Minkowski gauges and deviation measures

TL;DR

This paper introduces a framework for deriving deviation measures using Minkowski gauges of acceptance sets, offering new interpretations and characterizations relevant to finance and risk management.

Contribution

It establishes a novel connection between Minkowski gauges and deviation measures, filling a gap by linking acceptance sets with deviation measures and providing dual characterizations.

Findings

Minkowski gauge-based deviation measures can be derived from acceptance sets.

The framework generalizes deviation measures for convex, stable, and radially bounded sets.

Dual characterizations involve polar sets and support functionals.

Abstract

We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In doing so, we provide a new interpretation for such measures, namely, that they quantify how much one must shrink or deleverage a position for it to become acceptable. In particular, the Minkowski Deviation of a set which is convex, stable under scalar addition, and radially bounded at non-constants, is a generalized deviation measure. Furthermore, we explore the relations existing between mathematical and financial properties attributable to an acceptance set, and the corresponding properties of the induced measure. Hence, we fill the gap that is the lack of an acceptance set for deviation measures. Dual characterizations in terms of polar sets and support functionals are provided.