The paper will investigate the possibility of the formation of a speculative bubble in the U.S. stock markets after the 2007-2008 crises. The work is initiated by the observation of historically high equity prices in the stock markets as pointed out by Professor Robert J. Shiller in his new edition of his book ‘Irrational Exuberance’. Until recently the development of a methodology which can be used to detect bubble in real time has been challenging and previous work has mainly focused on ex-post econometric tests in time series that contain only one bubble. The paper uses the recent methodology developed by Philips et. al. that fixes for these deficiencies and enable real time monitoring of statistically significant exuberance and multiple collapsing bubbles within a single time series. Both NASDAQ and S&P 500 indexes are considered and investigation regarding exuberance that cannot be explained by the market fundamentals is conducted.