# Relationship between portfolios at $t=0$ based on $t=T$

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 $$t=0$$?

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.