Liquidity, Risk Measures, and Concentration of Measure

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 (ρ(λX)) λ≥0 , where ρ is a convex risk measure and X a random variable, and we call such a curve a 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 γ, deriving tractable necessary and sufficient conditions for concentration inequalities of the form ρ(λX) ≤ γ(λ), for all λ ≥ 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.

Concentration of dynamic risk measures in a Brownian filtration

“…where l is a given positive, increasing and convex function and ρ : L 2 → (−∞, +∞] is a risk measure. This type of inequalities first appeared in the work of Bobkov and Götze [3] and have recently been thoroughly studied by Lacker [19] in the context of liquidity risk in finance. Given a future uncertain loss X the inequality (1.1) yields an upper bound on the liquidity risk profile (ρ(λX)) λ≥0 of X.…”

“…Refer to [9,16] and the references therein for further development. Let us mention that Lacker [19] gave an integrability criterion for the concentration property of static risk measures including optimized certainty equivalent and shortfall risk measures. In the case where Ω is equipped with a metric δ, Bobkov and Ding [2] gave integral criteria on δ from which the concentration property for optimized certainty equivalent risk measures with power-type loss functions can be derived.…”

“…The question is then for which risk measures and which claims can the liquidity risk profile be controlled and bounded? see [19], where the author further gives an integrability condition under which (1.1) holds for shortfall risk measures. The goal of this note is to derive the bound (1.1) for dynamic risk measures in the Brownian setting.…”

“…While we can hardly overstate the importance of these inequalities, see e.g. the survey of Gozlan and Léonard [15], let us mention that Lacker [19] pointed out some interesting consequences of inequalities of the form (1.3): as mentioned above, they provide bounds on the liquidity risk profile of the financial derivative with payoff X and, on the other hand, they give new types of transportation inequalities and new descriptions of the concentration of measure phenomenon. In the same vein, a dual formulation of the inequality (1.3) leads us to a 1 ρ * 0,T denotes the convex conjugate of ρ 0,T defined as ρ * 0,T (Z) := sup X∈L 2 (E[ZX] − ρ 0,T (X)), Z ∈ L 2 .…”

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Motivated by liquidity risk in mathematical finance, D. Lacker [19] introduced concentration inequalities for risk measures, i.e. upper bounds on the liquidity risk profile of a financial loss.

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