For most qualified HomeBuyers, (Non-Veterans,) the program with the lowest down payment requirement currently is the FHA or Federal Housing Administration. The FHA Insures mortgages made by banks and Mortgage Bankers and allows for a down payment of only 3.5%.
After FHA, Conventional financing through either FannieMae or FreddieMac allows for a minimum down payment of 5%. The Lender will need to obtain approval from a Private Mortgage Insurance company (PMI) to complete the loan approval.
The best way to have your questions about down payment and mortgage qualifications answered is to locate a local Mortgage Banker with at least 15 years experience. Mortgage professionals like myself who work for local Mortgage Bankers are Licensed as opposed to the registered status of sales people at regular banks. Thus you will receive a higher level of service which is just what a First Time HomeBuyer needs in the Bronx.
You can verify the License of any Licensed Mortgage Loan Originator at nmls.consumeraccess.org
Hope this helps!