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\title{Derivatives Lab}
\author{Session 12}
\date{October 29, 2012}
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\begin{enumerate}

\item Compute an ensemble of standard Brownian paths $W(t)$ over the
interval $[0,1]$ partitioned into $N=500$ time steps.  Plot the
empirically determied mean and standard deviation of the ensemble as a
function of time.

\item Similarly, compute an ensemble of geometric Brownian paths
\[
  S(t) = \exp((\mu - \tfrac12 \, \sigma^2)\, t + \sigma \, W(t))
\]
with $\mu=0.05$ and $\sigma=0.3$ and plot mean and standard deviation as
a function of time on the interval $[0,1]$.

\item Compute the corresponding stock price paths which
underlie the binomial tree model using the parameters from Problem~2
and compare their mean and standard deviation with those obtained from
geometric Brownian motion.

\item Use the paths so obtained in a Monte--Carlo valuation of a
European call option with $K=0.9$, time of maturity $T=1.0$ and risk
free rate $r=\mu$.  Compare your result against the Black--Scholes
price by plotting the deviation from the Black--Scholes price against
the number of samples in a doubly logarithmic plot.

What is the order of the Monte--Carlo method as a function of the
number of samples?

\end{enumerate}

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