Bitcoin nature is such that it can not be stable because it is speculative and also decentralized. This is why the price of bitcoin is volatile. Currency that can be stable if it is centralized and controlled by the government. Even if bitcoin finally is globally accepted and replaced fiat money but still remains decentralized it will never be stable but continue to be volatile.
Bitcoin's nature is "A peer to peer to peer electronic cash system" that is the vision Satoshi gave to it from the very beginning.
What does it mean? First, it is cash, second, it is used for transactions/payment, speculation is just the temporary phenomenon as bitcoin still in its teenage.
So if it can be used for transaction/payment: 1) Its total market cap should be huge, much more than current $100B, maybe ten times more, which means the price of bitcoin will keep going up; 2) It tends to be stable & atcs as currency function or we can neglect its volatility when we use it as a payment method.
Of course it will not replace fiat currency as national digital currency is on its way to us, bitcoin will act as the real global currency, more than others.