Miguel A. Ruiz-Ortiz (main author); J. Gómez-Larrañaga; Jesús Rodríguez-Viorato

The Topological Data Analysis (TDA) has had many applications. However, financial markets has been studied slightly through TDA. In this paper we present a quick review of some recent applications of TDA on financial markets and propose a new turbulence index based on persistent homology --- the fundamental tool for TDA --- combining ideas of Gidea et al. (2020), and Baitinger and Flegel (2019), and seems to capture critical transitions on financial data based on our experiment with SP500 data before 2020 stock market crash in February 20, 2020, due to the COVID-19 pandemic. We review applications in the early detection of turbulence periods in financial markets and how TDA can help to get new insights while investing and obtain superior risk-adjusted returns compared with investing strategies using classical turbulence indices as VIX and the Chow et al. (1999) index based on the Mahalanobis distance. Furthermore, we include an introduction to persistent homology so the reader could be able to understand this paper without knowing TDA.