Of course you can trade the US financial markets from Europe.
You can open an account at TD Ameritrade
Opening a foreign brokerage account requires:
-a signed application (online);
-a W-8BEN (Certificate of Foreign Status) form
-a copy of a bank/financial statement, translated into English, with the name and
address matching the ones on the application;
-and a copy of each account owner?s current, valid passport.
Each account owner must fill out a separate W-8BEN form. The W-8Ben form must be
mailed back to us, not faxed. All other forms may be faxed (816-243-3769 ) or
mailed. All account applications are subject to review and TD Ameritrade reserves
the right to deny any account application. Submission of an application does not
guarantee that an account will be established.
Please mail to one of the following addresses.
REGULAR MAIL
TD AMERITRADE, Inc.
PO Box 2229
Omaha, NE 68103-2717
U.S.A.
OVERNIGHT & REGISTERED MAIL
TD AMERITRADE, Inc
1005 N Ameritrade Place
Bellevue NE 68005
U.S.A.
Of course you don't have to use TD Ameritrade but it is the company I know of and the above was a message they sent me when asking a question similar to yours. I wanted to open an account from the Caribbean.
Good Luck.