I guess the reason why bitcoin has a stable price is because miners are mining sometimes or even stop mining at all. The supply of coins freeze and the price would definitely freeze to. Correct me if im wrong but i really believe mining and the price are linked. If one stop moving then the other one will follow.
The mining process didn't stop and will not in the near future. What make the price relatively stable is the supply equal to demand. If the mining activity will cease then a new era for bitcoin will start.