Non-linear Leverage E ects

We introduce the concept of ordinal pattern dependence between time series and show in an explorative study that both types of this dependence show up in real world nancial data. The classical way to capture the leverage e ect in models for stock markets is to assume a negative correlation between the two datasets which is constant in time (e.g. Barndor Nielsen and Shepard (2002)). However, there is strong evidence that this e ect is not constant, but itself evolves in time. It seems that there are periods where the e ect is weaker and sometimes it even seems to be turned around , i.e., there is a positive correlation between the two datasets (Carr and Wu (2007)). Taking these empirical ndings into account, more sophisticated models where suggested. The correlation structure was modeled by a deterministic function or was made state space dependent. In Veraart and Veraart (2010) the correlation structure is itself a stochastic process on [−1; 1]. Instead of proposing even more complicated models we introduce a rather simple approach to analyze whether there is a dependence structure between two datasets. In order to capture the zikzak of the datasets we use so called ordinal patterns. This method was developed by Bandt (2005) and Emonds et al. (2007) in order to handle time-series with several thousands of data points which appear in medicine, meteorology and nance (cf. Keller and Sinn (2005)). We compare the two datasets from this point-of-view. On some occasions, as an example we will consider the S&P 500 and the corresponding volatility index VIX, a dependence structure of this kind seems to be more likely to be found in real data than the dependence modeled by the classical approach via correlation (cf. Whaley (2008)).