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We discuss liquidity risk from a pure risk-theoretical point of view in the axiomatic context of coherent measures of risk. We propose a formalism for liquidity risk that is compatible with the axioms of coherency. We emphasize the difference between 'coherent risk measures' (CRM) ρ(X ) defined on portfolio values X as opposed to 'coherent portfolio risk measures' (CPRM) ρ(p) defined on the vector space of portfolios p, and we observe that in the presence of liquidity risk the value function on the space of portfolios is no longer necessarily linear. We propose a new nonlinear 'Value' function VL(p) that depends on a new notion of 'liquidity policy' L. The function VL(p) naturally arises from a general description of the impact that the microstructure of illiquid markets has when marking a portfolio to market. We discuss the consequences of the introduction of the function VL(p) in the coherency axioms and we study the properties induced on CPRMs. We show in particular that CPRMs are convex, finding a result that was proposed as a new axiom in the literature of so called 'convex measures of risk'. The framework we propose is not a model but rather a new formalism, in the sense that it is completely free from hypotheses on the dynamics of the market. We provide interpretation and characterization of the formalism as well as some stylized examples.
Acerbi et al. (Wed,) studied this question.