I am an international student studying at Villanova University in Philadelphia, PA. I am very interested in trying my luck at the U.S stock market. However, I do not work on campus and so do not have a Social Security Number. Is there any way around this using which I can trade stocks using an online account?
You need an ITIN. Follow the instructions on the IRS page to apply. You might be better off getting an on-campus employment authorization and getting an SSN, though, as the ITIN process is not really convenient.