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.
Bitcoin doesn't have a stable price at all.And even if it seems to be stable,it is only for a very small duration and also a small variation seems to exist at that price also.Miners have nothing to do with stability of bitcoin price.