} \section*{Problem Statement} Consider a stock whose price today is \$50. Suppose that over the next year, the stock price can either go up by 10\%, or down by 3\%, so the stock price at the end of the year is either \$55 or \$48.50. The interest rate on a \$1 bond is 6\%. If there also exists a call on the stock with an exercise price of \$50, then what is the price of the call option? Also, what is the replicating portfolio? \section*{Solution} The payoffs are $f(SU) = \max( SU-50,0) = 5$ and $F(SD) = \max(SD - 50,0) = 0$. A replicating portfolio must satisfy: \begin{eqnarray*} 55 \phi + \exp(0.06) \psi &=& 5\\ 48.50 \phi + \exp(0.06) \psi &=& 0. \end{eqnarray*} Solving this system we obtain that $\phi = 0.769$ and $\psi = -35.14$, so we own $0.769$ shares of stock, and short (borrow) $35.14$ bonds. The value of the portfolio and therefore the option is then $\phi S + \psi = 3.32$. We can also figure the value from the risk-neutral measure, namely $$ \pi = \frac{\exp(0.06) - 0.97}{1.1 - 0.97} = 0.706434 $$ and $$ 1 - \pi = 0.293565 $$ and so $$ V = \exp(-0.06) (\pi 5 + (1 - \pi) 0) = 3.32. $$ \HCode{} Back to main section with problem statement \HCode{} \HCode{

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