Risk measuring under liquidity risk

Abstract: We present a general framework for measuring the liquidity risk. The theoretical framework defines a class of risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defined as the risk that a given security or a portfolio of securities cannot be easily sold or bought by the financial institutions without causing significant changes in prices. The new risk measures present some differences with respect to the standard risk measures. In particular, they are increasing monotonic and convex cash sub-additive on long positions. The contrary, in certain situations, holds for the sell positions. For the long positions case, we provide these new risk measures with a dual representation. In some specific cases also the sell positions can be equipped with a dual representation. We apply our framework to the situation in which financial institutions break up large trades into many small ones. Dual representation results are also obtained. We give many practical examples of risk measures and derive for each of them the respective capital requirement. As a particular example, we discuss the VaR measure.