Copula-Based Distances for Comparing Financial Time Series

Summary

This document explains copula-based measures for comparing financial return series by separating marginal distributions from dependence. It presents Spearman’s rho as a rank-based dependence measure and contrasts it with Pearson correlation, which captures linear association and can be sensitive to noise or unsuitable for distributions without finite second moments. It then describes Generic Parametric Representation (GPR) and Generic Non-Parametric Representation (GNPR) distances, which combine distributional and dependence information through a tunable weight.

GPR is presented as a faster proxy when means and variances dominate, with a stated limitation for heavy-tailed data. GNPR estimates the distance from observed samples, approximating distributions with histograms and using one-dimensional optimal transport for the distribution component. The weight can emphasize distribution, dependence, or a mixture; selecting it is difficult in unsupervised work without a clear objective. The material gives formulas and application examples, but no comparative empirical results. These distances support asset comparison and clustering, while the choice of weighting and distribution approximation affects interpretation.

Key ideas

• Spearman’s rho measures rank dependence and can capture monotonic relationships beyond linear correlation.
• The proposed distance combines similarity in marginal distributions with similarity in dependence.
• GPR offers a parametric approximation that can be inadequate for heavy-tailed data.
• GNPR estimates distances from samples and uses histogram-based distribution estimates with optimal transport.
• The weighting between distribution and dependence information is a modeling choice that is difficult to optimize without a clear objective.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.