Sparse Index Tracking via Topological Learning

ArXiv ID: 2310.09578 “View on arXiv”

Authors: Unknown

Abstract

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 $\ell_1$ and $\ell_2$ 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.

Keywords: Topological Data Analysis, Persistence Homology, Vietoris-Rips Filtration, Index Tracking, Sparse Portfolios, Equities (Indices)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper heavily employs advanced mathematics, including Topological Data Analysis (TDA), persistence homology, and Vietoris-Rips filtration, placing it in the high math complexity category. The empirical rigor is strong, supported by a 23-year out-of-sample backtest on S&P 500 data with comparisons to Elastic-Net and SLOPE, and it addresses practical concerns like transaction costs and turbulent market performance.

  flowchart TD
    A["Research Goal: Sparse Index Tracking via TDA"] --> B["Data: S&P500 23 Years"]
    B --> C["Methodology: Vietoris-Rips Filtration"]
    C --> D{"Persistence Homology Analysis"}
    D --> E["Topology-Based Parameter Learning"]
    E --> F["Sparse Portfolio Optimization"]
    F --> G["Outcomes: Robust Tracking"]
    G --> H["Results: Superior vs Elastic-Net & SLOPE"]
    H --> I["Key Finding: Enhanced Performance in Turbulence"]