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.
I really do not know where you are seeing the stable price from in this case ?
The price has been fluctuating pretty much, right from the onset, and still doing that. Mining is just one piece of the technical puzzle; it is not like it certainly affects so much in the market.
Most of the times, I would say it is actually the price of the market that affects mining in the long run, as it may not be considered profitable and some may bail out, with the hash rates dropping and then those who are in get to have more incentives, that is just the way it works. So, there is nothing stable here, neither does mining affect price in anyway.