# Systemic risk measures with markets volatility

## TL;DR

This paper introduces a novel framework for systemic risk measurement that incorporates market volatility by using a variable-exponent space, providing a more nuanced understanding of financial stability.

## Contribution

It develops a new systemic risk measure in the variable-exponent Bochner-Lebesgue space, capturing market volatility through a stochastic exponent and deriving its dual representations.

## Key findings

- The framework effectively models market volatility within systemic risk measures.
- Dual representations of the risk measure are explicitly derived.
- Examples demonstrate the applicability of the theoretical results.

## Abstract

Systemic risk measures are crucial for the stability of financial markets, yet classical formulations fail to capture the complexity of market volatility. We propose a new framework for systemic risk measurement on the variable-exponent Bochner-Lebesgue space $L^{p(\cdot)}$, where the exponent $p(\cdot)$ is a random variable rather than a deterministic constant parameter, thereby inherently encoding latent market volatility. By constructing suitable deterministic auxiliary functions and single-firm risk measures, we decompose the quantification of systemic risk in $L^{p(\cdot)}$ into two sequential steps, ultimately deriving its dual representations. Several examples are provided to illustrate the theoretical results.