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\title{Derivatives Lab}
\author{Session 21}
\date{November 27, 2012}
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\def\e{\mathrm e}
\def\d{\mathrm d}

Suppose $S_i$ for $i=0,\dots, N$ denotes time series data which
we believe behaves like geometric Brownian motion.  Then estimates for
the parameters $\mu$ and $\sigma$ can be obtained in the following
way. 

Consider the log-returns
\[
  r_i = \ln S_{i+1} - \ln S_i \,,
\]
then compute the sample mean
\[
  \bar r = \frac1N \sum_{i=0}^{N-1} r_i
\]
and sample variance
\[
  \sigma_r^2 = \frac1{N-1}
  \sum_{i=0}^{N-1} (r_i - \bar r)^2 \,.
\]
Then the estimates for $\sigma$ and $\mu$ are given by
\[
  \hat \sigma = \frac{\sigma_r}{\sqrt{\Delta t}} 
  \qquad \text{and} \qquad
  \hat \mu = \frac{\bar r}{\Delta t} + \frac{\hat \sigma^2}2 \,.
\]

\begin{enumerate}
\item Generate a large number of sample geometric Brownian paths with
fixed $\mu$ and $\sigma$.  For each, compute the estimates $\hat \sigma$
and $\hat \mu$.

\begin{enumerate}
\item Draw a histogram (command \texttt{hist}) for the
distribution of $\hat \sigma$ and $\hat \mu$.

\item It is known that the variance of the estimate for $\sigma$ is
approximately
\[
 \operatorname{Var}[\hat \sigma] = \frac{\hat \sigma^2}{2 N} \,.
\]
Does your statistics from part (a) reproduce this result?

\item What is the variance of $\hat \mu$?  Is it large or small?

\end{enumerate}

\item Modify your code in such a way that the number of points on the
geometric Brownian path is $N=2^k+1$ so that the number of log-returns
is a power of two.  Now repeat the estimation of $\sigma$ for a single
geometric path over large time steps of length $2^i \, \Delta t$ where
$i = 0, \dots, k-1$ and $\Delta t$ is the time step of the original
geometric Brownian path.  Plot the $\hat \sigma$ vs.\ the log of the
number of sample points (\texttt{semilogx}).

Describe, in writing, how this result changes if
\begin{enumerate}
\item you add Gaussian noise to the geometric Brownian motion;
\item you add a high frequency periodic perturbation?
\end{enumerate}

\item Perform a QQ-plot vs.\ the normal distribution for each of the
three cases considered above.

Discuss the result in writing.

\item Plot the autocorrelation function for each of the three cases
considered above.

Discuss the result in writing.

\end{enumerate}

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