Liquidity, risk measures, and concentration of measure

TL;DR

This paper develops a quantitative framework for modeling liquidity risk using convex risk measures, linking the shape of risk profiles to tail behavior and deriving bounds through concentration inequalities and transport inequalities.

Contribution

It introduces a systematic approach to bound liquidity risk profiles via concentration inequalities, connecting risk measures with transport inequalities and exploring time consistency properties.

Findings

Bounded liquidity risk profiles using concentration inequalities.

Established dual representations related to transport inequalities.

Connected time consistency of risk measures with tensorization of concentration inequalities.

Abstract

Expanding on techniques of concentration of measure, we develop a quantitative framework for modeling liquidity risk using convex risk measures. The fundamental objects of study are curves of the form $(\rho(\lambda X))_{\lambda \ge 0}$, where $\rho$ is a convex risk measure and $X$ a random variable, and we call such a curve a \emph{liquidity risk profile}. The shape of a liquidity risk profile is intimately linked with the tail behavior of the underlying $X$ for some notable classes of risk measures, namely shortfall risk measures. We exploit this link to systematically bound liquidity risk profiles from above by other real functions $\gamma$, deriving tractable necessary and sufficient conditions for \emph{concentration inequalities} of the form $\rho(\lambda X) \le \gamma(\lambda)$, for all $\lambda \ge 0$. These concentration inequalities admit useful dual representations related to transport inequalities, and this leads to efficient uniform bounds for liquidity risk profiles for large classes of $X$. On the other hand, some modest new mathematical results emerge from this analysis, including a new characterization of some classical transport-entropy inequalities. Lastly, the analysis is deepened by means of a surprising connection between time consistency properties of law invariant risk measures and the tensorization of concentration inequalities.