Abstracts

Tim Boonen (Invited Speaker)

Title: (No-)Betting Pareto-Optima under Rank-Dependent Utility

Abstract: In this talk, I discuss a pure-exchange economy with no aggregate uncertainty, and I characterize in closed-form and in full generality Pareto-optimal risk-sharing allocations between two agents who maximize rank-dependent utilities (RDU). I then derive a necessary and sufficient condition for Pareto-optima to be no-betting allocations (i.e., deterministic allocations - or full insurance allocations). This condition depends only on the probability weighting functions of the two agents, and not on their (concave) utility functions. Hence with RDU preferences, it is the difference in probabilistic risk attitudes given common beliefs, rather than heterogeneity or ambiguity in beliefs, that is a driver of a bet. As by-product of our analysis, I answer the question of when sunspots matter in this economy.

Mario Ghossoub (Invited Speaker)

Title: Bowley vs. Pareto Optima in Reinsurance Contracting

Abstract: The notion of a Bowley optimum has gained recent popularity as an equilibrium concept in problems of risk sharing and optimal reinsurance. In this paper, we examine the relationship between Bowley optimality and Pareto efficiency in a problem of optimal reinsurance, under fairly general preferences. Specifically, while we show that Bowley-optimal contracts are indeed Pareto efficient (hence providing a first welfare theorem), we also show that only those Pareto-efficient contracts that make the insurer indifferent between suffering the loss and entering into the reinsurance contract are Bowley optimal (hence providing only a partial second welfare theorem). We interpret the latter result as indicative of the limitations of Bowley optimality as an equilibrium concept in this literature. We also discuss relationships with competitive equilibria, and we provide an illustrative example.

Michael Kupper (Invited Speaker)

Title: Markovian Semigroups under Model Uncertainty

Abstract: Abstract: When considering stochastic processes for the modelling of real world phenomena, a major issue is so-called model uncertainty. In this talk, we present two ways to incorporate model uncertainty into a Markovian dynamics. One approach considers parameter uncertainty in the generator of a Markov process while the other considers perturbations of a reference model within a Wasserstein proximity. We show that, in typical situations, these two a priori different approaches lead to the same convex transition semigroup. In the second part, we focus on semigroups of convex monotone operators on spaces of continuous functions and their behaviour with respect to Gamma-convergence. In particular, we discuss the role of Lipschitz sets and provide a comparison result. Based on joint works with Daniel Bartl, Jonas Blessing, Robert Denk, Stephan Eckstein, Sven Fuhrmann and Max Nendel.

Fabio Maccheroni (Invited Speaker)

Title: Star-shaped Risk Measures

Abstract: In this paper monetary risk measures that are positively superhomogeneous, called star-shaped risk measures, are characterized and their properties studied. The measures in this class, which arise when the controversial subadditivity property of coherent risk measures is dispensed with and positive homogeneity is weakened, include all practically used risk measures, in particular, both convex risk measures and Value-at-Risk. From a financial viewpoint, our 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.

From a decision theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker.

Joint work with Erio Castagnoli, Giacomo Cattelan, Claudio Tebaldi, and Ruodu Wang.

Emanuela Rosazza-Gianin (Invited Speaker)

Title: Fully-dynamic risk measures: time-consistency, horizon risk, and relations with BSDEs and BSVIEs

Abstract: In a dynamic framework, we identify a new concept associated with the risk of assessing the financial exposure by a measure that is not adequate to the actual time horizon of the position. This will be called horizon risk. We clarify that dynamic risk measures are subject to horizon risk, so we propose to use the fully-dynamic version. To quantify the horizon risk we introduce the h-longevity as an indicator. We investigate these concepts together with other properties of risk measures as normalization, restriction property and different formulations of time-consistency. We also consider these concepts for fully-dynamic risk measures generated by backward stochastic differential equations (BSDEs), backward stochastic Volterra integral equations (BSVIEs) and families of these. Within this study we provide new results for BSVIEs: a converse comparison theorem and the dual representation of the associated risk measures.

Based on a joint work with Giulia Di Nunno.

Daniel Bartl

Title: Statistical estimation of stochastic optimization problems and risk measures

Abstract: We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems from an iid sample. This procedure is the first one that exhibits the optimal statistical performance in heavy tailed situations and also applies in highdimensional settings. We discuss the portfolio optimization problem and the estimation of risk measures.

Joint work with Stephan Eckstein and Shahar Mendelson.

Julian Hölzermann

Title: Pricing Interest Rate Derivatives under Volatility Uncertainty

Abstract: In this talk, we study the pricing of contracts in fixed income markets under volatility uncertainty in the sense of Knightian uncertainty or model uncertainty. The starting point is an arbitrage-free bond market under volatility uncertainty. The uncertainty about the volatility is modeled by a G-Brownian motion, which drives the forward rate dynamics. The absence of arbitrage is ensured by a drift condition. Such a setting leads to a sublinear pricing measure for additional contracts, which yields either a single price or a range of prices and provides a connection to hedging prices. Similar to the forward measure approach, we define the forward sublinear expectation to simplify the pricing of cashflows. Under the forward sublinear expectation, we obtain a robust version of the expectations hypothesis, and we show how to price options on forward prices. In addition, we develop pricing methods for contracts consisting of a stream of cashflows, since the nonlinearity of the pricing measure implies that we cannot price a stream of cashflows by pricing each cashflow separately. With these tools, we derive robust pricing formulas for all major interest rate derivatives. The pricing formulas provide a link to the pricing formulas of traditional models without volatility uncertainty and show that volatility uncertainty naturally leads to unspanned stochastic volatility.

Felix-Benedikt Liebrich

Title: Law invariance vs. heterogeneity in risk sharing

Abstract: 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. The first part of the talk is therefore devoted to the question if this heterogeneity can be mitigated because there is more than one reference measure.

​​​​Unfortunately, we shall see that for wide classes of functionals, reference measures are unique unless the functionals are (i) constant, or (ii) only dependent on the essential supremum and essential infimum of the argument. We then return to the risk sharing problem. The individual risk assessments in our study are assumed to be consistent with the respective second-order stochastic dominance relations, but not necessarily convex. Nevertheless, we shall provide and thoroughly discuss a simple sufficient condition for the existence of Pareto optima.

Max Nendel

Title: A decomposition of general premium principles into risk and deviation

Abstract: We provide an axiomatic approach to general premium principles in a probability-free setting that allows for Knightian uncertainty. Every premium principle is the sum of a risk measure, as a generalization of the expected value, and a deviation measure, as a generalization of the variance. One can uniquely identify a maximal risk measure and a minimal deviation measure in such decompositions. We show how previous axiomatizations of premium principles can be embedded into our more general framework. We discuss dual representations of convex premium principles, and study the consistency of premium principles with a financial market in which insurance contracts are traded. The talk is based on joint work with Frank Riedel and Maren Diane Schmeck.

Alesandro Sgarabottolo

Title: Asymptotic Parametrization of Wasserstein Balls

Abstract: In mathematical finance and actuarial science, the assessment of risk is closely related to matching the distribution of the underlying risk factors in an accurate way; a major issue in this process is the so-called model uncertainty or epistemic uncertainty. The latter refer to the impossibility of perfectly capturing the randomness of future states in a single stochastic framework. We explore a static setting that takes into account model uncertainty in the distribution of a risk factor. We allow for perturbations around a baseline model, measured via Wasserstein distances, and we try to understand to which extent this form of probabilistic imprecision can be parametrized. This leads to a risk measure which incorporates a safety margin with respect to nonparametric un- certainty and still can be approximated through parametrized models. In the last step we discuss extensions and open questions related to the order of the approximation and dierent forms of the risk measure. The talk is based on joint work with Max Nendel.