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\begin{document}
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\section{Introduction}
\subsection{American Barrier Options}
\begin{frame}
\begin{definition}
American barrier options are modified forms of standard call or put American options
that characterized by strike price, barrier level and a cash rebate associated to crossing
the barrier level.
\end{definition}
The four main types of barrier options are:
\begin{itemize}
\item Down and In
\item Down and Out
\item Up and In
\item Up and Out
\end{itemize}
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\subsubsection{Approximate the conditional value}
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\begin{itemize}
\item Approximate $C_j(S_j)$ using $\{e_1(S_j),e_2(S_j),\dots,e_m(S_m)\}$, the first $m$ basis function:
\begin{equation*}
 C_j(S_j) = \sum _{i=1}^m \alpha _i(S_j) e_i(S_j) = \alpha^m(S_j)  . e^m(S_j)
\end{equation*} 
$\alpha ^m(S_j) =(\alpha _1(S_j),\alpha_2(S_j),\dots,\alpha_m(S_j))$ and $e^m = (e_1,e_2,\dots,e_m)$.
\end{itemize}
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