The Latics will also pay further £10,000 instalments should the 19-year-old reach 15, 30, 45 and 60 first-team appearances for the club.
If John Sheridan's side win promotion to the Championship in the next two years, a lump sum of £25,000 will also be payable, providing Stephens has played in at least half of the fixtures.
As part of the deal, the Shakers will also be entitled to 20 per cent of any transfer fee should the player be sold in the future.
Chief executive Alan Hardy told the club's official website: "We're relatively happy considering the initial demands we received from Bury.
"I suppose out of the two parties, they'll be the more disappointed."
The teenager has failed to make a first-team appearance for Sheridan's side since moving to Boundary Park in the summer.