U.S. Accounting - Ask Avmar
Starting to pay tax
As a non-resident, foreign national you would not be able to get a US SSN. The tax reporting ID number that you would require (if you have US source income) would be an ITIN (see the following for more official information).
As alluded to by Tom, your tax liability depends on your tax domicile/residence (not necessarily the same "place" as you are actually living/working). Consult an expert for SA regarding possible non-resident liability (globally, non-resident tax issues tend to be somewhat complex - so take all "amateur" advice with a grain of salt, including mine).
The following articles touch on the issues you may need to consider regarding a US stock trading account.
This is the section of Publication 519 mentioned above that references Stock Trading (look under the heading label "Other Trade or Business Activities" - just below the "Non-Resident Aliens" heading) .