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\title{Derivatives Lab}
\author{Session 6}
\date{September 20, 2011}
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\begin{enumerate}

\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 2 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.  

\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 Suppose that level coupon bonds of all maturities, coupon rates,
and par values are freely tradeable at their fair market price.  
Suppose you wish to immunize a future liability at the end of period
$n$.  Even if interest rates never change, a portfolio which is
immunized at the beginning of the first period, the Macaulay duration
will drift, so the portfolio will need to be rebalanced.  You may
proceed as follows.
\begin{itemize}
\item Take a bond with maturity $2n-2$ (or maturity $1$ if
$n=1$).\footnote{The initial instructions suggested taking maturity
$2n$.  However, in this case, it is possible that the coupon rate
becomes negative.  Can you prove that this does not happen for any
maturity between $n$ and $2n-2$?}
\item Ensure that $\text{MD}=n$ by adjusting the coupon rate $c$.
\item Match the liability by adjusting the par value $F$.
\item At the end of each period, sell the bond at its fair market
price and buy a new bond, immunized according to the above procedure.
\item Do you gain or lose money relative to meeting the liability by
buying a zero coupon bond?
\item Repeat the analysis with random changes of the interest rate
within each period.
\end{itemize}

\item Look on the internet for yield rates of Greek treasury bonds.
Plot a curve showing the assumed default probability vs.\ years from
now under the analysis that in the event of default, no money is repaid.

\end{enumerate}


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