Pricing bitcoin
When Bitcoin is a better offering than existing financial products and services, more folks start using Bitcoin. Which, in turn, drives up the buying pressure and therefore the price of Bitcoin.
Any time you see the price go up independent of increased adoption for real-world use cases, the increase is insubstantial.
So what's a "fair" price for Bitcoin?
Bitcoin displacing current players
This is a matter of opinion, and if you could do this perfectly you'd be very wealthy. But I believe the best (rough) framework for pricing Bitcoin is to estimate the market capitalization that Bitcoin has earned based on the products/service it is displacing.
Bitcoin has a lot of promise but it's not yet displacing a lot of big players. So you have die-hard believers thinking it's work $100k+ per Bitcoin and people that see actual usage of Bitcoin and estimate closer to $1k - 10k.
International Remittance
The international remittance market is worth ~$700B with about ~$600B of transfers per year at the time of writing. (source) Some argue Bitcoin is superior to current players in this market. If Bitcoin was clearly better than the current systems then both of those values would be $0.
There is a time lag between Bitcoin being better and actually displacing those market participants, but you can think of the "value" of Bitcoin in this context as the % of international remittances would be better served by Bitcoin multiplied by the total value of the market.
Bitcoin isn't clearly better in all cases yet, but based on my understanding (via anecdotes from immigrants who send money home, and raw research on the industry) Bitcoin is clearly better than older players like Western Union in a lot of cases. Mostly because of how high their fees are compared to buying Bitcoin and sending it to a family member overseas. But Wise (formerly TransferWise) has considerably lower fees than Western Union and better convenience than Bitcoin.
So the market cap of Bitcoin that comes from displacing international remittance is some % of times Bitcoin is better than existing providers, call it 30%. That'd make Bitcoin's value based on remittances to be ~$210B (it's currently ~$310B). That 30% number is where it takes research and expertise in that industry.
You can think of the total value of Bitcoin as the sum of all the market caps of all the systems Bitcoin is displacing times % of the time Bitcoin is a better solution in that system.
Bitcoin doesn't just compete with international remittance companies, there are many things that Bitcoin is capable of (payment network, a messaging protocol, store of value, etc). Bitcoin just isn't better than all existing players in all of these markets, but as it does improve the value of Bitcoin will increase apace.
Here are some things Bitcoin has a decent shot at displacing in the near future.
Gold / inflation hedge / store of value
Good stores of value have a few distinct traits. Their price is pretty stable, they're physically dense, they have a high stock to flow ratio, they're easily divisible, they're easily transmissible, they're easy to protect against thieves, and they're easy to prove ownership. (best outlined in The Bitcoin Standard).
Gold has traditionally been the best at this, but transferring and storing gold became too slow for the velocity of money leading the world to get off the Gold Standard.
Now the US Dollar is currently the best at this.
Bitcoin has a better stock to flow ratio than the dollar and gold. Bitcoin is far denser (you can hold $1B in your hand). Bitcoin is much more easily transferable, and it's trivial to prove ownership. Dollars are pretty easy to transfer, and pretty easy to prove ownership, but it's recently stock to flow ratio has plummeted which has made Bitcoin look better.
The last change that needs to happen is for Bitcoin's price volatility to drop. Which will happen as more adoption happens for other use cases. Currently Bitcoin is less volatile than the S&P 500 (source). Then we'll start to see proof in action that Bitcoin is better than gold and the US Dollar for this use case.
There's about ~$11T in gold in the world right now (source) and therefore $11T of market cap for Bitcoin to capture.
Payment network / Visa
Merchants pay ~3% fee for all transactions that aren't done in cash (physical, ACH, or wire) using providers like Visa, Mastercard, Amex.
With protocols like Taro and RBG and using networks like Lightning and Liquid we are entering a world where merchants will have a competing payment providers that use Bitcoin under the hood.
It'll soon be possible to have instant cash settlement in any currency (via stablecoins on Bitcoin Layer 2 solutions) with fees far lower than 3%. Lightning currently allows small and medium transactions (a penny to hundreds of dollars) to settle in seconds for less than a penny. Once we can have stablecoins on Bitcoin, merchants won't even need to hold or even understand Bitcoin to benefit from these solutions.
When this has been proven out, restaurants, grocery stores, and other merchants will be begging people to use these solutions to pay them instead of using a credit or debit card. That'll increase demand for Bitcoin since it's what's under the hood.
As of the time of writing the market caps of Visa + Mastercard + American Express are $750B combined. The current market penetration is likely less than 1% but it'll grow rapidly with what's on the near-term roadmap.
National currencies
Bitcoin doesn't just compete with private companies, it will also compete whole national currencies and international institutions.
Many national currencies are more volatile than Bitcoin and so citizens are starting to use Bitcoin as an alternative in those countries (source).
There will come a time where Bitcoin has a more stable purchasing power than the US Dollar (if/when it becomes a better store of value), at that moment it'll be a no-brainer for countries that want off the US Dollar-based institutions to issue their currency backed by Bitcoin instead of Dollars.
This won't happen with the major currencies like the Yen, the Yuan, the Ruble, the Pound, the Euro for a looonnnng time, but there's a lot of the world that isn't using these currencies. In that case the entire value of money supply of each currency that makes the switch to Bitcoin backed will add to the market cap of Bitcoin.
The endgame
Bitcoin has a shot at being a better store of value, medium of exchange, and unit of account than the US dollar, which currently dominates all international institutions.
At that moment the product/service (and therefore "market cap") Bitcoin would be displacing is the total amount of US Treasuries in circulation ($40T as of time of writing).
For reference that would be a per-bitcoin price of $1.9M.
Build don't speculate
It's going to be a long road, even if we do end up in a future where Bitcoin dominates these markets. I do think a Bitcoin future is a better one. A future that displaces a lot of corrupt (source: how the dollar keeps countries poor) and colonial systems (source).
Instead of focusing on the price of Bitcoin and speculating, we need to keep building the protocols, tools, and companies that will allow Bitcoin to flourish.