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I am reading a derivation of the forward price $F$ of a forward contract. I think the author uses a circular argument to assume that "the value of the forward at inception is equal to 0" because the value of the forward at inception is equal to 0 only when the forward price is $S(0)e^{rT}$. When the forward price is not $S(0)e^{rT}$, the forward value should not be zero to prevent from an arbitrage opportunity. Can someone confirm it? enter image description here

Edit: Here is my derivation. Consider two portfolios; one is long one forward contract for the price K at time T and short 1 unit of the underlying asset. The other one is made of borrowing $Ke^{-rT}$ cash. Then two portfolios have the same value at the maturity. Assuming no arbitrage opportunity, they should, by LOOP, have the same value at time t $\leq$ T, thus we can write the equation. $$F(t) = S(t) - Ke^{-r(T-t)}$$ where F(t) is the value of the forward contract, S(t) is the price of the underlying asset. By taking t = 0, $F(0) = S(0) - Ke^{-r(T)}$. That shows $F(0)$ is equal to zero only when $K = s(0)e^{r(T)}$. When the contractual price $K$ is not equal to $s(0)e^{r(T)}$, the forward price $F(0)$ should not be zero to prevent an arbitrage opportunity. A simple example is the value of a contract that requires to buy one unit of the asset for price 0 at time T is definitely not zero.

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    $\begingroup$ I don't think it is circular. The value at inception of the forward contract is $0$ because neither of the counterparties pays anything to the other one to enter the contract. Perhaps what is lacking is a proper definition of a forward contract (a contract with no initial flow and a final flow of $S(T) - F$). $\endgroup$ – Antoine Conze Jan 29 at 15:11
  • $\begingroup$ @AntoineConze I see your point. Knowing either value of the forward or contractual price can determine the other. And the value of the forward contract is just the price of buying the forward contract at t = 0. Since neither of the counterparties pays anything to the other one, the value of the contract is zero, and we can derive a fair contractual price. That makes a lot of sense. $\endgroup$ – YellowRiver Jan 30 at 2:11
  • $\begingroup$ @AntoineConze There are senarios that one party has to pay for a forward contract to enter a possition, correctly? $\endgroup$ – YellowRiver Jan 30 at 2:15
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    $\begingroup$ Yes. If you entered a contract with an upfront premium of $p$ then the fair "forward price" in the contract would be $F = (p + S(0))e^{rT}$. As a real life example think of a 1 period swap with a fixed rate $\neq$ market swap rate. $\endgroup$ – Antoine Conze Jan 30 at 12:57

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