ABSTRACT
This chapter discusses the main modeling ideas and analytical approaches for the optimal execution and optimal placement problems within one exchange. The major modeling premise behind the optimal execution problem is that any trading strategy, especially one that involves a large amount of buying or selling within a short period of time, would have an impact on the stock price. There are natural continuous-time counterparts of these models, which can be solved directly by using the Calculus of Variations technique instead of the continuous-time dynamic programming. Modeling stock prices by a random walk or Brownian motion in conjunction with additive impact of large stock sales has been a widely used framework to solve the optimal execution problem. Guo and Zervos have considered the optimal execution problem with a Geometric Brownian Motion model and with a multiplicative price impact.
