A high-low-based omnibus test for symmetry, the Lévy property, and other hypotheses on intraday returns

We consider different models for intraday log-returns: Lévy models, symmetric models, and Lévy processes subjected to independent continuous time-changes. For these models, we show bivariate interchangeability of intraday upand downside volatility ratios which are built using daily high-low-prices. Using conditional inference permutation tests on bivariate interchangeability, we develop an omnibus test for the abovementioned models. Empirically, we find strong evidence against intraday returns belonging to these model classes, as we reject bivariate interchangeability of the volatility ratios for half of the components of the DJIA, two thirds of the S&P 500 shares and almost all stocks of the German DAX.