# Proof for long-forward price at time t

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I tried to rigorously prove the value of long-forward price at time t. The outline of the proof is given on slide 22 of "The introduction the derivatives" slides.

I am quite close to the correct answer except that my answer has the incorrect sign. I have attached my workings as a pdf.  Proof.pdf (0,3 MB)

Any idea of where I went wrong?

answered May 2, 2017 by (4,010 points)
selected May 9, 2017 by Rowan

Hi Conor

In the your proof, the amount of money you wish to borrow at time $$t$$ should be the present value of the difference between what you are going to receive from the long Forward and what you are going to pay for the short Forward.

i.e. The present value of $$F_{0,T} - F_{t,T}$$.

That should then give you the correct answer.

commented May 13, 2017 by (1,160 points)

If we borrow the PV of $$F_{0,T} - F_{t,T}$$. then surely we would have to pay back $$F_{0,T} - F_{t,T}$$.  i.e. have a cashflow of -($$F_{0,T} - F_{t,T}$$)=$$F_{t,T} - F_{0,T}$$? which would then make our overall cashflow at T 2($$F_{t,T} - F_{0,T}$$)? which is then non-zero and makes the arbitrage argument not work?

commented May 14, 2017 by (4,010 points)

Hi Dean

Thanks for pointing out my error.

I have relooked at the proof. The amount which is borrowed is indeed supposed to be $$F_{t,T} - F_{0,T}$$ .

From what I can see, the reason why Conor was getting the wrong sign in the final step is because he was adding the cashflows at time $$t$$ to get zero instead equating them to each other. If the cashflows at all other times are equal, then the values of each component at time $$t$$ should be equal to each other. They should not add up to zero.