# Copula functions

A neat way of quantifying dependence structures between random variables. Useful in, e.g. Quantitative Risk Management.

The trick is simple: Informally, you look at the marginal iCDF of each of $$n$$ variables, and fiddle with the joint distribution of those marginals on $$[0,1]^n$$. (That’s assuming variables are absolutely continuous w.r.t some underlying measure space; distribution with atoms are more fiddly.)

This is a good trick, although I need to sit down and think it through. I would like to better understand:

• the relationship between the underlying event space and the instrumental one we “sort of” construct in copula modeling.
• Is any information lost with non-monotonic coupling in a copula model?
• conditional copulas and how they work
• the occasionally-mentioned relationship between copula entropy and mutual information.

## Elliptical

For now, see elliptical distributions.

## Vine copulas

Hierarchical graphical models, AFAICS.

## References

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———. 2012. Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications. Series in Quantitative Finance, v. 4. London : Hackensack, NJ: Imperial College Press ; World Scientific.
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Nelsen, Roger B. 1999. An Introduction to Copulas. Lecture Notes in Statistics 139. New York: Springer.
Patton, A J. 2009. “Copula-Based Models for Financial Time Series.” In Handbook of Financial Time Series, 767–85. Berlin, Heidelberg: Springer Berlin Heidelberg.
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Schmidt, Thorsten. 2006. “Coping with Copulas.” In Copulas from Theory to Applications in Finance.
Shaw, William T. 2006. Journal of Computational Finance, July.
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Watts, Samuel. 2016. “The Gaussian Copula and the Financial Crisis: A Recipe for Disaster or Cooking the Books?” 25.
Whelan, Niall. 2004. Quantitative Finance 4: 339–52.
Wu, Florence, Emiliano Valdez, and Michael Sherris. 2007. Communications in Statistics - Simulation and Computation 36 (5): 1019–34.

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