  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.
