The United Arab Emirates’ rulers will soon achieve a ­distinction they’ve long coveted. Their once sleepy stock exchanges will be home to companies valued at $1 trillion. This outsize  success—the markets already ranked No. 17 in the world at the end of March, ahead of Brazil and Spain—relies on the recent chart-topping performance of the Abu Dhabi Securities Exchange, run by the UAE’s biggest and wealthiest city-state.