Mathematical Modelling of Liquidity Risk in Financial Markets

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In financial markets, an asset such as a stock, a bond or a commodity is said to be illiquid when large quantities cannot always be traded at any desired time (buyers and sellers are not always available), when the added cost of trading large quantities, or liquidity premium, is non-negligible, or when the impact of a large transaction on prices persists over time. Liquidity becomes a risk factor when its level evolves randomly in time. In recent years, the availability of limit order book data has made liquidity risk the focal point of many trading desks around the world. From a mathematical modelling point of view, only the last two dimensions of illiquidity have been considered in details in the academic literature. Furthermore, these two notions have usually beentreated very differently and few connections have been made between them. Many times in the literature, it has been found that a realistic model can lead to realistic solutions only if it properly integrates all dimensions of liquidity. The broad objective of this research program is to develop a useful mathematical model of the three dimensions of liquidity based on quantities that can be observed or estimated in practice and investigate the relations and the impact of each dimension by considering various pricing and optimal portfolio problems.