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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
    Commented Oct 12, 2017 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
    Commented Oct 12, 2017 at 23:06
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Think in this way.

want to buy Futures. yes. buy it. now have obligation to pay at expiry so you need money at expiry so buy bond today which will pay F contract price. but to buy Future and buying bond we need funds so short sell stock. and then adjust cost and benefit.

in Nutshell Forward+Bond-s=0 today

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