Lotsa typical ET responses. My suggestion if you want to get on a major trading desk is to be persistant. I would get the name of the person in charge of the specific area of trading you're looking to break in . Send a 1 page resume with a short cover letter stating that you will contact this person next week.
Next week comes, call once pre-market (around 7- 7:30 AM) and once after the mkt. always leaving your name (this way at least he will know ur name). Do this for the week.
By chance if you get him on the phone, you have about 2 mins. to sell yourself and to set up an interview (so think very carefully about what you are going to say). I would refrain from telling him ur track record over the phone. Basically just say you have always wanted the opportunity to beak in on a successful trading desk such as _____. Tell him you're willing to do whatever it takes (12 hrs/day 6/wk etc...), and want to set up a meeting so he can see what youre all about.
If you cant get him on the phone after the first week of calling, put on your best suit and go there in person (either premkt, lunch or aftrmkt). Do that every other day for the week. This should at least get u in the door.
The point of doing it this way is that the Man. Dir. of trading usually worked his way up through the trenches and will def respect your persistance and tenacity, at least enough to interview you.
Do this with ten firms and you should be able to get a few interviews and very possibly your dream job.
Best of luck and dont let anybody say you cant do it. Oh yeah, this approach works a lot better if you are young (early 20's).