Dealing with transaction costs in portfolio optimization: online gradient descent with momentum

Outperforming the markets through active investment strategies is one of the main challenges in finance. The random movements of assets and the unpredictability of catalysts make it hard to perform better than the average market, therefore, in such a competitive environment, methods designed to keep low transaction costs have a significant impact on the obtained wealth. This paper focuses on investing techniques to beat market returns through online portfolio optimization while controlling transaction costs. Such a framework differs from classical approaches as it assumes that the market has an adversarial behavior, which requires frequent portfolio rebalancing. This paper analyses critically the known online learning literature dealing with transaction costs and proposes a novel algorithm, namely Online Gradient Descent with Momentum (OGDM), to control (theoretically and empirically) the costs. The existing algorithms designed for this setting are either (i) not providing theoretical guarantees, (ii) providing a bound to the total regret, conditionally on unrealistic assumptions or (iii) computationally not efficient. In this paper, we prove that OGDM has nice theoretical, empirical, and computational performances. We show that it has regret, considering costs, of the order [EQUATION], T being the investment horizon, and has Θ(M) per-step computational complexity, M being the number of assets. Furthermore, we show that this algorithm provides competitive gains when compared empirically with state-of-the-art online learning algorithms on a real-world dataset.