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 :
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?