The Fed, if not the ECB, also has full powers of regulation of banks, so, via prudential regulation and directives it could prevent the banks from using the freedom they get from QE to invest in stock and commodity markets. They don't do this because their function, especially the New York Fed and the Board of Governors in D.C., is to serve the financial sector by providing a screen of 'independence' and because they place the interests of the financial sector at least an order of magnitude above the interests of the broader economy. This has always been partly baked in the cake, but, with capture of government via financial sector campaign contributions, it has been locked down. As far as I can tell, the ECB does not even have a small fraction of the actual authority that the Fed has over the financial sector.

"It is not necessary to have hope in order to persevere."