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\title{Derivatives Lab}
\author{Session 20}
\date{November 28, 2011}
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\begin{enumerate}
\item Traders often like to know the sensitivity of the value of a
derivative to the change of parameters.  (These sensitivity measures are often
refereed to as ``The Greeks''.)  

Let us, as an example, look at 
\[
  \kappa = \frac{\partial C}{\partial \sigma}
\]
which is often referred to as ``Vega'' (not Greek).

\begin{itemize}
\item Derive a partial differential equation for $\kappa$.  What are
the boundary conditions?

\item Modify your Black--Scholes solver to find $\kappa$.

\item Compare to the explicit formula (which can be obtained by direct
differentiation of the Black--Scholes formula) for a European put or
call option, where
\[
  \kappa = S \, \sqrt T \, N'(x)
\]
where 
\[
  x = \frac{\ln(S/K)+(r + \sigma^2/2) T}{\sigma \sqrt T}
\]
and
\[
  N'(x) = \frac1{\sqrt{2\pi}} \, \e^{-x^2/2}
\]
is the probability density function of the standard normal distribution.
\end{itemize}



\end{enumerate}



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