I mean, IMO it was not necessary to go to negative prices. It would have also worked by setting the r param for example to -75% or so. The price would fall much, but never fall below zero. If options were involved in setting that decision (I don't know) then it very well could have been the fact of the unfair PUT payout of BSM. Again, I can't tell b/c I didn't follow those discussions then regarding below zero prices. My "FairPut" option pricing model on the other hand is intended only for a normal equity market where prices never can be negative.
If the put payout is unfair compared to the call payout, why does long stock plus long put replicate the [same strike] call payout? I realize I am wasting my time trying to help you understand options. So I'll give you one last piece of advice: Either scrape together enough money to open an account and start selling those overpriced standard puts; or open an unregulated crypto derivatives exchange from your mom's basement featuring standard calls and "fair" puts. Edit: to be clear, if a fairput has the same price as a standard put, no one will buy the standard calls as they would get a better payout than the call with long stock and long the put. Similarly, just banning the standard puts on your exchange will do you no good, as people will just replicate the put with the call plus short stock, and arb that synthetic standard put against the fairput.
Because it incidentally just fits, maybe? As said in prev postings, I'm not an expert on synthetics (yet), and that discussion here is also not primarily on synthetics, so let's please stay on topic.
Remember: BSM pricing is correct, as far as I can say. The only error in BSM is in its payout at expiration for the PUT side.
SO the coder at the end of the day for a trader what exactly will your product look like like? In simple terms can you give an example and the reasoning why it would be different from normal PUT option?
That's easy to explain: In a market with only CALL and FairPUT options the payout for FairPUT is the same as the payout for the CALL for the same z distance from K. Ie. like in a fair game: you get the same reward regardless which side you bet, regarded you bet correct. Ie. if CALL and FairPUT cost the same (depends on the params), then you can expect to get the same reward in both cases, in case you win. Ie. in a 50:50 chance for both sides, the same reward is paid out for each side. And: if CALL and PUT cost the same, then this is an indicator that both have the same probability for expiring ITM. But since both have the same probability and also do cost the same, then consequently both should also give the same payout. Q.E.D. In contrast to that: with BSM PUT this is not the case! And that's its prime error! Everybody can verify that fact with a BSM calculator (calculating twice: once for t, and then setting t near 0 to indicate expiration), or just doing simple arithmetics in the head. Ie. in BSM the PUT is discriminated over the CALL; ie. PUT is not equally treated, as the PUT earns much less than the CALL, even if both have the same probability for expiring ITM and do cost the same... (This analysis is comparing Long CALL and Long FairPUT with each other, not their Short versions, though the relation is the same with their Short versions as well)
You ingored every advice thrown your way. So just my 2cents here: Ignore that Equities have a Chi Square Distribution. Cliff Asness did his Phd Thesis under Fama (EMH) and basically went against the grain by describing how markets move more rapidly on a sudden down move than an up move, so think about the more expensive Put as the human element you are missing in your purley mathematic equation. You might also wanna look into Bachelier who has become revived since Oil Prices went negative. If anyone would try to implement your Puts on Equities, he/she/they would be broke within a fragment of a second, since HFT Algos would Arbitrage the S*** out of your Put.