Optimal execution with non-linear transient market impact

We study the problem of the optimal execution of a large trade in the propagator model with non-linear transient impact. From brute force numerical optimization of the cost functional, we find that the optimal solution for a buy programme typically features a few short intense buying periods separated by long periods of weak selling. Indeed, in some cases, we find negative expected cost. We show that this undesirable characteristic of the non-linear transient impact model may be mitigated either by introducing a bid-ask spread cost or by imposing convexity of the instantaneous market impact function for large trading rates; the objective in each case is to robustify the solution in a parsimonious and natural way.