Mathematics of Market Microstructure

29 February - 30 May 2024

Thursdays, 10:15 - 12:00

Location: HG G 43

First lecture: 29 February

Abstract

One of the main goals of the Market Microstructure Theory is to understand the temporary and permanent impacts of trade on the asset price and how the price-setting rules of market makers or exchanges evolve in time. While the temporary impacts arise as a result of inventory considerations of the market makers or dealers, the permanent impacts are due the asymmetric information in the market. Traders who are concerned with the price impacts of their trades, mainly due to the execution of large orders, are called strategic. The models that motivate the content of the course consider the equilibrium among strategic and non-strategic traders, and market makers.

Our primary focus will be the Kyle model as well as the Glosten-Milgrom model, which are the canonical models of market microstructure that analyse in an equilibrium framework the price impact of trades and other liquidity measures. The analysis of the equilibrium in continuous time requires tools from stochastic filtering, potential theory of Markov processes (e.g. h-transforms and Markov bridges) and the theory of enlargement of filtrations. This course will develop the fundamental techniques from Markov processes theory that are needed for the study of such equilibria. Familiarity with Brownian motion, stochastic differential equations, Poisson process, and basic notions from martingale theory will be assumed.

The topics that will be covered in the course include (most of) the following:

1. Brief overview of canonical models of market microstructure and the notion of equilibrium. Kyle's model in discrete time and related stability issues for its numerical solution.
2. A quick introduction to Markov processes and their nomenclature. One dimensional diffusions on the real line and their potential theory. 
3. Static Markov bridges, h-transforms and their SDE representation.
4. Initial and progressive enlargement of filtrations.
5. Kyle model in continuous time with static signals.
6. Innovations approach to filtering and uniqueness of solutions of Kushner-Stratonovich equations.
7. Glosten-Milgrom model in continuous time and its relation with the Kyle model.
8. Kyle model with dynamic signals and dynamic Markov bridges. 
9. Kyle equilibrium as an ill-posed inverse problem and its alternative reformulations.
10.  Insider trading with legal penalties and quadratic BSDEs.
11.  Connections with the Schroedinger problem.
12.  Kyle model with options: an application of optimal transport.
13.  Further extensions (risk aversion, stochastic volatility, high-frequency traders) and open problems.