Bitcoin will not be able replace fiat for now because of several factors. Among them are unstable bitcoin exchange rates and international laws protecting bitcoins dont' yet.
Yes there are a lot of factors that bitcoin really can't be considered for replacing fiat. Even the price is getting higher and higher nowadays still that is not going to matter because in the end the government is the one that will be followed and law makers are coming from the. But let's see one day we will be having paperless transactions.
It's not about the price but on the adoption, fiat is adopted by all people in the world and even users of bitcoin is using it regularly, with the success of bitcoin will just make the transaction faster and will also increase the circulation of fiat, it's like fiat to bitcoin and then bitcoin back to fiat so at the end of the day all bitcoins will be converted to fiat in order to enjoy its value.
bitcoins value will be derived from how many people use it. when the user base of bitcoin becomes large enough, its conversion price to fiat currency will naturally stabilize and we wont see any crazy price swings like we did in the past.