  This paper employs Topological Data Analysis (TDA) to detect extreme events
(EEs) in the stock market at a continental level. Previous approaches, which
analyzed stock indices separately, could not detect EEs for multiple time
series in one go. TDA provides a robust framework for such analysis and
identifies the EEs during the crashes for different indices. The TDA analysis
shows that $L^1$, $L^2$ norms and Wasserstein distance ($W_D$) of the world
leading indices rise abruptly during the crashes, surpassing a threshold of
$\mu+4*\sigma$ where $\mu$ and $\sigma$ are the mean and the standard deviation
of norm or $W_D$, respectively. Our study identified the stock index crashes of
the 2008 financial crisis and the COVID-19 pandemic across continents as EEs.
Given that different sectors in an index behave differently, a sector-wise
analysis was conducted during the COVID-19 pandemic for the Indian stock
market. The sector-wise results show that after the occurrence of EE, we have
observed strong crashes surpassing $\mu+2*\sigma$ for an extended period for
the banking sector. While for the pharmaceutical sector, no significant spikes
were noted. Hence, TDA also proves successful in identifying the duration of
shocks after the occurrence of EEs. This also indicates that the Banking sector
continued to face stress and remained volatile even after the crash. This study
gives us the applicability of TDA as a powerful analytical tool to study EEs in
various fields.
