What will Frank copula tell me? As we all know, there are several copula functions, each with its own ability to describe specific dependency structure. I wonder what the Frank copula can tell me. For example, Clayton copula is a lower tail dependency function; that is, the lower values are correlated more strongly than the larger values. However, I could not understand or interpret the plot of Frank. What I will learn from it?
 A: More generally, how do we choose which copula model to use in a given problem? The main guiding principle I learned is to choose a copula model based on the dependence structure of the variables.
Specifically regarding the Frank copula, I'll first introduce two concepts you may know (at a relatively intuitive level, I don't want to start using lots of mathematical notation). A copula is called comprehensive if the copula allows for any dependence structure from full negative rank correlation to full positive rank correlation and also allows for independence. A copula is called archimedean is you can basically model all the dependency of the variables through a generator function.
It can be shown that the only two comprehensive achimedean copulas are Clayton and Frank, so if you want those two properties, you have two choices (I think Nelsen's An Introduction to Copulas has this result). Now how do they differ? Well Clayton is asymmetric, and has more dependence on the negative tail, so you use it when the variables are likely to be jointly low values, but not high. In contrast, Frank is symmetric in the dependence structure, so you use it when the variables are equally likely to be jointly low or high values.
Frank's copula was introduced by Genest (1987), and if you're interested in more motivation for it, I definitely suggest starting by reading that paper:
Genest, Christian. "Frank's family of bivariate distributions." Biometrika 74.3 (1987): 549-555.
