  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.
