What else do you propose would set its value? It's not exactly correlated to precious metal so the exchange value is really the only definition isn't it?
The value is set just like for other currencies. Really no difference there, no matter how much some people are trying to say that there is some "real" value behind other currencies. Nothing has a "more real" value behind them, not even gold. It's all based on how limited the supply of it is, and the value of the transactions done it in. That's how all currencies get valued.
The "problem" with Bitcoin right now is that much of its new value is based on rumors and hype. But it's base value (say that of ~$15 or so) is pretty real, as real as it can be. But most or all of the difference between $15 and its current price of $25 or whatever it is now, seems to have been made mostly on hype, and I hope it ends soon. And this is only happening right now because the transaction value is still pretty tiny of around $200 million or so. Get that to $200 billion or $2 trillion, and then this shouldn't happen as often or with as much variability.
Other currency markets are millions of times larger than bitcoin. The entire bitcoin market could be cornered and manipulated by a single modestly wealthy individual.
That's correct. My point is that the original theory of exchange rates had to do with "natural" demand and supply. If a country produces great cheese, and you want to buy that cheese, demand for the currency to buy that cheese is up. This does not apply any more, and many financial markets have little to do with real economic utility. Alas, when gravity hits - and it always does - people are still shocked.