  Using a large set of daily US and Japanese stock returns, we test in detail
the relevance of Student models, and of more general elliptical models, for
describing the joint distribution of returns. We find that while Student
copulas provide a good approximation for strongly correlated pairs of stocks,
systematic discrepancies appear as the linear correlation between stocks
decreases, that rule out all elliptical models. Intuitively, the failure of
elliptical models can be traced to the inadequacy of the assumption of a single
volatility mode for all stocks. We suggest several ideas of methodological
interest to efficiently visualise and compare different copulas. We identify
the rescaled difference with the Gaussian copula and the central value of the
copula as strongly discriminating observables. We insist on the need to shun
away from formal choices of copulas with no financial interpretation.
