Risk Measuring Under Liquidity Risk

Abstract

We present a general framework for measuring the liquidity risk. The theoretical framework de fines risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defi ned as the risk that a given security or a portfolio of securities cannot be easily sold or bought by the financial institutions without causing signi ficant changes in prices. The new risk measures are decomposed into two terms, one measuring the risk of the future value of a given position in a security or a portfolio of securities, and the other the initial cost of this position. Within the framework of coherent risk measures, the risk measures applied to the random part of the future value of a position in a determinatesecurity present some diff erences with respect to the standard riskmeasures. In particular, they are increasing monotonic and convex cash subadditive 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 speci fic cases also the sell positions can beequipped with a dual representation. By using convex risk measures, 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 for both positions in securities and portfolios. We give many practical examples of risk measures and derive for each of them the respective capital requirement, and show that our setting can also be applied to contingent securities. As aparticular example, we discuss the VaR measure.