This property is required because the empirically observed densities of returns contrast with the Gaussian model [see Pagan 1996]. This rejection results from two stylised facts. First, large price changes appear more frequently than the normal density would lead to expect. Second, there are indications of significant asymmetry in stock returns. In other words, negative and positive price changes do not have the same probability. These two stylised facts are also apparent in implied volatilities. The plot of the volatilities and their corresponding strike prices shows a U-shaped or inverted J- shaped relation. In the literature, this empirical observation has been termed the smile.
