Symmetry alteration of ensemble return distribution in crash and rally days of financial markets

We select the $n$ stocks traded in the New York Stock Exchange and we form a statistical ensemble of daily stock returns for each of the $k$ trading days of our database from the stock price time series. We study the ensemble return distribution for each trading day and we find that the symmetry properties of the ensemble return distribution drastically change in crash and rally days of the market. We compare these empirical results with numerical simulations based on the single-index model and we conclude that this model is unable to explain the behavior of the market in extreme days.