# Optimal execution with rough path signatures

## TL;DR

This paper introduces a signature-based method for approximate optimal execution in financial markets, leveraging rough path theory to handle complex price dynamics and impact functions, with strong numerical validation.

## Contribution

It develops a novel signature method framework for optimal execution, capable of approximating solutions in complex rough path settings, including cases without known exact solutions.

## Key findings

- Accurately approximates known solutions in benchmark cases
- Demonstrates flexibility and accuracy through numerical experiments
- Outperforms standard execution strategies in unknown-solution scenarios

## Abstract

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.