Recent technological advances have created new opportunities for trading in financial markets. However, the effects of these changes are two-fold. On the one hand, they increase the competition and provide additional liquidity to the market, which, in particular, manifests itself in reduced bid-ask spreads. On the other hand, the new technology introduces opportunities for the better-equipped participants (e.g. the high-frequency traders) to adjust their positions rapidly, in order to take advantage of the trading needs of other investors. The latter may result in a sudden reduction of the market liquidity, which was well demonstrated by the so-called flash crash of 2010. As financial processes affect all parts of modern life, it is crucial for society to be able to analyze and control the financial stability of markets. The investigator develops tools for quantitative analysis of the tradeoff between the liquidity-providing role of strategic traders and the liquidity risk they generate. Such tools can be used to predict and avoid future liquidity crises, as well as to test the potential effects of new financial regulation (e.g. a transaction tax or limits on trading frequency).
The investigator develops a rigorous mathematical framework for modeling market microstructure and liquidity risk by analyzing the real-world system of market participants (agents) who interact with each other through trading. The macroscopic properties of this system are described by the so-called limit order book, whose shape and dynamics arise endogenously from the actions of individual agents, rather than being taken as an input to the model. The approach is based on the methods of mean-field games, which allow for the analytically tractable description of an equilibrium in stochastic games with a large number of interacting agents. One of the challenges of the project is the complicated dependence structure between the dynamics of individual agents. For example, any realistic model of the above system requires that the agents interact through their control values, rather than through their states. This artifact introduces an additional constraint to the classical forward-backward system describing a mean-field game model, making the analysis more complicated. Another mathematical challenge is due to the fact that the control process of each agent takes values in the space of measures, which represent the limit orders submitted by the agent. As a result, the use of infinite-dimensional analysis is required to obtain an analytic characterization of the solutions to the associated optimization problems. Finally, an important problem arising in this line of study is the convergence of the proposed discrete time mean-field games to the continuous time limit. In particular, in order to address this problem, the investigator extends the existing results on discrete time approximation of the Hamilton-Jacobi-Bellman equation to the setting that allows for measure-valued controls. The project provides a natural framework for quantifying the tradeoff between the liquidity-providing role of the strategic players (e.g. the high-frequency traders) and the liquidity risk they generate. In particular, the resulting models can be used to obtain real-time predictions of the potential liquidity crises (e.g. flash crashes), as well as to test the implications of new financial regulation.