Systemic Risk Measures with Market Volatility

Systemic risk measures are crucial for the stability of financial markets. However, traditional frameworks, such as those relying on fixed exponent spaces, rest on an assumption of globally consistent moment conditions, which fails to capture the time-varying and state-dependent nature of financial market volatility. In this paper, we propose a new framework for systemic risk measurement on the variable-exponent Bochner–Lebesgue space Lp(·), where the exponent p(·) is a random variable rather than a deterministic constant parameter. The variable exponent p(·) can be regarded as an endogenous risk-sensitivity adjuster. This design imbues the systemic risk measure with inherent adaptability, enabling it to move beyond applying a fixed scale across all market states and instead allowing its criterion to adjust dynamically in response to the evolving nature of risk amid shifts in market conditions. By constructing suitable deterministic auxiliary functions and single-firm risk measures, we decompose the quantification of systemic risk in Lp(·) into two sequential steps, ultimately deriving its dual representations. Several examples are provided to illustrate the theoretical results.