We present here two examples of a large deviations principle where the rate function is not strictly convex. This is motivated by an example from mathematical finance, and adds a new item to the zoology of non strictly convex large deviations. For one of these examples, we also show that the...

We study the shapes of the implied volatility when the underlying distribution has an atom at zero. We show that the behaviour at small strikes is uniquely determined by the mass of the atom up to the third asymptotic order, under mild assumptions on the remaining distribution on the positive...