Exercise Sheet 2


 1. Consider a forward contract on dividend-paying asset S.

    (a) The contract is entered into at time t and initially has a value of zero. At time T
        (maturity) the holder must buy the asset for an amount F (the forward price).
        Suppose that it is known at time t that a dividend of D will be paid at time td
        (i.e. D is a known amount to be paid at a known time). Determine the forward
        price by means of an arbitrage argument. You may assume that the risk free
        interest rate is constant r.
    (b) The contract is the same as in (a) except that the dividend paid at the time td
        is a known fraction of the share price, i.e. the dividend payment is DStd , for
        some D ∈ [0, 1]. In this case the exact payment cannot be known at time t < td
        since Std is not known at that time. Determine the forward price by an arbitrage
        argument. [Hint: you may take the view of the writer of the contract and re-
        invest the dividend by purchasing the asset at the instant it is paid you - you
        may assume that you can purchase fractions of shares].

 2. Consider the following table of $/   exchange rates:
     Maturity     forward $/      rate       US interest rate
      (days)                             (quarterly compounded)
     0 (spot)          0.008213                    N/A
        90             0.008301                    2.5%
       180             0.008524                    2.6%
       360             0.008947                   2.75%
   Calculate the theoretical continuously compounded rates in Japan.
 3. The S&P500 was at 827.55 on 27 February 2003. Assume that its dividend yield was
    3.1% per annum compounded continuously and assume that the 90-day c.c. rate was

    (a) What was the 90-day forward price on 27 February 2003?
    (b) Suppose that the next day the S&P500 was at 835.23. What was the value of
        that forward contract (entered into on 27 Feb), given that the 89-day c.c. rate
        was 3.0%.
     (c) Suppose instead that the 89-day c.c. rate was 2.9% (while the S&P500 was at
         835.23). Without doing any calculations, was the value of the forward contract
         greater or less than your answer in (b)? Confirm your answer with a calculation.

4. A bank offers a corporate client a choice between borrowing cash at 11% p.a. and
   borrowing gold at 2% p.a. The interest rate on the gold must be repaid in gold, i.e. if
   the client borrows 100 ounces, he must repay 102 ounces in one years time. Suppose
   the risk-free rate is 9.25% p.a. and that the storage cost for gold is 0.5% p.a. Which
   is the better deal? Explain. You may assume that the interest rates of 11%, 9.25%
   and 0.5% are compounded continuously.

5. A US-based exporter knows that she will receive 1 million Swiss Francs in 3 months
   time. She decides to hedge her exposure to forex risk by using forex futures on the
    (a) Should she go long or short on futures?
   (b) Suppose the dollar/franc and dollar/euro futures rates are 0.8 and 1.25, respec-
       tively, that their volatilities are 25% and 31%, respectively, and that the correla-
       tion between the Swiss franc and the euro is 0.9. If each future is on a nominal
       of 10 000 euro, how many futures contracts should she have a position on for an
       optimal hedge.
6. A door handle manufacturer wants to hedge an anticipated purchase of one ton of
   brass in one years time. She chooses to use copper futures. The spot price of brass
   is currently $5 000 per ton and the futures price of copper is $6 000.00 per ton. The
   volatilities of the spot price of brass and the copper futures price are 30% and 33%,
   respectively. Suppose that the manufacturer uses h futures on copper to hedge the
   purchase of one ton of brass. Use Excel to draw graphs of the variance of the portfolio,
   which is long in copper futures and short in brass, as a function of h for the cases when
   the correlation is 0.8, 1.0, -1.0 and 0. In each case, determine whether the minimum
   of your graph corresponds to the theoretical optimal hedge ratio. For which value of
   the correlation is copper futures hedging impossible for brass.
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