Liquidity Risk Theory and Coherent Measures of Risk

Abstract

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 which is compatible with the axioms of coherency. We emphasize the difference between 'coherent risk measures',(CRM) defined on portfolio values as opposed to 'coherent portfolio risk measures' (CPRM) defined on the vector space of portfolios, and we observe that in presence of liquidity risk the value function on the space of portfolios is no more necessarily linear. We propose a new nonlinear 'Value' function which depends on a new notion of 'liquidity policy' based on a general description of the micro-structure of illiquid markets and the impact that this micro-structure has when marking to market a portfolio. We discuss the consequences of the introduction of the function 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 example.