Deriving a Dividend-Paying Asset Forward Price by Replication
Summary
The document derives the fair delivery price for a three-year forward on thirty dividend-paying assets using a replication argument. Buy the assets at spot by borrowing their cost, invest the interim dividends, and compare the resulting time-three cash flows with delivery under the forward. The loan balance grows at the continuously compounded risk-free rate; the invested dividends reduce the amount needed to repay it. This yields a forward price equal to the financed spot purchase less the future value of the dividends.
The worked example specifies dividends at two dates and a continuously compounded rate, and the accepted answer gives a numerical delivery price. It also clarifies the cash-flow logic behind the law of one price. The result assumes the stated financing and reinvestment rate applies, and omits practical frictions such as transaction costs, taxes, and dividend uncertainty. A second answer sketches another replication but is less explicit about the derivation.
Key ideas
- Replication finances the asset purchase by borrowing its spot cost.
- Interim dividends are invested until the forward maturity and offset the financing balance.
- The fair forward delivery price is financed spot value minus the accumulated value of dividends.
- The worked result relies on the stated continuous rate and reinvestment assumptions.
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Full text
# How to compute the forward price using a replicating portfolio?
# How to compute the forward price using a replicating portfolio?
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?
## Answer by nbbo2 (score 3, accepted)
https://quant.stackexchange.com/a/36409
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 $K=32038.626$
## Answer by NIKETAN KOTADIYA (score 0)
https://quant.stackexchange.com/a/74683
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 todayShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.