Does anyone know how difficult it would be to come out of Chicago as an undergrad and get a position in Trading Fixed Income in NYC?
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How difficult? There aren't many people that become FI traders straight out of undergrad. IMHO your goal should be to obtain a FI "analyst" position and use it as a stepping stone to trading. I'm not saying that it can't be done but "entry-level" FI trading positions are usually reserved for those with an grad degrees in a quantitative discipline or people that have proven themselves via an analyst program.
Univ of Chicago is a solid school. For starters check out the BB firms' websites and see which ones recruit on campus.