Except Bitcoin has no value. That is implicit based on how volatile the trading price for it is. Actual currency has value based on a very complicated myriad of processes, economics, supply and demand, etc... The value goes up and down generally based on some form of cause and effect, supply and demand.
So,
exactly like all other currencies, then?
Bitcoin has the same
value as any currency that people trade in (intrinsically none, but practically whatever expectation the market has today that someone, somewhere, will want some tomorrow - if only because the market expects that someone, somewhere, will still want some the day after tomorrow...).
What it lacks is the
stability you get from having a large pool of guaranteed users, due to some entity with whom they do business (usually a government) accepting only BTC in payment (typically of taxes).
The
value of any currency is mostly a feedback loop. For any fiat currency, it is
only a feedback loop.
Stability is a function of how unlikely it is that a large new injection of money will occur, and of how likely it is that someone will want that particular currency more than any other.
When these two factors are balanced, a currency is stable. Demand for US Dollars has a hard floor, because Uncle Sam will accept no other currency in payment of taxes; It has a fairly smooth pattern above that floor, because the world trade in USD is fairly constant in the short to medium term; And the supply of USD is matched to the demand curve fairly well by the inflation management policies of the Federal Reserve. Indeed, "inflation management policies" is synonymous with "currency stability policies".
BTC is volatile because it lacks a hard demand floor, it is too small a player to have fairly smooth demand, and nobody is trying to match supply to demand.
The same would be true of gold, if not for the much larger demand pool for gold caused by its widespread use as a hedge against instability. And of that sounds circular, it's because it is. Gold is a good investment because everyone thinks that gold is a good investment.
If enough people are convinced (and remain convinced) that BTC is a good investment, then it is. If and when they stop being, it won't be.
Just like gold.