It depends how you measure value. By stabilise I mean stops growing in value by 50%/yr with big short term swings of 80%.
As it matures and gets close to it's ultimate value, volatility will naturally reduce.
Once it is used as the unit of account, everything else will fluctuate in value relative to bitcoin, which has more stable fundamentals than anything else on earth (fixed/zero issuance, liquidity etc), but this will be decades in the future when it's dollar value will be 8 or 9 figures in today's money
As it matures and gets close to it's ultimate value, volatility will naturally reduce.
Once it is used as the unit of account, everything else will fluctuate in value relative to bitcoin, which has more stable fundamentals than anything else on earth (fixed/zero issuance, liquidity etc), but this will be decades in the future when it's dollar value will be 8 or 9 figures in today's money