Abstracts

Michael Kupper (Long Talk)

Title: Discrete approximation of risk-based pricing under uncertainty

Abstract: We discuss the limit of risk-based prices of European contingent claims in discrete-time financial markets under volatility uncertainty when the number of intermediate trading periods goes to infinity. The limiting dynamics are obtained using recently developed results for the construction of strongly continuous convex monotone semigroups. We connect the resulting dynamics to the semigroups associated to G-Brownian motion, showing in particular that the worst-case bounds always give rise to a larger bid-ask spread than the risk-based bounds. Moreover, the worst-case bounds are achieved as limit of the risk-based bounds as the agent’s risk aversion tends to infinity. The talk is based on joint work with Jonas Blessing and Alessandro Sgarabottolo.

Anna Aksamit (Long Talk)

Title: Robust Duality for multi-action options with information delay

Abstract: We show the super-hedging duality for multi-action options which generalise American options to a larger space of actions (possibly uncountable) than {stop, continue}. We put ourselves in the framework of Bouchard & Nutz model relying on analytic measurable selection theorem. Finally we consider information delay on the action component of the product space. Information delay is expressed as a possibility to look into the future in the dual formulation. This is a joint work with Ivan Guo, Shidan Liu and Zhou Zhou.

Christian Laudagé (Short Talk)

Title: Scalarized utility-based multi-asset risk measures

Abstract: In this talk, we present a dual representation for a map that generalizes classical risk measures and has not been analysed so far. It unifies dual representations for well-known risk measures. As a special case, we introduce a risk measure that simultaneously allows to minimise hedging costs and maximize expected utility in the presence of a risk constraint. We call it the scalarized utility-based multi-asset (SUBMA) risk measure. It is a combination of risk measures using multiple eligible assets and so-called optimal expected utility (OEU) risk measures. The latter are introduced in Geissel et al. (2018). 

For the SUBMA risk measure we state the following results: If the utility function has constant relative risk aversion and the risk constraint is coherent, then the SUBMA risk measure is coherent. In a one-period financial market setup, we present a sufficient condition for the SUBMA risk measure to be finite-valued. Finally, we derive results about the existence of optimal payoffs.

Ruodu Wang (Long Talk)

Title: Star-shaped and quasi-star-shaped risk measures

Abstract: The class of star-shaped risk measure can be obtained by relaxing all of subadditivity, convexity and positive homogeneity from coherent risk measures. We discuss their representation results. This class includes all practically used risk measures, in particular, both convex risk measures and Value-at-Risk (VaR), and their robust versions. From a financial viewpoint, the relaxation of convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive delegation, or robust aggregation mechanisms. In the context of cash-subadditive risk measures, we introduce and characterize the property of quasi-star-shapedness, with the help of quasi-convexity and quasi-normalization. A main example of quasi-star-shaped risk measures is the Lambda-VaR. These results highlight a stream of work on the representation of risk measures without convexity and quasi-convexity.

Fabio Bellini (Long Talk)

Title:  Duet expectile preferences

Abstract: We present a novel decision-theoretic framework for the axiomatization of expectiles. As in Savage's theory, we consider an economic agent having a preference relation on a space of acts, represented by measurable functions on a suitable measurable space. We introduce a novel axiom that we call Co-loss Aversion (CA), that prescribes that adding an act sharing common adverse scenarios (that is, as we say, being a Co-loss) is the least preferred situation among equally preferred acts. Our main result is to show that the axiom of Co-loss Aversion combined with two additional mild axioms leads to a representation of the preferences in the form of a novel functional called Duet Expectile, whose definition involves two a priori different probability measures. We discuss properties of Duet Expectiles and connections with fundamental concepts such as probabilistic sophistication, risk aversion and uncertainty aversion. Joint work with Tiantian Mao, Ruodu Wang and Qinyu Wu.

Roger Laeven (Long Talk)

Title: A Rank-Dependent Theory for Decision under Risk and Ambiguity

Abstract: This talk discusses the axiomatization, in a two-stage setup, of a new theory for decision under risk and ambiguity. Our theory extends the rank-dependent utility model of Quiggin (1982) for decision under risk to risk and ambiguity, reduces to the variational preferences model when linear in probabilities, and is dual to variational preferences when affine in wealth in the same way as the theory of Yaari (1987) is dual to expected utility. As a special case, we obtain a preference axiomatization of a decision theory that is a rank-dependent generalization of the popular maxmin expected utility theory. We characterize ambiguity aversion in our theory. Based on joint work with Mitja Stadje.

Peyman Mohajerin Esfahani (Long Talk)

Title: Nonlinear Distributionally Robust Optimization

Abstract: This talk focuses on a class of distributionally robust optimization (DRO) problems where, unlike the growing body of the literature, the objective function (risk) is potentially non-linear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present both theoretical and computational challenges. Motivated by this, we propose an alternative notion for the derivative and corresponding smoothness based on Gateaux (G)-derivative for generic risk measures. We then propose a G-derivative based Frank-Wolfe (FW) algorithm for generic non-linear optimization problems in probability spaces and establish its convergence under the proposed notion of smoothness in a norm-independent manner. We use the set-up of the FW algorithm to devise a methodology to compute a saddle point of the non-linear DRO problem.

This talk is based on https://arxiv.org/abs/2306.03202.

David Criens (Short Talk)

Title: Propagation of Chaos under Model Uncertainty

Abstract: A classical problem of statistical physics is the understanding of large interacting particles systems. The difficulty of the analysis on the microscopic level increases heavily with the number of particles. When the influence of interaction with a single particle decreases with the number of particles, it is reasonable to expect that IID properties propagate over time, i.e., that particles started in IID configurations are asymptotically IID. This phenomenon is called ``propagation of chaos’’. Thanks to this, for many questions one can change the point of view to the macroscopic world and study the so-called mean field limit. 

In this talk, we make a step into the direction of propagation of chaos under model uncertainty. Namely, we aim to understand a system of interacting particles with volatility uncertainty and its macroscopic picture. The particle system is given by a version of Peng’s $G$-expectation with drift interaction and the proposed mean field limit is a McKean-Vlasov version of the $G$-expectation. Propagation of chaos is studied on the level of uncertainty sets, initiating the concept of set-valued propagation of chaos, and in terms of convergence of the value functions. If time allows, extensions to general $G$-diffusions and infinite dimensional settings are outlined. 

Partially based on joint work with Moritz Ritter.

Stephan Eckstein (Short Talk)

Title: Statistically optimal and adversarially robust estimation of the expected shortfall

Abstract: We shed new light on the statistical estimation of the expected shortfall. First, we prove that the frequently used plug-in estimator has poor statistical properties whenever the distribution of losses is heavy-tailed. We propose a simple alternative, a median-of-means estimator, and prove that it recovers the best possible statistical properties (dictated by the central limit theorem) under modest assumptions. On top of its statistical merits, we prove that this estimator is robust against adversarial perturbations of the data. This robustness can be seen as a strong stability property under uncertainty about the data used for estimation. For both plug-in and median-of-means estimator, our analysis reveals the importance of continuity of the quantile function of the loss distribution around the level α used for the expected shortfall. Based on joint work with Daniel Bartl.

Zachary Van Oosten (Short Talk)

Title: Partially Law-Invariant Risk Measures

Abstract: I will introduce the concept of partial law invariance, generalizing the idea of law-invariant risk measures. This new concept is motivated by practical considerations of decision-making under uncertainty. I will present a characterization of partially law-invariant coherent risk measures via a novel representation formula, which has little resemblance to the classical formula for law-invariant coherent risk measures. A notion of strong partial law-invariance will be introduced, allowing for a representation formula akin to the classical one. I will give some examples of the new risk measures, including partially law-invariant versions of the Expected Shortfall and the entropic risk measures, and illustrate their applications in risk assessment under model uncertainty.

Tolulope Fadina (Long Talk)

Title: A framework for measures of risk under uncertainty

Abstract: A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable but also on various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles that quantify jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalised risk measure. We present a series of relevant theoretical results. The worst-case, coherent, and robust generalised risk measures are characterised via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law-invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, we provide some characterization results under strong law invariance.

An Chen (Long Talk)

Title: Optimal Payoffs under Smooth Ambiguity

Abstract: We study optimal payoff choice for an investor in a one-period model under smooth ambiguity preferences, also called KMM preferences as they were proposed by (Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73 (6), 1849–1892). In contrast, to the existing literature on optimal asset allocation for a KMM investor in a one-period model, we also allow payoffs that are non-linear in the stock price. Our contribution is threefold. First, we characterize and derive the optimal payoff under KMM preferences. Second, we demonstrate that a KMM investor solves an equivalent problem to an investor under classical subjective expected utility (CSEU) with adjusted second-order beliefs. Third, in a setting of a log-normal market asset under drift and volatility uncertainty, we reveal that ambiguity leads to optimal payoffs that are no longer necessarily long in the market asset. 

(joint with Steven Vanduffel and Morten Wilke)

Corrado De Vecchi (Short Talk)

Title: Pricing insurance contracts with an existing portfolio as background risk

Abstract: We develop and investigate a premium principle that explicitly takes into account the impact of a new risk on some insurer's existing portfolio. Specifically, we propose the notion of indifference premium for a new risk conditioned on an existing portfolio acting as background risk. The resulting premium rule, which in our case depends on the joint distribution of the new risk and the existing portfolio, is analyzed in detail with respect to its mathematical properties. In order to underline the differences between our approach and the literature on law-invariant premium rules, special attention is given to the indifference premium behaviour with respect to some well-known dependence concepts. Axiomatic and continuity properties of the proposed indifference premium rule are also investigated. To demonstrate the practical relevance of our approach, we consider a portfolio of exchangeable risks and investigate the role of the portfolio’s dimension on the price of a risk to be added. This illustrates the (limits of) diversification benefits under this flexible assumption on the joint distribution of a sequence of risks.

Giulio Principi (Short Talk)

Title: Antimonotonicity for Preference Axioms: The Natural Counterpart to Comonotonicity

Abstract: Comonotonicity (“same variation”) of random variables minimizes hedging possibilities and has been widely used in many fields. Comonotonic restrictions of traditional axioms have led to impactful inventions in decision models, including Gilboa and Schmeidler’s ambiguity models. This paper investigates antimonotonicity (“opposite variation”), the natural counterpart to comonotonicity, minimizing leveraging possibilities. Surprisingly, antimonotonic restrictions of traditional axioms often do not give new models but, instead, give generalized axiomatizations of existing ones. We, thus, generalize: (a) classical axiomatizations of linear functionals through Cauchy’s equation; (b) as-if-risk-neutral pricing through no-arbitrage; (c) subjective probabilities through bookmaking; (d) Anscombe-Aumann expected utility; (e) risk aversion in Savage’s subjective expected utility. In each case, our generalizations show where the most critical tests of classical axioms lie: in the antimonotonic cases (maximal hedges). We, finally, present cases where antimonotonic restrictions do weaken axioms and lead to new models, primarily for ambiguity aversion in nonexpected utility.

Steven Vanduffel (Long Talk)

Title: Robust distortion risk measures

Abstract: We study for any given distortion risk measure its robustness to distributional uncertainty by deriving its largest (smallest) value when the underlying loss distribution has a known mean and variance and furthermore lies within a ball – specified through the Wasserstein distance - around a reference distribution. We employ the technique of isotonic projections to provide for any distortion risk measure a complete characterisation of sharp bounds on its value and obtain analytic bounds in the case of Range-Value-at-Risk. We extend our results to account for uncertainty in the first two moments and provide an application to model risk assessment. 

Joint work with Carole Bernard and Silvana Pesenti.