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\title{Derivatives Lab}
\author{Session 6}
\date{September 25, 2012}
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\begin{enumerate}

\item Show how you can construct a portfolio which is equivalent to
the issue of a zero coupon bond at period $i$ with maturity $j$ at
today's implied forward rate $S(i,j)$.  (Such a portfolio is called a
\emph{forward contract}.)

\item Show that, under continuous compounding, the implied forward
rate equals
\[
  S(i,j) = \frac{j \, S(j) - i \, S(i)}{j-i} \,,
\]
so that the implied single-period forward rate reads
\[
  S(i,i+1) = (i+1) \, S(i+1) - i \, S(i) \,.
\]
\emph{Hint:} The discount factor over $n$ periods is $\exp(-n \,
S(n))$.

Extend your program from Session 5, Exercise 4 to also plot the
implied one-period forward rate into the same coordinate system as
computed by the above formula, and the ``instantaneous forward rate''
given in the ECB table.  Comment on what you see.

\item Look up yield curve data for bonds of a European corporation of
your choice.  Extract the spot rates from the yield curve data.

\item For the spot rates of Question 3 and the ECB yield curve
extracted earlier, compute and visualize the implied probability of
corporate default as a function of time.

\end{enumerate}

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