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Copula Sampling and Fitting for Pairs Trading

Article Hudson & Thames
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Summary

This technical article explains how to sample from and fit bivariate copulas, which model dependence between two variables separately from their marginal distributions. Sampling from a fitted copula can help compare simulated quantile pairs with historical observations, but it does not generate a sequential price path: the copula treats observations as independent draws and contains no time-order information. For Archimedean copulas, sampling uses a generator-based procedure; for Gaussian and Student-t copulas, it transforms draws from the corresponding multivariate distribution. A mixed copula is sampled by first selecting a component according to its weight.

For fitting, the article describes mapping marginal observations to quantiles with empirical CDFs and estimating dependence through Kendall’s tau, with maximum likelihood used for some parameters. It also outlines an EM approach for mixtures, whose flexibility can capture tail dependence but makes estimation harder. Model fit can be compared on one dataset with log-likelihood or information criteria, while evaluation across datasets lacks a widely accepted standard. Limits include nonstationary marginals, thin-data issues with stepwise ECDFs, slow or unstable optimization, and difficulty distinguishing similar mixture components. The methods calibrate dependence; they do not by themselves define a trading strategy.

Key ideas

  • A copula models dependence between variables separately from their marginal distributions.
  • Copula samples are useful for quantile comparisons but do not preserve time-series sequence.
  • Empirical CDFs can map marginal observations to quantiles before estimating copula dependence.
  • Mixed copulas can represent richer tail behavior, but their fitting can be unstable or slow.
  • Stationarity and model evaluation remain important limitations when fitting copulas to financial data.

Tags

Pairs tradingStatisticsArbitrage

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