Some Mathematical Aspects of Market Impact Modeling

Abstract

Market impact models describe the feedback of trading strategies on the underlying asset prices. The resulting feedback effects lead to interesting questions of stability and regularity, which are closely related to the existence and behavior of strategies for optimal order execution. In this paper, we give a survey on some recent results that were obtained in this context. In the first part, we explain in particular the stochastic control approach to the maximization of the expected utility of revenues in the Almgren-Chriss framework. In the second part, we describe stability issues that arise when market impact is allowed to be transient.

Keywords: Market Impact Model, Optimal Order Execution, Hamilton-Jacobi-Bellman Equation, Finite-Fuel Control, Transient Price Impact, Price Manipulation, Transaction-Triggered Price Manipulation, Fredholm Integral Equation, Potential Theory, Capacitary Distribution