Like any other product, its price will go to stability with the expansion of its spread among users and not be in the hands of a single authority capable of manipulating it. Just like with all traditional currencies that we consider stable.
The biggest problem, in my view, that will face this stability resulting from a wide distribution of a very limited amount of bitcoin in its total number, is that it does not accept splitting into more than eight decimals. So imagine if one satoshi equal to 10$ (likely possible t happen sooner than expected) then how becomes possible for the community to share the held while we intend bitcoin to replace Fiat money.