Large deviations built on max-stability

TL;DR

This paper extends large deviations theory to max-stable monetary risk measures, establishing a duality with the Laplace Principle and illustrating it with the asymptotic shortfall risk measure.

Contribution

It introduces a large deviation principle for max-stable risk measures, linking risk concentration to classical large deviations results.

Findings

Max-stable risk measures satisfy a Laplace Principle under tightness.

The LDP is equivalent to the LP for max-stable measures.

The asymptotic shortfall risk measure exemplifies the theory.

Abstract

In this paper, we show that the basic results in large deviations theory hold for general monetary risk measures, which satisfy the crucial property of max-stability. A max-stable monetary risk measure fulfills a lattice homomorphism property, and satisfies under a suitable tightness condition the Laplace Principle (LP), that is, admits a dual representation with affine convex conjugate. By replacing asymptotic concentration of probability by concentration of risk, we formulate a Large Deviation Principle (LDP) for max-stable monetary risk measures, and show its equivalence to the LP. In particular, the special case of the asymptotic entropic risk measure corresponds to the classical Varadhan-Bryc equivalence between the LDP and LP. The main results are illustrated by the asymptotic shortfall risk measure.