In passing I’ve seen what look like suggestions that European banks could simply buy debt from their home countries and park it at the ECB. Thus, arbitraging the difference between the overnight rate and the rate on sovereign debt.
In its simple form, however, this will not work because the Sovereign debt is marked-to-market. Thus, if you buy primary bonds at 3% but they then trade in the secondary market at 6%, the value of the bonds as collateral at the ECB will decline to the 6% price, which on top of you have to take a haircut.
This dynamic is actually why you can have well subscribed auctions for Italian debt, but at very high yields. Its good stuff to have but only at the price it trades at in the secondary.
In order to keep this backend problem from biting the bank basically needs to make a commitment to the government that is big enough that it calms market fears about the ability of the government to roll over debt.
However, if a bank can make such a guarantee then the market yield on primary and second debt will collapse.
So, you actually need some sort of scheme to profit from this arbitrage. You need to be able to capture the yield decline that comes from making the Lender of Last Resort guarantee.
This type of scenario is what I tried to sketch here.