Optimal Execution of Foreign Securities: A Double-Execution Problem with Signatures and Machine Learning

Abstract

We employ the expected signature of equity and foreign exchange markets to derive an optimal double-execution trading strategy. The signature of a path of a stochastic process is a sequence of real numbers that provides a full description of the evolution of the process. The double-execution strategy maximises the wealth in the domestic currency of an investor who liquidates a block of shares in a foreign stock market and the proceeds from the sale are exchanged into the investor's domestic currency. Our approach is model agnostic because we do not specify the dynamics of the market. We prove that the optimal strategy is a linear combination of the terms in the expected signature of the market and employ high-frequency data from Nasdaq and for various currencies to compute the signature of the market. Our results show that the signature-based double-execution strategy considerably outperforms various benchmarks that use single-execution strategies (one for the shares and one for the currency). The outperformance is more pronounced for the illiquid stocks and the illiquid domestic currencies we study.