You can get some of the Hori sticks for around $100. I have a Madcatz tournament edition stick and it is fantastic. I'm still godawful at fighting games when it comes to serious competition, but the TE sticks are very durable and precise. Unfortunately they run as high as $160, you'd have to snag one on sale. Something like this should be all right, I have no personal experience with them though.
Madcatz SE sticks (Fight Stick, Brawlstick, etc.) are good and fit in your budget. You can also easily swap out the parts for real arcade parts if you decide to get more serious about your stick down the line. I think the Brawlstick has better parts in it than the SFIV Fight Stick did. Not sure if they've done another SE since the Brawlstick.
If you want a higher quality stick, I recommend Qanba or Eightarc. They work on both 360 and PS3/PC, if you play on either of those other two. Honestly, though, any tournament quality stick as as good as another aside from those two working on both systems without additional modding. They'll run higher than your $120 max, though. Generally $150 or so. You can occasionally find them cheaper if you stick it out, though. Sometimes you'll see Amazon try to clear stock of something during a video game Gold Box sale, usually a Madcatz TE. You could see them as low as $100 around that point. I think they were selling the Soul Calibur V TE for $105 last time. --- Even at his most powerless, man's existence is never without meaning. My Backloggery: http://www.backloggery.com/daverost
As mentioned above the hori soul calibur stick is in you budget. Amazon.com has great prices for many budgets. I got the tekken 6 limited edition wireless fightstick bundle from amazon. also a hori stick. It was on sale when i bought it for $50. I bought two and gave one as a gift . I see its $100 now at amazon. Comes with tekken 6 and character art book --- PSWiiPC60DS - Deus Ex, Crysis, Skyward Sword, Black ops 2, Farcry 3 Forza 4, Rage, InFamous 2, Darksiders 2, Vanquish, NBA2K13, Halo 4, Skyrim, Borderlands