TITLE: financial calculus put option
QUESTION [0 upvotes]: Consider a 1-year European put (right to sell) with a strike price of 52 on a stock whose current price is 50. We suppose that there is only time step at which the stock moves either up by 20% or down by 20%. We also suppose that the risk free interest rate is 5%. What is the value of this option?

REPLY [1 votes]: The stock can move up to 60 or move down to 40.
Let the price of the option be $t$. The option value at maturity is 0 if the stock goes up, and $52-50*(1-20\%) = 12$ if the stock goes down.
You can make this option by a portfolio of stock and risk-free bond. Let you invest $s$ of this stock now and $b$ in risk free bond.
You have to solve
$$\left\{\begin{array}{l}
60s + 1.05b = 0\\
40s+ 1.05b = 12
\end{array}\right.\\\left\{\begin{array}{l}
s=-0.6\\
b=34.285714
\end{array}\right.\\$$
I.e. with this set of $s$ and $b$, the value of portfolio at maturity is the same as that of the option. Therefore the price of this option should also be the same as the present value of this portfolio, which is $34.285714 - .6\times 50$.
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