1. Don't transfer to your name, but continue paying under the present owner's name. Separately you get a letter from the owner saying that vehicle had been sold to you, including transfer papers. He also have to give a letter to the Finance company saying, he has no objection of releasing original vehicle registration book to you when the deal is completed. You can also write an agreement with a lawyer, so no risk.
2. If you transfer it under your name, normally it will be a new leasing facility as I know. Not a good idea.