Simple, isn’t it? Mortgage refinance rates are the lowest rates, homeowners can get when trying to refinance their mortgages. Maybe so, but obtaining the very best mortgage refinance rate may prove to be a little more difficult. ... For this amount, the lender may charge you between $3,000 and $6,000. However, some lenders may offer zero points at a higher interest rate, which may significantly reduce your initial costs, although your payments may be somewhat higher.