Mathematical models of liquidity risk and applications to finance

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A financial asset is said to be liquid when large quantities of it can be easily traded at any desired time (i.e., a buyer or seller is readily available to trade with), when the added cost of trading large quantities is negligible, when a trade has little impact on prices, or when the price impact of a large transaction decays rapidly over time. The liquidity of an asset becomes a risk factor when the extent to which the asset is liquid evolves randomly and unpredictably over time. The long-term goal of my Discovery research program is to model the different notions of liquidity, and investigate the relations between each of its dimensions by studying various pricing and portfolio problems in finance in the context of liquidity risk. Many financial markets are based on limit order books. A limit order is an order to buy or sell that is sent to the market specifying price and quantity, but for which there may not currrently be any counterparty to accept it. A limit order book keeps track of all limit and market orders entered, modified, deleted and matched in the system. It offers a direct snapshot of liquidity in which all four dimensions of liquidity can potentially be quantified. Three main objectives associated with the current proposal include: 1. To give a realistic mathematical model of liquidity risk in the context of optimal execution of large orders in a limit order market. The goal of the optimal execution problem is to find the optimal way to divide a large order to buy or sell into smaller transactions in order to minimize expected liquidity costs and market risks related to the variability of prices. Optimal execution techniques allow portfolio managers to considerably reduce transaction costs, and financial institutions have increasingly relied on them in recent years to rein in costs. 2. To explore the consequence of illiquidity in the problem of options hedging and market making. Options are interesting from both a financial and mathematical point of view as their value depends on the underlying asset on which they are defined. The market risk of options can be partially hedged by trading its underlying. By recognizing that trading of the underlying can be done through limit and market orders in a limit order book, the classical problem of options hedging takes an interesting direction due to the mathematical and practical complexity of order books modelling. 3. Management of liquidity risk by firms, banks and other financial institutions. The liquidity of assets and the costs of borrowing are important considerations for firms that must make decisions regarding dividend policy, capital structuring, especially in the face of possible bankruptcy. The outcome of my research will offer new methods of proofs in the field of stochastic control and original modelling ideas of financial markets with liquidity risk. Newly developped numerical methods will have the potential to be applied to a large range of related control problems.