Exchange rates are set by supply, and demand. Bitcoin has through technical means controlled the supply part of the equation, but how can it control demand? So fluctuations in demand will cause changes. In the growth period, Bitcoin can't be stable, and it would be impossible for any currency to do so without extremely precise statistics on how a currency was being used and control over inflation (so you could, say, adjust supply exactly to match fluctuations in demand).
In the long run, most of the obvious equilibriums for Bitcoin are very far away from whatever it's currently trading at. If Bitcoin fails and becomes playmoney, then it's overpriced by hundreds of times; if it succeeds and replaces any sovereign country's currency, then it's undervalued by hundreds or thousands or millions of times. So in the long run, you must expect it to not be 'stable', but to change drastically - in some direction.
(If this isn't convincing, try applying the double reversal test Nick Bostrom devised for combating the status quo bias (http://www.nickbostrom.com/ethics/statusquo.pdf) and anchoring (http://en.wikipedia.org/wiki/Anchoring). Suppose that Bitcoin were at $2 rather than $4. Would you be saying 'obviously the right price for Bitcoin is $4, so I should buy some'? I doubt it. )
When Bitcoin is supposed to be stable is when demand for it is not varying; supply will be stable by the Bitcoin architecture, and hence, the overall equation wouldn't change in that situation. This differs from fiat currency in that you are only vulnerable to changes in the demand part of the equation - there is no Fed or other cabal that can hurt you by manipulating the supply part of the equation.
Imagine Amazon starts accepting bitcoins. The rate will probably skyrocket to more than 100 USD.
Actually, digital currency is inevitable, it's only a matter of time till there's some global currency not controlled by one government. Bitcoin is a pretty good candidate.
In the long run, most of the obvious equilibriums for Bitcoin are very far away from whatever it's currently trading at. If Bitcoin fails and becomes playmoney, then it's overpriced by hundreds of times; if it succeeds and replaces any sovereign country's currency, then it's undervalued by hundreds or thousands or millions of times. So in the long run, you must expect it to not be 'stable', but to change drastically - in some direction.
(If this isn't convincing, try applying the double reversal test Nick Bostrom devised for combating the status quo bias (http://www.nickbostrom.com/ethics/statusquo.pdf) and anchoring (http://en.wikipedia.org/wiki/Anchoring). Suppose that Bitcoin were at $2 rather than $4. Would you be saying 'obviously the right price for Bitcoin is $4, so I should buy some'? I doubt it. )
When Bitcoin is supposed to be stable is when demand for it is not varying; supply will be stable by the Bitcoin architecture, and hence, the overall equation wouldn't change in that situation. This differs from fiat currency in that you are only vulnerable to changes in the demand part of the equation - there is no Fed or other cabal that can hurt you by manipulating the supply part of the equation.