The Danish model has another critical and innovative feature. Holders can retire their own mortgages by purchasing the same face amount of mortgage bonds at the prevailing market price. To prepay a mortgage by purchasing bonds, the home owner must give advance notice of several weeks to the MCI [mortgage credit institutions], which designates by lottery the specific bonds to be purchased. Thus, if rising interest rates or other factors cause mortgage bonds to trade at a discount, home owners can reduce the principal or retire the whole mortgage by purchasing an appropriate mortgage bond at a discount.
That passage is from Robert Pozen's new and notable Too Big to Save? How to Fix the U.S. Financial System.
You can't do this in the United States. You can pay off your mortgage but the "face value" of that transaction does not vary with market conditions. In essence the Danish system creates a new contingent claims market for homeowners who do not understand how to use interest rate futures and options. De facto, the homeowner receives some implicit insurance against the prospect of negative equity in the home.
Here is The Economist on the Danish model. Denmark also allows for speedy repossession of property, in case of default. Here is a more general discussion of the Danish model, which emphasizes transparency. Mortgage finance is conducted by explicitly designated institutions and originators retain a financial interest in the loan, even following securitization. The emphasis is on plain vanilla products. Here is Wikipedia on the Danish mortgage model. Here is a longer study.