

\subsection{\bf Stock Market Example of}
\subsection{\bf Multiple Monte Carlos --- II}

\begin{slide}
\begin{itemize}
\item Now we realize that we need to combine the two approaches and
integrate over both $x_i$ and $y_j$!

\item Then naively this is hopeless as it requires
$$
n=n_f \cdot n_g = 10^8 \left( {\sigma_f\,\sigma_g\over
{\mu_f\,\mu_g}}\right)^2 \hbox{points}.
$$

\item But this is regarding joint Monte Carlo independently, but
rather we should think about problem as 
$$
K = \int f(x_1 \ldots)\, g(y_1 \ldots)\, dx_1 \ldots dy_1
\ldots
$$
where fresh $x_i$, $y_j$ are chosen each point.  
\end{itemize}
\end{slide}

