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\author{Jeremy Morris}
\title{Project 2}

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The purpose of this project is to demonstrate the difference in price between American and European options.  The following parameters will be used for the duration of the project ($r = 0.05, T=0.25, S_{0} = 23, N= 50$).
\begin{enumerate}
\item First, the binomial model will be used to price American call and put options on a stock that pays no dividend.  The difference between the American and European options will be demonstrated using a table of values representing the risk-neutral price for each.  In the case of a put option, we have the following table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|} \hline
K & $\sigma$ & Am Price & Eu Price & Diff\\ \hline
21 & 0.10 & 0.00795 & 0.00769 & 0.000257\\
21 & 0.15 & 0.06461 & 0.06299 & 0.001614\\
21 & 0.20 & 0.17379 & 0.17029 & 0.003499\\ \hline
23 & 0.10 & 0.35347 & 0.32487 & 0.028597\\
23 & 0.15 & 0.57444 & 0.54691 & 0.027524\\
23 & 0.20 & 0.79793 & 0.77117 & 0.026758\\ \hline
25 & 0.10 & 2.00000 & 1.73023 & 0.269773\\
25 & 0.15 & 2.00909 & 1.85223 & 0.156854\\
25 & 0.20 & 2.12116 & 2.02132 & 0.099834\\ \hline
\end{tabular}
\end{center}
Notice that the difference between the prices is always positive, so that the American put option is always more expensive than the European put option.  This demonstrates that there is an optimal time to exercise the American option before the expiration date.  The next table shows values for American and European call options.
\begin{center}
\begin{tabular}{|l|l|l|l|l|} \hline 
K & $\sigma$ & Am Price & Eu Price & Diff\\ \hline
21 & 0.10 & 2.2685 & 2.2685 & 0\\
21 & 0.15 & 2.3238 & 2.3238 & 0\\
21 & 0.20 & 2.4311 & 2.4311 & 0\\ \hline
23 & 0.10 & 0.6106 & 0.6106 & 0\\
23 & 0.15 & 0.8326 & 0.8326 & 0\\
23 & 0.20 & 1.0569 & 1.0569 & 0\\ \hline
25 & 0.10 & 0.0407 & 0.0407 & 0\\
25 & 0.15 & 0.1628 & 0.1628 & 0\\
25 & 0.20 & 0.3318 & 0.3318 & 0\\ \hline
\end{tabular}
\end{center}
Here there is never a differnce in price demonstrating that there is never an optimal time to exercise the option before the expiration date.

\item Next, consider a stock that pays a dividend as in exercise 2.10.  Assume that $A_{n} = \delta \Delta t$ where $\delta = 0.06$ and $\Delta t$ is the time step.  The following table shows values for American and European call options.
\begin{center}
\begin{tabular}{|l|l|l|l|l|} \hline 
K & $\sigma$ & Am Price & Eu Price & Diff\\ \hline
21 & 0.10 & 2.0000 & 1.9352 & 0.064845\\
21 & 0.15 & 2.0492 & 2.0146 & 0.034583\\
21 & 0.20 & 2.1691 & 2.1439 & 0.025196\\ \hline
23 & 0.10 & 0.4286 & 0.4231 & 0.005480\\
23 & 0.15 & 0.6586 & 0.6526 & 0.005991\\
23 & 0.20 & 0.8872 & 0.8809 & 0.006328\\ \hline
25 & 0.10 & 0.0196 & 0.0195 & 0.000164\\
25 & 0.15 & 0.1121 & 0.1115 & 0.000635\\
25 & 0.20 & 0.2589 & 0.2577 & 0.001240\\ \hline
\end{tabular}
\end{center}
Now there is a difference in price, demonstrating that there is now an optimal time to exercise the American call option.

\end{enumerate}

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