Two different flavours of complexity in financial data

We discuss two elements that define the complexity of financial time series: one is the multiscaling property, which is linked to how the statistics of a single time-series changes with the time horizon; the second is the structure of dependency between time-series, which accounts for the collective behaviour, i.e. the market structure. Financial time-series have statistical properties which change with the time horizon and the quantification of such multiscaling property has been successful to distinguish among different degrees of development of markets, monitor the stability of firms and estimate risk. The study of the structure of dependency between time-series with the use of information filtering graphs can reveal important insight on the market structure highlighting risks, stress and portfolio management strategies. In this contribution we highlight achievements, major successes and discuss major challenges and open problems in the study of these two elements of complexity, hoping to attract the interest of more researchers in this research area. We indeed believe that with the advent of the Big Data era, the need and the further development of such approaches, designed to deal with systems with many degrees of freedom, have become more urgent