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I have two portfolios $V$ and $U$ given by

$$ V(S,t) = C-P \\ U(S,t) = S-Ee^{r(t-T)} \\ $$ where $P$ and $C$ denote a put and call option with the same maturity time $T$ and strike price $E$, respectively.

The pay-off function for $V$ is a linearly increasing function that intersects the horizontal $S$-axis at $E$. This is identical to the pay-off function for $U$ at time $t=T$.

So, this means that at $t=T$ the value of $V$ and $U$ are identical. But based on this, can I say anything about the relationship of the two portfolios at $t=0$?

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If the value of $V(S,t)$ and of $U(S,t)$ is identical at $t=T$, then the value/price at $t=0$ should be the same too. Otherwise there is arbitrage.

Imagine $V(S,T) = U(S,T) = X_T $ for some unknown $X_T$ but $V(S,0) > U(S,0)$ then we apply sell high and buy low. We sell $V(S,0)$ and buy $U(S,0)$ and we have a gain of $x :=V(S,0)-U(S,0)$. We can put this on a bank account. At $t=T$ the portfolio is worth: $$ x(1+rT) + V(S,T) - U(S,T) = x(1+rT) + X_T - X_T = x(1+rT) >0 $$ for sure.

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  • $\begingroup$ That makes good sense. Of course they have to be be the same for all other times as there would be arbitrage! $\endgroup$ – Tyler D Oct 1 at 7:45
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    $\begingroup$ Right, otherwise one could again buy low and sell high. I agree. $\endgroup$ – Ric Oct 1 at 9:59

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