On the dependence structure between S&P500, VIX and implicit Interexpectile Differences

We study the dependence structure between the S&P500, the VIX Index, and implicit Interexpectile Differences, that are an alternative measure of implied volatility based on the notion of implicit expectile, recently introduced in Bellini et al. [Implicit expectiles and measures of implied volatility. Quant. Finance, 2018a, 18, 1851–1864]. After filtering the time series of the marginals by ARMA-(E)GARCH models, we fit several parametric families of copulas to the pairwise joint distribution of the residuals, in order to investigate the presence of radial asymmetry and asymptotic tail dependence. We find a negative dependence between S&P500 and both implied volatility indices and a positive dependence between VIX and Interexpectile Differences. The best fitting copulas seem relatively stable over time and display both asymmetry and strong tail dependence, in accordance with the leverage effect.