Put Call derivation using two approaches [ Some confusion of getting different results ]

Question

I understand the put-call parity and am trying to derive the results based on a CME article in their education section and also the Wikipedia explanation in the Black Scholes model where Put-call parity is derived for European call and put.

As per Black Scholes equation and put call parity Url : https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model

C(s,t) - P(s, t) = D(F-K) [ D discount factor, F is forward/future price, K is strike ].

=> C-P=D.F - D.K

=> C-P=S - Ke^(-r(T-t)) [ Future discounted at risk free rate should be spot and Discount factor for strike is e^(-r(T-t))

C-P=S-K*e^(-r(T-t)) --------------- [ A ]

As per CME Url :

https://www.cmegroup.com/education/courses/introduction-to-options/put-call-parity.html

The put-call parity is C - P = F - K [ Assume here that there is no convexity issue and F is either future/forward price ].

Using earlier result from put call parity We can write this as

F - K = S - PV(K) [ This is also available at https://www.investopedia.com/terms/p/putcallparity.asp ]

=> F - K = S - K*e-(r(T-t)) since discount factor based on continuous compounding.

=> F = S + K[1-e^-(r(T-t))] -------------- [ B ]

We know that Future is Spot plus the cost of carrying. And the cost of carrying funding of strike price K is K*e^(-r(T-t))

Hence Future = Spot + Cost of carrying position

=> F = S + K*e^(-r(T-t)) should be the result. But as per B it is different.

Why is this the case? Where is the mistake in my understanding?

Answer 1

The put-call parity from CME, C - P = F - K, is not correct. I think CME is making it simple. You need to discount the right-hand side. Then, you will get the same put-call parity as Wikipedia.

Answer 2

I talked to a resource at CME on this. It turns out that the equation

C - P = F - K

is true from the perspective of instantaneous market traded prices. I.e If the prices observed in the market did not hold this equation true for a reasonable period of time, There a trader can have an arbitrage opportunity. So In theory they are wrong ( i.e from a literature perspective ). They are correct only from an instantaneous market observed equilibrium price perspective on a given contract (i.e its future and option prices ).