Black-Scholes-Merton and the Original Put-Call Parity
A) It gives robust arbitrage constrains on pricing put versus call. Clearly understood by Nelson 1904...and possibly much earlier, (put-call parity very diffusely indicated by De La Vega 1688).
B) but also as a tool to create calls out of puts, puts out of calls and straddles out of calls or puts for the purpose of hedge options with options. In other words more than simply arbitrage constrain, but a very important tool to transfer risk in optimal and robust way between options. See Nelson (1904) and Reinach (1961).
In Black-Scholes-Merton world only point A is of importance, point B has no significance and is even inconsistent with the BSM model as it means supply and demand for options will play a role. If no pure arbitrage possibilities according to A then there is no need to use put-call parity as described under B.
Example: If you as market maker has loads of customers coming and wants to buy puts in BSM world you can simply manufacture them risk-free based on BSM dynamic delta replication argument. You would not care if there were someone you could get lots of calls from (except in pure arbitrages situation). In real world where dynamic delta replication fails to remove most risk it would be important to get hold of calls if available and convert them to puts from a risk reduction perspective. And if not available you would need to rise put price to get paid for risk you not can hedge (even if you delta hedge to remove some risk).
The original invention and USE of the put-call parity is fully consistent and even "predicts" that supply and demand for options will affect option prices. The BSM way of reducing importance of put-call parity to a pure arbitrage constrain is not consistent with this view.
Any knowledgeable option trader or market maker knows the importance of taking into consideration supply and demand for options. Option traders that don’t are doomed to blow up sooner or later. Delta hedging to remove risk (but not to argue for risk neutral valuation under a series of unrealistic assumptions that fails in practice) as described before BSM are fully consistent with original USE (and knowledgeable traders use today) of put-call parity.
You cannot derive BSM formula (consistent with BSM model) based on the original intentions of put-call parity without loosing some of the original ideas behind the USE and intentions behind the put-call parity, but possibly you can derive the more robust Bachelier-Thorp formula in this way? (will naturally not be consistent with BSM model, which is the good news). Most option formulas before BSM are anyway consistent with all the intentions behind the put-call parity. As soon as you NOT claim you can remove most risk with dynamic delta replication then supply and demand for options will play important role, and put-call parity will play a role also as a tool for "hedging/risk transfer" in addition to its use as a pure arbitrage constrain.
BSM is consistent with the arbitrage argument behind the put-call parity, but not with important aspects of its original intentions as used back then and today in the marketplace.
In addition to this BSM enforces flat vol smile, relays on the Gaussian, uses a hedging argument that only works at the university campus but fails big time in the market. Traders though they where using it and fudging it, but reading the ancient partly forgotten and ignored history it is clear knowledgeable option traders are relaying on other more robust principles and formulas.
