You can make yourself an entire set of balsa bits using brass tubes. I use dowel rods inside and turn or sand one end down to fit my drill press chuck. Simple, cheap and works great.
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No, it isn't hard at all to chuck up the brass tube. As this responder has found out, hardwood dowels make a simple stem. Epoxy in place and voila. For balsa, I've filed a saw tooth pattern on the edge of the tube.

Fostner bits are possibly simpler and readily available at hardware stores, so that a great approach. But in a pinch, when you need something right now, you can create the tool yourself