At this moment, you need about $400-800 million dollars to attack the $44 billion Bitcoin chain (and in PoW too, you can get some or even all of them back by short selling in the right moment).
This $800 million sounds about right, but can you source or show how you calculated this number?
This makes PoW seem way less secure than PoS.
PoW is also a huge pointless electricity drain on the world, already using more electricity than Iceland, the country.
If prices go 1000x higher as we like, it'll be 1000 Icelands, or entire continents of power just for PoW.
PoW therefor seems to have no future.
There is no fixed amount that will allow you to attack Bitcoin as it depends on how long you run the attack. It is a different story on how much you would have to spend on the initial costs for buying the mining rigs.
If you already have the mining hardware then it can simply cost you as much as it costs the entire network to mine, which should be about the same as the amount they get from it, which is 12.5 bitcoins per 10 minutes on average, which is about $47000 currently per 10 min.
As for the initial costs, they are about 700 million to match the hashrate of the network, which is about 8 million TH/s, with the Bitmain's Antminer S9-14TH/s that costs 1310 USD.