Journal Article

Technical Note-The Joint Impact of F-Divergences and Reference Models on the Contents of Uncertainty Sets

Kruse, Thomas, Schneider, Judith C., Schweizer, Nikolaus

2019

Overview

Any quantitative model (e.g., in financial risk management) must rely on modeling assumptions and is thus prone to model risk. In “The Joint Impact of F Divergences and Reference Models on the Contents of Uncertainty Sets,” the authors reassess the “robustness” approach to model risk. In this approach, model risk is defined in a nonparametric way. Calculations under a reference model are contrasted against worst case scenarios over all alternative models within a maximal “divergence” from the reference model. The choices of the reference model and the divergence measure jointly shape the uncertainty set—and thus, the perceived severity of model risk. The authors argue that there is no single divergence measure that is suitable for all reference models. Instead, when choosing a divergence measure, properties of the reference model should be taken into account. This concerns in particular assumptions on tail risk made under the reference model. In the presence of model risk, it is well established to replace classical expected values with worst-case expectations over all models within a fixed radius from a given reference model. This is the “robustness” approach. For the class of F -divergences, we provide a careful assessment of how the interplay between reference model and divergence measure shapes the contents of uncertainty sets. We show that the classical divergences, relative entropy and polynomial divergences, are inadequate for reference models that are moderately heavy-tailed, such as lognormal models. Worst cases either are infinitely pessimistic or rule out the possibility of fat-tailed “power law” models as plausible alternatives. Moreover, we rule out the existence of a single F -divergence, which is appropriate regardless of the reference model. Thus, the reference model should not be neglected when settling on any particular divergence measure in the robustness approach.

• Publisher: INFORMS,Institute for Operations Research and the Management Sciences
• Subject: Analysis / Divergence / heavy tails / Kullback–Leibler divergence / Mathematical models / model risk / Operations research / Polynomials / Probability / Robustness / Uncertainty