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\title{Derivatives Lab}
\author{Session 23}
\date{December 4, 2012}
\maketitle

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\begin{enumerate}
\item Generate and plot an artificial time series for the ARMA(1,1)
process
\[
  Y_t - a \, Y_{t-1} = \mu + \varepsilon_t + b \, \varepsilon_{t-1}
\]
with $\mu = 1$, $a = 1/2$, $b = 2/3$, and $\epsilon_t$ is a i.i.d.\
sequence of Gaussian random variables with zero mean and standard
deviation $\sigma = 0.3$.

\item Using the time-series generated above, estimate the parameters
for the ARMA(1,1) above and compare with the model values.  You can
use the estimator

\url{http://thomas-cokelaer.info/software/spectrum/html/user/ref_param.html#spectrum.arma.arma_estimate}

If it is not already installed on your machine, you should be able to
follow the installation instruction on the right hand menu of the
documentation page.

\item Use the ARMA(1,1) estimator to estimate the volatility of your
discrete geometric Brownian motion from the previous task sheet.
Compare with the known model value and with your previous parameter
estimates. 
\end{enumerate}

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