Mathematical models of liquidity risk and applications to finance

当一项金融资产在任何预期的时间都可以很容易地进行交易(即，可以随时与买方或卖方进行交易)，当大量交易的额外成本可以忽略不计，当一笔交易对价格的影响很小，或者当一笔大额交易的价格影响随着时间的推移迅速衰减时，金融资产就被称为流动性资产。当资产的流动性程度随着时间的推移随机且不可预测地演变时，资产的流动性就成为一个风险因素。我的发现研究项目的长期目标是对流动性的不同概念进行建模，并通过在流动性风险的背景下研究金融中的各种定价和投资组合问题来调查其每个维度之间的关系。许多金融市场都是以限价指令为基础的。限价指令是一种买入或卖出指令，它被发送到市场，指定价格和数量，但目前可能没有任何交易对手接受它。限价指令簿会跟踪系统中输入、修改、删除和匹配的所有限价指令和市场指令。它提供了流动性的直接快照，其中流动性的所有四个维度都有可能被量化。与当前建议相关的三个主要目标包括：1.在限价订单市场中大额订单最优执行的背景下，给出一个现实的流动性风险的数学模型。最优执行问题的目标是找到一种最优方法，将大额买卖订单划分为较小的交易，以最小化预期流动性成本和与价格变化相关的市场风险。最优执行技术使投资组合经理能够大幅降低交易成本，近年来金融机构越来越依赖这些技术来控制成本。2.探讨流动性不足在期权套期保值和做市问题中的后果。从金融和数学的角度来看，期权都很有趣，因为它们的价值取决于定义它们的基础资产。期权的市场风险可以通过交易其标的来部分对冲。通过认识到标的的交易可以通过限制订单账簿中的限价和市场订单进行，由于订单账簿建模的数学和实践复杂性，期权对冲的经典问题采取了一个有趣的方向。3.公司、银行和其他金融机构对流动性风险的管理。资产的流动性和借款成本是公司必须就股息政策、资本结构做出决定的重要考虑因素，特别是在可能面临破产的情况下。本文的研究成果将为随机控制领域提供新的证明方法，并为具有流动性风险的金融市场提供独到的建模思路。新发展的数值方法将有可能应用于一系列相关的控制问题。

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.