-fin.MF↗

arXiv2022

Risk sharing under heterogeneous beliefs without convexity

We consider the problem of finding Pareto-optimal allocations of risk among finitely many agents. The associated individual risk measures are law invariant, but with respect to agent-dependent and potentially heterogeneous reference probability measures. Moreover, we assume that the individual risk assessments are consistent with the respective second-order stochastic dominance relations. We do not assume their convexity though. A simple sufficient condition for the existence of Pareto optima is provided. The proof combines local comonotone improvement with a Dieudonné-type argument, which also establishes a link of the optimal allocation problem to the realm of "collapse to the mean" results. Finally, we extend the results to capital requirements with multidimensional security markets.

q-fin.RM↗

arXiv2022

Model Uncertainty: A Reverse Approach

Robust models in mathematical finance replace the classical single probability measure by a sufficiently rich set of probability measures on the future states of the world to capture (Knightian) uncertainty about the "right" probabilities of future events. If this set of measures is nondominated, many results known from classical dominated frameworks cease to hold as probabilistic and analytic tools crucial for the handling of dominated models fail. We investigate the consequences for the robust model when prominent results from the mathematical finance literature are postulate. In this vein, we categorise the Kreps-Yan property, robust variants of the Brannath-Schachermayer Bipolar Theorem, Fatou representations of risk measures, and aggregation in robust models.

math.PR↗

arXiv2021

Law-invariant functionals that collapse to the mean: Beyond convexity

We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the "collapse to the mean" for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the "collapse to the mean" to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis.

q-fin.MF↗

arXiv2021

Separability vs. robustness of Orlicz spaces: financial and economic perspectives

We investigate robust Orlicz spaces as a generalisation of robust $L^p$-spaces. Two constructions of such spaces are distinguished, a top-down approach and a bottom-up approach. We show that separability of robust Orlicz spaces or their subspaces has very strong implications in terms of the dominatedness of the set of priors and the lack of order completeness. Our results have subtle implications for the field of robust finance. For instance, norm closures of bounded continuous functions with respect to the worst-case $L^p$-norm, as considered in the $G$-framework, lead to spaces which are lattice isomorphic to a sublattice of a classical $L^1$-space lacking, however, any form of order completeness. We further show that the topological spanning power of options is always limited under nondominated uncertainty.

math.PR↗

arXiv2018

Risk sharing for capital requirements with multidimensional security markets

We consider the risk sharing problem for capital requirements induced by capital adequacy tests and security markets. The agents involved in the sharing procedure may be heterogeneous in that they apply varying capital adequacy tests and have access to different security markets. We discuss conditions under which there exists a representative agent. Thereafter, we study two frameworks of capital adequacy more closely, polyhedral constraints and distribution based constraints. We prove existence of optimal risk allocations and equilibria within these frameworks and elaborate on their robustness.

q-fin.RM↗

arXiv2017

Model Spaces for Risk Measures

We show how risk measures originally defined in a model free framework in terms of acceptance sets and reference assets imply a meaningful underlying probability structure. Hereafter we construct a maximal domain of definition of the risk measure respecting the underlying ambiguity profile. We particularly emphasise liquidity effects and discuss the correspondence between properties of the risk measure and the structure of this domain as well as subdifferentiability properties. Keywords: Model free risk assessment, extension of risk measures, continuity properties of risk measures, subgradients.

q-fin.RM↗