Introduction to Optimal Execution

The developments in electronic markets have led to the diversification of trading activity, and traders need to manage the liquidity risk carefully. This problem is called the optimal execution problem and has become a significant issue among financial mathematicians, economists, and practitioners. This chapter aims to overview how the current financial market works and how one can analyze and build the algorithms for an “optimal execution strategy.” The first section gives a review of current financial markets, which leads to the basics for constructing a model of optimal execution from the viewpoints of market microstructure. In particular, I clarify the system of the “limit order book,” which includes an exposition about how traders place orders and influence the market. Also, this section presents the basic concepts of “large trader” and “market impact,” on top of which most execution models are built. The succeeding sections explain how one can incorporate market impact in modeling and formulate an execution problem through a fundamental model posed by Almgren and Chriss (J. Risk 3:5–39, 2000 [2]). I then describe an extensive model with a moderate change in market impact modeling, discussed in Ohnishi and Shimoshimizu (Quant. Financ. 20:1625–1644, 2020 [35]). These models embody the foundation of algorithms for optimal execution strategies.