A stock-market company of the same size as Bitcoin now, would have a market-specialist overseeing an orderly market. This chap would be able to go long/short on considerable margin and be free of immediate delivery requirements.
A normal trader woud have to make good on his short positions and deliver borrowed stock on the settlement day.
If the market became too volatile, or one side of the market became illiquid (eg a lack of supply), the specialist would be quite capable of physically closing the market until a more balanced order book could be produced. It's quite common for major SP500 stocks to not open for several hours (or sometimes whole days) whilst the specialist exercises his god-like powers to restore a balanced order book.
Bitcoin's behaviour from here is unique in that there will never be an official, centralised specialist with these powers. The result is likely going to be increased volatiltity.
Which part of Bitcoin do you see as analogous to a "stock market company"? To my mind, the core network (providing payment/settlement services) is more akin to CSDs/Custodians in the securities market.
By contrast, the likes of mt.gox, etc., are the analogue to exchanges. And there's nothing to stop a wannabe market maker entering into a "Bitcoin borrowing" arrangement with a large passive holder of BitCoins and then setting up on mt.gox to sell (these borrowed coins) into other people's buy orders, and vice versa. Sure... this requires a large passive pool of Bitcoins that can be borrowed on demand but would serve the same function, no?
