Several people have claimed you cannot derive the Black-Scholes-Merton formula from the put-call parity. This is correct because the BSM formula (model and methodology) is NOT consistent with the original use, derivation and current market use of the put-call parity.
The Black-Scholes-Merton formula is only consistent with a academic version of the put-call parity. Using the continuous time dynamic delta hedging argument of BSM and getting the option formula consistent with the CAPM put strong constrains on what type of put-call parity that is consistent with the BSM formula. In the BSM world the put-call parity must have the same volatility for each strike. This is not consistent with the original use and development of the put-call parity. Black-Scholes-Merton are putting unnecessary and unrealistic constrains on the put-call parity.
Myth: Put-call parity was first described by Stoll 1969 in Journal of Finance.
Fact: From the known and available literature the put call parity was diffusely touched upon as early as in the 1688 by De-La Vega, and possibly earlier. It was fully described in much more detail by Nelson (1904) that refer to Higgins (1902). Neither Stoll nor Black, Scholes or Merton had any references to Higgins 1902, Nelson 1904, Bronzin 1908 etc. ...
The put-call parity in its original form as described by for example Nelson 1904 was an extremely robust arbitrage constrain that was practical usable and described in the way it still is used by the option market today. The volatility could be different for each strike and in this way be consisted with the fact that also supply and demand for options themselves played an important role and play an important role in option pricing and hedging.
The fact is the Black-Scholes-Merton model is not consistent with the original form of the put-call parity. Only the option formula as used by knowledgeable option traders that developed from a series of inventions by option traders and researches pre-Black Scholes, the Bachelier-Thorp formula is consistent with the original put-call parity.
Please take the time to read some of the ancient masters before you make up your conclusion. Read Bachelier, Nelson 1904.... Bronzin 1908 and several other German sources are under translation to English. Many more references given in our paper: Haug and Taleb 2008 "Why We Never Have Used the Black-Scholes-Merton Option Pricing Formula.
Please stop ignoring the Arrow of time and what can be done in the real market place versus fantasy assumptions that in some cases put strong unrealistic restrictions on well-established methodologies, like the put-call parity. The put-call parity came long long before Black-Scholes-Merton....and they should have tried to make their model consistent with how traders used the put-call parity not to put theoretical restrictions on the put-call parity.
When something not is consistent with academic models then many academics (luckily not all) are happy to put in some non-scientific fudge to save their current view of the model-world and claim it robust instead of seeking the truth. The volatility smile is NOT consistent with BSM but fully consistent with the ORIGINAL put-call parity that came long before both BSM and their strongly restricted version of the put-call parity. Stop ignoring the Arrow of time.
Yes we cannot derive BSM consistent from the original put-call parity because BSM is contradicting the essence of the original put-call parity. We can however derive the market formula (Bachelier-Thorp, the way knowledgeable traders use the pricing formula ) based on the original put-call parity as it originally was developed and still used today together with other robust PRACTICAL arbitrage constraints.
The put-call parity as used by BSM is NOT
C - P = S - X*exp(-rt).
BSM put-call parity is obviously on a restricted form (even if not stated directly in their paper), it is of this type:
C (sigma_X) - P(sigma_X) = S - X*exp(-rt),
That is sigma must be same for all strikes, not stated directly, but everything else will be inconsistent with BSM model, and even if it was consistent with put-call parity with volatility smile that is not enough, but a minimum. For example if you had a stochastic vol model and assumed you still could remove most risk with dynamic delta hedging, the model and argument would blow up in your face for jumps in market, and we have jumps in the market, I am interested in what can be used in the market.
You cannot price up options based on put-call parity and other robust arbitrage principles as well as supply demand of options and now enforce strict BSM methodology on it without changing whole valuation/pricing process to enforce flat vol smile. BSM is consistent with put-call parity, but enforces flat vol smile on everything used in relation to BSM including the put-call parity, this was not original use of put-call parity that was consistent with supply demand based option pricing and not with flat-vol smile.