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\author{Jeremy Morris}
\title{Project 3}
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In general, we will use the following as paramters for this project : $r = 0.05, T=0.25, N=50, S_{0} = 23$ and $\sigma = 0.15$.
\begin{enumerate}
\item \textit{Consider the following option : if the stock at $T$ is greater than or equal to \$25, you recieve \$1, otherwise you recieve nothing.  Price this option.}

The price of this option is $0.192$.

\item \textit{Use the same binomial tree to price a call on a call option.  With this compound option, at time $T_{1} = 0.25$, the holder has the right to pay $K_{1} = 3$ to receive a call option with a strike $K_{2} = 25$ that expires at $T_{2} = 0.5$.}

The price of this option is $0.0066$.  The calculation is not very complicated, just calculate the call prices out to $T_{2} = 0.5$ and at $T_{1} = 0.25$ swap the prices for their intrinsic value against the new strike $K_{1} = 3$.

\item \textit{Consider this put option with $K = 21$ that expires at $T = 0.25$.  You are allowed to exercise at $t = (0,0.05,0.1,0.15,0.2,0.25)$.  Extend your previous American option pricer to price this put option and compare the price with those of the European put and standard American put.}

The prices for the three options are as follows:
\begin{center}
\begin{tabular}{|c|c|c|} \hline
European & Burmuda & American \\ \hline
0.0629 & 0.0637 & 0.0646 \\ \hline
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Then we get the prices that we expect, namely that the Burmuda option price falls between the European and American prices.

\end{enumerate}

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