They simply borrow money from the exchanges to make those trades, but the trouble is, the market movement is basically random enough that 20%, 10%, 5% price changes in the wrong direction can completely wipe out your position because that's basically the "risk" for the exchange to place such positions on you.

Exchanges are simply giant order books anyway so if there was an exchange that was secretly holding open positions, that would be a scandal since they would basically be trading with itself and creating a fake trading volume.