How can we trust big exchanges?
There are multiple exchanges each with its own BTC/USD (or other currency pair) , so arbitrage between exchanges is what keeps prices among exchange inline.

There are short term disruptions when those providing arbitrage are out of cash (for buying) and a lesser disruption for moving bitcoins (for selling).  The selling limitation is smaller because bitcoin transactions confirm in about an hour.

There are permanent price deltas between exchanges due to the cost of moving funds and trading fees.  For instance, it is hard to get USDs to BTC-e, and therefore the selling there can be relatively far below the exchange rate at Mt. Gox -- a difference as much as 10%.  But the direction up and down are generally in sync thanks to arbitrage.