The skew pattern of implied volatility in the DAX index options market

The aim of this paper is twofold: to investigate how the information content of implied volatility varies according to moneyness and option type, and to compare option-based forecasts with historical volatility in order to see if they subsume all the information contained in historical volatility. The different information content of implied volatility is examined for the most liquid at-the-money and out-of-the-money options: put (call) options for strikes below (above) the current underlying asset price, i.e. the ones that are usually used as inputs for the computation of the smile function. In particular, since at-the-money implied volatilities are usually inserted in the smile function by computing some average of both call and put implied ones, we investigate the performance of a weighted average of at-the-money call and put implied volatilities with weights proportional to trading volume. Two hypotheses are tested: unbiasedness and efficiency of the different volatility forecasts. The investigation is pursued in the Dax index options market, by using synchronous prices matched in a one-minute interval. It was found that the information content of implied volatility has a humped shape, with out-of-the-money options being less informative than at-the-money ones. Overall, the best forecast is at-the-money put implied volatility: it is unbiased (after a constant adjustment) and efficient, in that it subsumes all the information contained in historical volatility.