Step 1: Make a live center
I ripped apart an old battery drill, a cheap skil drill from the thrift store for a dollar, Out of the pieces you should be able to save a chuck with shaft and a thrust bushing.
Now bolt a scrap of hardwood to your drill press table and bore a hole through the wood slightly smaller than the shaft of the drill chuck. For example, the chuck shaft was slightly over 5/16th of an inch so I bored a 5/16ths inch hole. Here i'm using a scrap of Mahogany.
Step 2: Making a great fit
Step 3: Almost done
Step 4: Turning
Use a mallet to knock the drive countersink into the work piece and then slip the piece between the chucks and lift the table so the work piece is securely held. Turn the piece by hand a couple of times to make sure it's centered. Now take a deep breath and turn the drill press on at medium speed.
Provided the wood did not fly out you can start rounding it with the shurform blade, once it's round, shaping can be done with the rasps.
and yes I really want real lathe now.