The research involves examining how financial markets, particularly the NASDAQ and S&P 500 indices, react when under stress, as well as applying advanced time series techniques in an attempt to predict crashes. Accurate prediction of crashes is important due to the tremendous impact financial market collapses, including the 2008 and COVID-19 epidemics, have on the worldwide economy. To model non-linear market dynamics, the study combines dynamic GARCH extensions and wavelet-based time series decomposition with ARIMA and GARCH models to forecast market volatility. The sample period ranged from January 2021 to August 2024, with total observations of 787 and 921 for the S&P500 and NASDAQ, respectively. The selection of the ARIMA and GARCH models was confirmed by the ADF and PP tests to determine whether the time series is stationary. The GARCH model with the GARCH effect of 0.912741 has most certainly accommodated the volatility clustering phenomenon, due to which an episode of high (low) volatility was followed by another episode of the same kind and successive spikes in the volatility, especially in the case of NASDAQ. The volatility persistence of the S&P 500 was lower (0.6785330 GARCH effect). For a relatively small level autoregressive table, the forecasts demonstrate that the variance of S&P 500 substantially increases in high volatility periods for most by up to 0.006. The NASDAQ was somewhat more persistent, as indicated by a variance of 0.00024. These findings illustrat...