Optimal liquidation of foreign currencies when FX rates follow a generalised Ornstein-Uhlenbeck process

Abstract

In this article, we consider the case of a multinational company realizing profits in a country other than its base country. The currencies used in the base and foreign countries are referred to as the domestic and foreign currencies respectively. For its quarterly and yearly financial statements, the company transfers its profits from a foreign bank account to a domestic bank account. Thus, the foreign currency liquidation task consists formally in exchanging over a period T a volume V of cash in the foreign currency f for a maximum volume of cash in the domestic currency d. The foreign exchange (FX) rate that prevails at time t is denoted X_d/f(t) and is defined as the worth of one unit of currency d in the currency f. We assume in this article that the natural logarithm of the FX rate $x_{t}=\log X_{d/f}(t)$ follows a discrete generalized Ornstein-Uhlenbeck (OU) process, a process which generalizes the Brownian motion and mean-reverting processes. We also assume minimum and maximum volume constraints on each transaction. Foreign currency liquidation exposes the multinational company to financial risks and can have a significant impact on its final revenues, since FX rates are hard to predict and often quite volatile. We introduce a Reinforcement Learning (RL) framework for finding the liquidation strategy that maximizes the expected total revenue in the domestic currency. Despite the huge success of Deep Reinforcement Learning (DRL) in various domains in the recent past, existing DRL algorithms perform sub-optimally in this task and the Stochastic Dynamic Programming (SDP) algorithm – which yields the optimal strategy in the case of discrete state and action spaces – is rather slow. Thus, we propose here a novel algorithm that addresses both issues. Using SDP, we first determine numerically the optimal solution in the case where the state and decision variables are discrete. We analyse the structure of the computed solution and derive an analytical formula for the optimal trading strategy in the general continuous case. Quasi-optimal parameters of the analytical formula can then be obtained via grid search. This method, simply referred to as ”Estimated Optimal Liquidation Strategy” (EOLS) is validated experimentally using the Euro as domestic currency and 3 foreign currencies, namely USD (US Dollar), CNY(Chinese Yuan) and GBP(Great British Pound). We introduce a liquidation optimality measure based on the gap between the average transaction rate captured by a strategy and the minimum rate over the liquidation period. The metric is used to compare the performance of EOLS to the Time Weighted Average Price (TWAP), SDP and the DRL algorithms Deep Q-Network (DQN) and Proximal Policy Optimization (PPO). The results show that EOLS outperforms TWAP by 54%, and DQN and PPO by 15 − 27%. EOLS runs in average 20 times faster than DQN and PPO. It has a performance on par with SDP but runs 44 times faster. EOLS is the first algorithm that utilizes a closed-form solution of the SDP strategy to achieve quasi-optimal decisions in a liquidation task. Compared with state-of-the-art DRL algorithms, it exhibits a simpler structure, superior performance and significantly reduced compute time, making EOLS better suited in practice.