Hello, I have not seen a lot of examples of applications of Fundamental copulas and Implicit copulas. Is that something we should know for the exam?
I don't think there are any examples of the applications in the Act Ed notes.
An implicit copula example is the Gaussian copula and the student - t copula. Uses;-
Gaussian Copulas have zero tail dependence and are used in the pricing of Synthetic CDO with underlings made up of a basket of CDS. You should remember some Xi Li guy!
Student - t copula is used in a similar way for the valuation of synthetic CDO, Hull and White show that a good fit to the market is obtained when marginal distributions for F (a common factor affecting defaults for all companies) and Zi (a factor affecting only company i) follow Student t distributions with four degrees of freedom. They call this the double t copula.
The problem is the joint estimation of the covariance matrix and the degrees of freedom for the student - t, however you can use the degrees of freedom that results in a good fit to the tails in your data.
As for fundamental copulas, these aren't really copulas. They are special cases in which marginals can be "coupled" depending on the associated dependence...
...for example set rho to 100% for the Gaussian Copula (try bivariate case) and you will get the "minimum" = "comonotonic" = lower Frechet-Hoeffding bound COPULA! Set the correlation to 0 and you should get the independence copula.
Proof;-
C(u,v) = Phi_1 ( Phi^-1(u) , Phi^-1(v) ) = P ( X < Phi^-1(u) and X < Phi^-1(v) ) = Phi ( min ( Phi^-1(u), PHi^-1(v) ) ) = Phi ( Phi^-1( min ( u,v ) ) ) = min (u ,v)
ALSO see Sweeting page 303 it shows how x is a monotonic transformation of y i.e X=f(Y) for a minimum copula and gives the sterling example if you want to see an example in which the Frechet - Hoeffding copulas can OCCUR...
....the answer to your question is they are axioms that describe what is going on, not a tool for any use. But maybe Tutors can help here if I am wrong.
