Honest chances for S&T internship (I have read everything! Please advise)

Hey there everyone,

First of all, let me say this is a fantastic site to stumble across. Even just scrolling through forum topics, there is a "oasis" of information that you'd never think of asking about in the first place...thanks to everyone who is actively involved.

Anyways, I did a year at the top music university in Canada for Opera (yes, opera) and then transferred into Economics at another very well known Canadian university. Their business school is actively recruited, but not the Arts/Science faculty. Over this past academic year I fast tracked second and third year, so although technically I'm a junior I am really a sophomore/junior combination with the new program.

I have a 3.8/4.0 GPA, I'm actively involved in freelance music outside school, clubs inside school, and I have this years award for best undergraduate writing piece (research essay) in the economics department.

I don't, however, have any internship experience and I'm looking to get some with a top bank. Specifically in S&T. I was wondering if I should be cold contacting HR people, or whether with my academic credentials and personal history I have a shot at interviews by simply applying off the websites when the positions are posted?

What do you more experienced folk suggest I do? I've been trading for a few years now on my own, but I'd like to do it in an institutional setting.

Thanks!