Abstract
We study an optimal execution problem in consideration of market impact as some regular stochastic control problem. We focus on mathematical formulation of such an optimization problem and study some properties of the corresponding value functions. We formulate our optimal execution problem as a discrete-time model and describe the value function with respect to a trader's optimization problem. By shortening the intervals of execution times, we derive a value function of a continuous-time model and study some properties of them. We show that the properties of the continuous-time value function vary by the strength of market impact. Moreover we introduce some examples of this model, which tell us that the forms of the optimal execution strategies entirely change according to the amount of the security holdings. As one of consequences, we observe some phenomenon by using typical examples of our model, that “the form of market impact function is concave or S-shape.”
