I cant imagine bitcoin to become stable, maybe investors will not buy and holds bitcoin as an investment if this would become stable, for me there is no way for btc to become stable because it is decentralized and i never seen btc become stable from the past so it will never ever become stable.
There will always be some level of change in the price since it is open for trade 24/7 in the entire world, so every single person in the planet has the potential to change the price, all the time.
But, over time, these changes will be less and less since there will be more capital invested in it.
Even today it is relatively small, less than some public companies.