Dynamic risk measures for fluctuations in market volatility under Bochner-Lebesgue spaces

TL;DR

This paper develops dynamic risk measures within variable exponent Lebesgue spaces to better capture market volatility fluctuations, providing dual representations and addressing recent financial market complexities.

Contribution

It introduces risk measures on $L^{p(ullet)}$ spaces with random exponents, extending traditional models to account for market volatility fluctuations.

Findings

Established axioms for variable exponent risk measures

Derived dual representations for these risk measures

Enhanced modeling of market volatility in financial risk assessment

Abstract

Starting from the global financial crisis to the more recent disruptions brought about by geopolitical tensions and public health crises, the volatility of risk in financial markets has increased significantly. This underscores the necessity for comprehensive risk measures capable of capturing the complexity and heightened fluctuations in market volatility. This need is crucial not only for new financial assets but also for the traditional financial market in the face of a rapidly changing financial environment and global landscape. In this paper, we consider the risk measures on a special space $L^{p(\cdot)}$, where the variable exponent $p(\cdot)$ is no longer a given real number as in the conventional risk measure space $L^{p}$, but rather a random variable reflecting potential fluctuations in volatility within financial markets. Through further development of axioms related to this class of risk measures, we also establish dual representations for them.