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\title{Derivatives Lab}
\author{Session 19}
\date{November 22, 2011}
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\def\e{\mathrm e}
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\begin{enumerate}
\item Recall that the explicit form of the finite difference
approximation to the Black--Scholes equation reads
\[
  \frac{V_n^m - V_n^{m-1}}{\Delta t}
  + \frac{\sigma^2}2 \, 
    \frac{V_{n-1}^m-2 \, V_n^m + V_{n+1}^m}{\Delta X^2}
  + \biggl( r - \frac{\sigma^2}2 \biggr) \,
    \frac{V_{n+1}^m- V_{n-1}^m}{2 \, \Delta X}
  - r \, V_n^m = 0 \,.
\]
Further, recall the definition $S=\exp(X)$.

Write a code which uses the explicit finite difference scheme to price
a European call option.  (What are the boundary conditions?)

\item Show that the explicit code becomes unstable unless the time
step $\Delta t$ is rather small.

\item Modify your code to use the implicit finite difference scheme
\[
  \frac{V_n^{m+1} - V_n^{m}}{\Delta t}
  + \frac{\sigma^2}2 \, 
    \frac{V_{n-1}^m-2 \, V_n^m + V_{n+1}^m}{\Delta X^2}
  + \biggl( r - \frac{\sigma^2}2 \biggr) \,
    \frac{V_{n+1}^m- V_{n-1}^m}{2 \, \Delta X}
  - r \, V_n^m = 0 \,.
\]
Show that it is stable even when the time step $\Delta t$ is large.

\item Demonstrate the order of convergence of the implicit finite
difference method with the same number of meshpoints in $t$ and $X$
variable. 

\end{enumerate}



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