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

The joint distribution of stock returns is not elliptical