An Optimal Execution Problem with S-shaped Market Impact Functions

TL;DR

This paper investigates optimal trade execution strategies considering S-shaped market impact functions, revealing that optimal speeds are either zero or above a certain threshold, with specific strategies for small trades in Black-Scholes markets.

Contribution

It extends the optimal execution framework to S-shaped impact functions, analyzing the resulting Hamilton-Jacobi-Bellman equation and deriving new optimal strategies.

Findings

Optimal execution speed is either zero or exceeds a threshold $ar{x}_0$.

For small trades, the time-weighted average price strategy is optimal.

The analysis includes examples within the Black-Scholes model.

Abstract

In this study, we extend the optimal execution problem with convex market impact function studied in Kato (2014) to the case where the market impact function is S-shaped, that is, concave on $[0, \bar {x}_0]$ and convex on $[\bar {x}_0, \infty )$ for some $\bar {x}_0 \geq 0$. We study the corresponding Hamilton-Jacobi-Bellman equation and show that the optimal execution speed under the S-shaped market impact is equal to zero or larger than $\bar {x}_0$. Moreover, we provide some examples of the Black-Scholes model. We show that the optimal strategy for a risk-neutral trader with small shares is the time-weighted average price strategy whenever the market impact function is S-shaped.