Sparse Index Tracking via Topological Learning

In this research, we introduce a novel methodology for the index tracking problem with sparse portfolios by leveraging topological data analysis (TDA). Utilizing persistence homology to measure the riskiness of assets, we introduce a topological method for data-driven learning of the parameters for regularization terms. Specifically, the Vietoris-Rips filtration method is utilized to capture the intricate topological features of asset movements, providing a robust framework for portfolio tracking. Our approach has the advantage of accommodating both ℓ1 and ℓ2 penalty terms without the requirement for expensive estimation procedures. We empirically validate the performance of our methodology against state-of-the-art sparse index tracking techniques, such as Elastic-Net and SLOPE, using a dataset that covers 23 years of S&P500 index and its constituent data. Our out-of-sample results show that this computationally efficient technique surpasses conventional methods across risk metrics, risk-adjusted performance, and trading expenses in varied market conditions. Furthermore, in turbulent markets, it not only maintains but also enhances tracking performance.