Abstracts

Hans Föllmer (Leibniz Lecture)

Title: Risk Measures on Wiener Space

Abstract: After some general remarks on the structure and the role of convex risk measures, we will focus on risk measures on Wiener space that are defined in terms of entropy and Wasserstein metrics. We will then show how their rescaled versions admit a dual representation in terms of volatility and specific relative entropy, and that they allow us to quantify some risks in the fine structure of Brownian paths.

Ruodu Wang (Invited Speaker)

Title: Diversification quotients: Axiomatic theory and applications

Abstract: We establish the first axiomatic theory for diversification indices using six intuitive axioms – non-negativity, location invariance, scale invariance, rationality, normalization, and continuity – together with risk measures. The unique class of indices satisfying these axioms, called the diversification quotients (DQs), are defined based on a parametric family of risk measures. DQ has many attractive properties, and it can address several theoretical and practical limitations of existing indices. In particular, for the popular risk measures Value-at-Risk and Expected Shortfall, DQ admits simple formulas, it is efficient to optimize in portfolio selection, and it can properly capture tail heaviness and common shocks which are neglected by traditional diversification indices. When illustrated with financial data, DQ is intuitive to interpret, and its performance is competitive against other diversification indices.

Michael Kupper 

Title: Nonlinear semigroups and limit theorems for convex expectations

Abstract: Motivated by model uncertainty, we focus on semigroups of convex monotone operators on spaces of continuous functions. Starting with a family $(I(t))_{t\geq 0}$ of operators, the semigroup is constructed as the limit $S(t)f:=\lim_{n\to\infty}I(\frac{t}{n})^nf$ and is uniquely determined by the time derivative $I’(0)f$ for smooth functions. As an application of our results, we show that LLN and CLT type results for convex expectations can be systematically obtained by the so-called Chernoff approximation. Furthermore, the limit can be viewed as a dynamic risk measure, which is given as a G-expectation. The talk is based on joint works with Jonas Blessing, Robert Denk, Max Nendel and Sven Schweizer.

Thilo Meyer-Brandis 

Title: Collective Arbitrage and Super-replication

Abstract: We introduce novel notions of arbitrage and super-replication in case agents are allowed to cooperate by exchanging risk in order to increase optimal wealth. We accordingly establish versions of the fundamental theorem of asset pricing and of the pricing-hedging duality. We conclude by some examples illustrating the concepts.

Jan Streicher 

Title: An axiomatic approach to default risk and model uncertainty in rating systems

Abstract: In this talk, we deal with an axiomatic approach to default risk. We introduce the notion of a default risk measure, which generalizes the classical probability of default (PD), and allows to incorporate model risk in various forms. We discuss different properties and representations of default risk measures via monetary risk measures, families of related tail risk measures, and Choquet capacities. In a second step, we turn our focus on default risk measures, which are given as worst-case PDs and distorted PDs. The latter are frequently used in order to take into account model risk for the computation of capital requirements through risk-weighted assets (RWAs), as demanded by the Capital Requirement Regulation (CRR). In this context, we discuss the impact of different default risk measures and margins of conservatism on the amount of risk-weighted assets.

Georg Bollweg 

Title: Affine Processes under Parameter Uncertainty

Abstract: Sublinear expectations were introduced by Peng as generalization of linear expectations in classical probability theory. A sublinear expectation can be represented as upper expectation over a set of probability measures and thus provides a framework to study Knightian uncertainty. In this talk, we outline the construction of non-linear affine processes. To be precise, we specify a set of probability measures such that the coordinate mapping process can be regarded as affine process under parameter uncertainty. Further, we present some of its properties and discuss possible applications in life insurance.

Based on joint work with Francesca Biagini and Katharina Oberpriller.

Cosimo Munari (Invited Speaker)

Title: Fundamental theorem of asset pricing with acceptable risk in markets with frictions

Abstract: We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given market. Trading is subject to nonproportional transaction costs and portfolio constraints and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterization requires an appropriate extension of the classical Fundamental Theorem of Asset Pricing where the role of arbitrage opportunities is played by good deals, i.e., costless investment opportunities with acceptable risk-reward tradeoff. In particular, we highlight the importance of scalable good deals, i.e., investment opportunities that are good deals regardless of their volume. The talk is based on joint work with Maria Arduca.

Johannes Langner 

Title: Bipolar Theorems for Sets of Non-negative Random Variables

Abstract: We assume a robust, in general not dominated, probabilistic framework and provide necessary and sufficient conditions for a bipolar representation of subsets of the set of all quasi-sure equivalence classes of non-negative random variables without any further conditions on the underlying measure space. This generalises and unifies existing bipolar theorems proved under stronger assumptions on the robust framework.

Annika Kemper 

Title: A Principal-Agent Framework for Optimal Incentives in Renewable Investments

Abstract: We investigate the optimal regulation of energy production reflecting the long-term goals of the Paris Climate Agreement. We analyze the optimal regulatory incentives to foster the development of non-emissive electricity generation when the demand for power is served either by a monopoly or by two competing agents. The regulator wishes to encourage green investments to limit carbon emissions, while simultaneously reducing intermittency of the total energy production. We find that the regulation of a competitive market is more efficient than the one of the monopoly as measured with the certainty equivalent of the Principal's value function. This higher efficiency is achieved thanks to a higher degree of freedom of the incentive mechanisms which involves cross-subsidies between firms. A numerical study quantifies the impact of the designed second-best contract in both market structures compared to the business-as-usual scenario. In addition, we expand the monopolistic and competitive setup to a more general class of tractable Principal-Multi-Agent incentives problems when both the drift and the volatility of a multi-dimensional diffusion process can be controlled by the Agents. We follow the resolution methodology of Cvitanić et al. (2018) in an extended linear quadratic setting with exponential utilities and a multi-dimensional state process of Ornstein-Uhlenbeck type. We provide closed-form expression of the second-best contracts. In particular, we show that they are in rebate form involving time-dependent prices of each state-variable. (In collaboration with René Aïd & Nizar Touzi.)

Ariel Neufeld (Invited Speaker)

Title: Markov Decision Processes under Model Uncertainty

Abstract: In this talk, we introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle we obtain a local-to-global paradigm, namely solving a local, i.e., a one time-step robust optimization problem leads to an optimizer of the global (i.e. infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure.

Moreover, we apply this framework to portfolio optimization involving data of the S&P 500. We present two different types of ambiguity sets; one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

This talk is based on joint work with Julian Sester and Mario Šikić.

Thorsten Schmidt (Invited Speaker)

Title: Robust asymptotic insurance-finance arbitrage

Abstract: In most cases, insurance contracts are linked to the financial markets, such as through interest rates or equity-linked insurance products. To motivate an evaluation rule in these hybrid markets, Artzner et al. (2022) introduced the notion of insurance-finance arbitrage. In this paper we extend their setting by incorporating model uncertainty. To this end, we allow statistical uncertainty in the underlying dynamics to be represented by a set of priors P. Within this framework we introduce the notion of robust asymptotic insurance-finance arbitrage and characterize the absence of such strategies in terms of the concept of QP-evaluations. This is a nonlinear two-step evaluation which guarantees no robust asymptotic insurance-finance arbitrage. Moreover, the QP-evaluation dominates all two-step evaluations as long as we agree on the set of priors P which shows that those two-step evaluations do not allow for robust asymptotic insurance-finance arbitrages. Furthermore, we introduce a doubly stochastic model under uncertainty for surrender and survival. In this setting, we describe conditional dependence by means of copulas and illustrate how the QP-evaluation can be used for the pricing of hybrid insurance products. The talk is based on joint work with Katharina Oberpriller, Moritz Ritter.

Alexander Voß 

Title: Towards a Measurement of Cyber Pandemic Risk

Abstract: Based on classical models for network contagion, we analyze the spread dynamics of systemic cyber risks like the 2017 WannaCry and NotPetya incidents in a stylized fashion. Our findings show that topology-based measures are necessary to control the risk exposure of large-scale cyber systems with a more heterogeneous – and thus realistic – arrangement of network nodes. Building on this, we present the first steps towards an axiomatic approach to measuring cyber pandemic risks.

Tim Boonen 

Title: Optimal insurance under maxmin expected utility

Abstract: We examine a problem of demand for insurance indemnification, when the insured is sensitive to ambiguity and behaves according to the maxmin expected utility model of Gilboa and Schmeidler (1989), whereas the insurer is a (risk-averse or risk-neutral) expected-utility maximiser. We characterise optimal indemnity functions both with and without the customary ex ante no-sabotage requirement on feasible indemnities, and for both concave and linear utility functions for the two agents. This allows us to provide a unifying framework in which we examine the effects of the no-sabotage condition, of marginal utility of wealth, of belief heterogeneity, as well as of ambiguity (multiplicity of priors) on the structure of optimal indemnity functions. In particular, we show how a singularity in beliefs leads to an optimal indemnity function that involves full insurance on an event to which the insurer assigns zero probability, while the decision maker assigns a positive probability. We examine several illustrative examples, and we provide numerical studies for the case of a Wasserstein and a Rényi ambiguity set. This presentation is based on joint work with Corina Birghila and Mario Ghossoub.

Birgit Rudloff (Invited Speaker)

Title: Time Consistency of the Mean-Risk Problem

Abstract: Choosing a portfolio of risky assets over time that aims to maximize the expected return as well as minimize portfolio risk is a classical problem in mathematical finance and is referred to as the dynamic Markowitz problem (when the risk is measured by variance) or, more generally, the dynamic mean-risk problem. In most of the literature, the mean-risk problem is scalarized, and it is well known that this scalarized problem does not satisfy the (scalar) Bellman’s principle. Thus, the classical dynamic programming methods are not applicable. For the purpose of this talk we focus on the discrete time setup, and we will use a time-consistent dynamic convex risk measure to evaluate the risk of a portfolio. We will show that, when we do not scalarize the problem but leave it in its original form as a vector optimization problem, the upper images, whose boundaries contain the efficient frontiers, recurse backward in time under very mild assumptions. Thus, the dynamic mean-risk problem does satisfy a Bellman’s principle, but a more general one, that seems more appropriate for a vector optimization problem: a set-valued Bellman’s principle. We will present conditions under which this recursion can be exploited directly to compute a solution in the spirit of dynamic programming. Numerical examples illustrate the proposed method.

The obtained set-valued Bellman’s principle is not unique to the mean-risk problem, but helps also to solve other multivariate problems like Nash equilibria in dynamic games or dynamic risk measures in a market with frictions.