If you can get your hands on a set of left handed HSS (or similar) twist drill bits the job MAY be a whole load easier.
Take your time and get as central as possible with the pilot hole, go up a bit with the second, and then use the largest you dare. As it bites the left handed bit will be trying to screw the stud out, whilst developing a bit of heat down where you need it.
Plenty of penetrating oil too, I've been using 5 / 10% ATF in acetone since learning about on here a few years back. Butch rated the paraffin from the wash down tank in the same thread, as I recall..
If drilling into the stud the 'wrong' way does nothing, you've got the hole you were going to drill anyway
Cheers
Stef