They're peddled on the bond market, not the stock market.
And many securities are handled in just this way. Do you propose to eliminate the bond market (which is three times the size of the stock markets)?
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It was not previously the case where home mortgages were bundled and sold as if they were bonds. Banks used to own the mortgages and collect the payments on them for 30 years. It would not be all that difficult to prohibit the sale of bundled home mortgages.
They were sold from bank to bank as safe secured securities. They were only renamed 'sub prime mortgages' when this whole mess started to head south.
Variable rate mortgages, mortgages with huge baloon payments and somesuch crap are not necessary, either.