I was just pointing out that the ALLOY is HSS through and through. The hardening is not necessarily the same throughout (nor should it be). In theory, good bits should be hardened fully to just past the base of the flutes, leaving most of the remainder of the shank is a relatively soft state (there is a visible line on most of my cobalt bits, but that gets erased with most surface finishes on other types of bits). So, it SHOULD be soft enough to file, and therefore maybe machine (with some difficulty). But aside from doing this as a one-off in a pinch, buying a set to fit your drill is so much easier. Bits with 3/8" and even 1/4" reduced shanks are easily sourced.
Oh, and for absolute correctness, there was a mention about S&D bits above. Silver & Deming is generally accepted to only refer to 1/2" reduced shank bits in a fractional set from 1/2" to 1". That doesn't apply to 3/8" shank.