Indicator from the graph Laplacian of stock market time series cross-sections can precisely determine the durations of market crashes

Stock market crashes are believed to occur unpredictably and have profound negative impacts on the economy and society. However, there is no universally agreed-upon definition of stock market crashes, whether it is an actual market state (implying that there is a start and an end) or just a transition between two different states (implying that it is a point event). Conventionally, extreme events in the financial markets can be determined using various change-point detection methods. However, these methods typically rely on a model of the time series data and/or use sliding time windows. Expanding on our previous work, we propose an alternative way of defining market crashes as short states by utilizing information cross-filtering by two time windows of the time derivative of the maximum Laplacian spectral gap across filtration parameters 1.0≤ϵ≤1.8. When we applied this method to analyze the time derivative of the maximum spectral gap for S&P 500, Nikkei 225, SGX and TWSE, we found persistent peaks (found across different time window widths) associated with the COVID-19 crash starting in March 2020 and ending only in April 2020. These dates correspond roughly with the highest point before the crash and the lowest point after the crash seen in the indices. We also found non-persistent peaks (found only across short time windows or long time windows) before and after the COVID-19 crash. The explanations for these non-persistent peaks are peculiar to individual markets, and also particular market crashes such as the 2008 Global Financial Crisis. Based on this work, we argue that a definition of market crash in terms of a duration is more natural and perhaps more useful for risk management.