I did not buy the Chisox Hat, as I thought the dealer wanted far too much (as nice as it was). But I will say that your Pennant looks good. It would be nice if there was a tag stitched on the reverse... do you see any remnant of a cloth tag?
The materials and composition look to be authentic, and it could date to as early as the teens. Tough to pinpoint, as there is no date, and the "SOX" design is fairly generic in nature. If forced to guess, I'd say late teens/early twenties on this one. Of course most dealers would call it 1919, and then jack up the price accordingly
Great pickup, and a really attractive piece... congrats!