In this talk, we review the models of Algmren and Chriss, Obizhaeva and Wang, and Alfonsi, Fruth and Schied. We use variational calculus to derive optimal execution strategies in these models, and show that static strategies are dynamically optimal, in some cases by explicitly solving the HJB equation. We present general conditions under which there is no price manipulation in models with linear market impact. Finally, we present some new generalizations of the Obizhaeva and Wang model given in a recent paper by Gatheral, Schied and Slynko, again deriving explicit closed-form optimal execution strategies.

This is partially joint work with Alexander Schied and Alla Slynko.