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\title{Derivatives Lab}
\author{Sessions 10--11}
\date{October 15--16, 2012}
\maketitle

\begin{enumerate}


\item Recall that the price of a European Call option with current
stock price $S$, strike price $K$, annualized volatility $\sigma$,
annual risk-free interest rate $r$, and
maturity time $T$ is given by the Black--Scholes formula
\begin{equation*}
  C = S \, \Phi(x) - K \, \mathrm{e}^{-rT} \, \Phi(x-\sigma \sqrt T) \,,
\end{equation*}
where 
\begin{equation*}
  x = \frac{\ln(S/K) + (r + \sigma^2/2) \, T}{\sigma \sqrt T}
\end{equation*}
and $\Phi$ denotes the cumulative distribution function of the
standard normal distribution with mean zero and variance one.

Compare your call option prices from the binomial tree model against
those computed by the Black--Scholes formula.  Does the error scale
like a power of $N$?  (Plot the logarithm of the error vs.\ $N$.  Do
you obtain a straight line?)

\item Modify your binomial tree algorithm to price an American put
option (the holder may exercise the option at any time before
expiration).  Is the price of an American put higher or lower than
that that of a European put with otherwise identical parameters?

\item Look up stock option quotes for European call options on the
stock of a major corporation.  Plot the implied volatility vs.\ the
strike price, while the time to maturity is fixed.  (The applicable
interest rate is the spot rate for zero coupon bonds of the same
maturity.)

\end{enumerate}

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