I am on the stock market does not sound correct. I would say I am at the stock market.
An American input please. I always though that "the stock market" was the action of selling and buying and that the " stock exchange" was the building?
Difference between stock market and stock exchange: http://finance.zacks.com/difference-between-stock-market-stock-exchange-4028.html
Thanks Lng52 for the clarification. The infrastructure mentioned in your reference includes the building in which it operates, e.g. The NY Stock Exchange is I believe on Wall Street. So the only correct answer would seem to be "I am in the stock exchange".