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A stock is currently priced at 25. In 4 months it will be either 22 or 29. The risk-free rate is 6% per annum with continuous compounding. Let $S_\frac{4}{12}$ be the price of the stock in 4 months.

Compute the price of a derivative that pays you ${(S_\frac{4}{12}})^3$ dollars in 4 months

I am trying to understand how to compute this problem. I think that in order to get the derivative to the third power (or $S_ou^3d^0$), this would have to be a 2-step binomial tree. However, I have no idea what would be in between the $S_o$ and $S_ou^3d^0$. So, I don't think my assumption is correct.

I need help solving this please!

user_123945839432
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1 Answers1

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Where do you see two steps?

You have two branches - up (u) with the stock price of 29 and down (d) with 22. On the u branch your derivative price is 29^3 = 24,389, the d branch gives you 22^3 = 10,648. Because the u and d branches are assumed to have equal probability, your expected future value is the average of two, $FV = 17,518.5$.

Now, you need to discount this with your 4-month risk-free interest rate.

Thus your derivative's present value is

$PV = FV * e^{-rt} = 17,518.5* e^{-1/3 * 0.06} = 17,171.6$

Alex
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