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I post this question here as I didn't receive an answer in the Mathematics community.

I am trying to understand how replicating portfolios can help us determine fair prices.

Suppose we have a 3-year forward contract on 30 assets where each asset pays a dividend of \$15 in one year and \$10 in two years. Revenues are invested in zero coupon bonds maturing at $t=3$ years. The spot price now is $S_0=\$1000$ and the interest rate is 3 % compounded continuously.

Consider a portfolio A with 30 assets. Create a replicating portfolio B using a forward contract on 30 assets with forward price $K$ and bank deposits to determine $K$.

I calculate the future value of the original portfolio A of 30 assets to be $$30S_3+30(10 \cdot e^.03+15\cdot e^.06)$$ at $t=3$ years (the payments are invested in a bank). We do not have $K$ in here since we are not in a contract? I want this to equal the value of portfolio B in order to use the law of one price. The value of the forward contract of B is $30S_3-K$ and thus I need bank deposits to the value of $$30 S_3+30(10 \cdot e^.03+15\cdot e^.06)-(30S_3-K)=K+30(10 \cdot e^.03+15\cdot e^.06).$$Therefore I need to deposit $$Ke^{-3\cdot 0.03}+30\cdot(10 \cdot e^.03+15\cdot e^.06)e^{-3\cdot 0.03}$$ at $t=0$ years. By the law of one price, the portfolios are worth the same at $t=0$ so $$30S_0=Ke^{-3\cdot 0.03}+30\cdot(10 \cdot e^.03+15\cdot e^.06)e^{-3\cdot 0.03}$$ and so I get $$K=e^{3\cdot0.03}\cdot30 \cdot 1000-30\cdot(10 \cdot e^.03+15\cdot e^.06)=\$32038.27$$ Does my approach work?

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    $\begingroup$ Please spend a bit of time formatting your question properly. It will incite people to answer you a lot more. $\endgroup$ – SRKX Oct 12 '17 at 3:32
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It can perhaps be explained more straightforwardly as follows, in terms of what happens at various times:

1) At time 0 I (the arbitrageur) borrow $30\cdot S_0$ from a bank and use it to buy 30 assets.

2) At time 3 I still have the 30 assets, and I owe the bank $30\cdot S_0 \cdot e^{.09}$. I also have the dividends which have been temporarily invested and are now worth $30(10⋅e^{.03}+15⋅e^{.06})$ as you said. This is helpful in (partially) repaying the loan. I still need to come up with $$K =30\cdot S_0 \cdot e^{.09}-30(10⋅e^{.03}+15⋅e^{.06})$$ for loan repayment.

If you give this amount $K$ at time 3 and I deliver the 30 assets to you, I will break even. So this is the forward price which I/we should agree to today. Numerically the value I get is $K=32038.626$

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  • $\begingroup$ Are you considering a long forward? Because then at t=3 the value of the forward is $50S_3-K$ and the bank loan $-30S_0e^.09$. To replicate A, the $-K$ and $30S_0e^.09$ need to cancel. Do you mean that we use bank deposits at t=0? $\endgroup$ – user30523 Oct 12 '17 at 23:06

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