SAN FRANCISCO — As the price of Bitcoin has soared, the virtual currency has edged toward the mainstream.

Square, the fast-growing payments company run by the Twitter co-founder Jack Dorsey, has begun selling Bitcoins to ordinary consumers, and the Chicago Mercantile Exchange will soon allow banks to trade on the value of Bitcoin.

But if you want to see where the price of Bitcoin is actually determined in round-the-clock bidding, you have to go to a number of unregulated exchanges that often fly in the face of American and European laws.

These days, no exchange is bigger than Bitfinex, an opaque operation that provides no information on its website about where it is or who operates the company.