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Please explain why put call parity could be compared to the payoff of a long forward contract. ie. $C_E-P_E=V_X(0)$ where $C_E,P_E$ are the call/put premiums and $V_X(0)$ is the value of a long forward contract.

Also please explain why if strike price $X$is equal to the theoretical forward price $S(0)e^{rT}$ of the asset, then the value of the forward contract is $0$, and so $C_E=P_E$

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    $\begingroup$ For the first part draw the payoff diagrams of a call, of a short put and see if they add up to the payoff diagram of a forward. For the second part write the put call parity: $C-P=S-X e^{-rt}$ and substitute $X=S e^{rt}$ $\endgroup$ – Alex C Mar 27 '16 at 4:36
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The intuitive explanation is given in @Alex C's comment. You should stick to that if you understand it.

Yet, if you are more comfortable with a mathematical approach:

  • Payoff of being long a forward contrat with maturity $T$: $(S_T - X)$. Interpretation: at time $T$, you pay a certain price $X$ in exchange for which you receive the underlying $S_T$
  • Payoff of being long a call option struck at $X$ and with maturity $T$: $(S_T - X)^+$. Interpretation: at time $T$, you exercise if the underlying's value is greater than the strike price, in which case you earn the difference between the 2, otherwise you do nothing.
  • Payoff of being long a put option struck at $X$ and with maturity $T$: $(X - S_T)^+$. Interpretation: at time $T$, you exercise if the underlying's value is smaller than the strike price, in which case you earn the (absolute) difference between the 2, otherwise you do nothing.
  • Payoff of being long (+) call option struck at $X$ and short (-) a put struck at $X$ at maturity $T$, is the difference of the 2 previous ones: \begin{align} (S_T - X)^+ - (X - S_T)^+ &= (S_T - X)1\{S_T \geq X\} - (X - S_T)1\{S_T \leq X\} \\ &= (S_T - X)1\{S_T \geq X\} + (S_T - X)1\{S_T \leq X\} \\ &= (S_T - X) \end{align}

As you can see, being long a forward with 'strike' $X$ and being long a call/short a put with strikes $X$ give the same payoff at $T$: $(S_T-X)$.

By absence of arbitrage opportunity, these two strategies should therefore have exactly the same value today, that is: $$ C_E(S_0;T,X) - P_E(S_0;T,X) = V_X(0) $$ which is the famous put-call parity relationship, where $V_X(0)$ represents the value of being long a forward contract struck at $X$ and with maturity $T$ as seen of today, in other words $$ V_X(0) = P(0,T)(F(0,T)-X)$$ with $P(0,T)$ representing the discount factor applying to cash-flows paid at $T$ and $F(0,T)$ the fair forward value.

By setting $X$ (strike price) equal to $F(0,T)$ (forward value $S_0e^{rT}$ in the absence of dividends), then you see that $V_X(0) = 0$ and put-call parity re-writes as $$ C_E(S_0;T,X) - P_E(S_0;T,X) = V_X(0) = 0 $$ meaning $$ C_E = P_E $$

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