  We present a method for obtaining approximate solutions to the problem of
optimal execution, based on a signature method. The framework is general, only
requiring that the price process is a geometric rough path and the price impact
function is a continuous function of the trading speed. Following an
approximation of the optimisation problem, we are able to calculate an optimal
solution for the trading speed in the space of linear functions on a truncation
of the signature of the price process. We provide strong numerical evidence
illustrating the accuracy and flexibility of the approach. Our numerical
investigation both examines cases where exact solutions are known,
demonstrating that the method accurately approximates these solutions, and
models where exact solutions are not known. In the latter case, we obtain
favourable comparisons with standard execution strategies.
