An Optimal Execution Problem with S-shaped Market Impact Functions

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 $[{\bar{x}}_0,\mathrm{\infty })$ for some ${\bar{x}}_0\ge 0$. 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.