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Solvay Business School
Options et Marchés Spéculatifs
A.Farber
EXAM
January 23, 2002
QUESTION 1 – Interest Rate Futures.
The current date is January 1, 2002. Mr Eaglet, assistant treasurer at PC (People’sCar)
has just been informed that the company will borrow 10 million euros from the end of
March until the end of June. He has been asked to hedge the interest rate risk by using
the 3-month Euribor contract traded on LIFFE. The size of one contract is 1 million
euros and the tick is 25 euros. The underlying rate is 3-month Euribor rate (a simple
interest rate). The current quotation for the March contract is 96.74
(1.1) What position should PC take on the Euribor futures contract (long or short) and
on how many contracts? Explain.
(1.2) Suppose that one month later, the futures quote is 96.60. Calculate the gain (or
loss) on one contract for PC.
(1.3) Suppose that at maturity, the 3-month Euribor rate is 4.20%. Verify the
effectiveness of the hedge.
(1.4) Suppose now that the current three-month (March) and six-month (June) Euro
discount factors are 0.9913 and 0.9814 respectively. Would this create an
arbitrage opportunity? Explain.
QUESTION 2 – Interest Rate Swap
SubUS, SA, the French subsidiary of a US company, took 3 years ago a long position
on a 5 years interest rate swap. The notional amount of the swap is 50 million euros.
This interest rate swap has a remaining life of 2 years. SubUS, SA pays once a year 3
million euros and receives the 12-month EURIBOR rate. The current term structure of
interest rates (with continuous compounding) is:
Maturity
(year)
1
2
Yield on zero-coupon
(with continuous compounding)
4.00%
4.50%
(2.1) Calculate the value of this swap for SubUS by decomposing it into two bonds.
(2.3) Decompose the interest rate swap into FRAs. Calculate the value of each
individual FRA. Compare your result to your answer to question (2.1)
(2.3) Show how to value the swap based on the current 2-year swap rate.
(2.4) Show how to value the swap based on current forward interest rates.
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QUESTION 3 – Portfolio insurance.
Ms Jennifer Hera is working for Zeus Asset Management (ZAM), a money
management firm. She is to meet this afternoon Big Pension Fund (BPM), a potential
client. BPM is currently reviewing its asset allocation. It is considering investing in
stocks but, at the same time, wishes to have a guarantee that the future value of the
portfolio will not drop below some level. Ms Hera would like to convince BPM to use
portfolio insurance. Hopefully, ZAM would receive a mandate to implement this
strategy (for a fee, of course).
Suppose that the objective of BPM is to have a guarantee that, after 3 years, the
portfolio is worth at least € 1 million. In addition, BPM would receive 80% of the
increase in the Bel20 index (with dividend reinvested).
To explain how portfolio insurance works, Ms Hera has decided to focus her
presentation on a simple model based on the binomial option pricing model with one
step per year and no transaction cost. Assume that the volatility of the Bel20 index is
22.31%. The risk free interest rate (with continuous compounding) is 4%
(3.1) Draw a diagram showing the future value in 3 years of the portfolio as a function
of the level of the Bel20. Identify the option(s) hidden in this strategy.
(3.2) How much should BPM invest today in order to achieve its objective?
(3.3) How should the money be allocated between stocks and bonds?
(3.4) What should BPM do in year 1 if the Bel20 index goes up?
QUESTION 4 – Option on a stock index
Anita Hull (a distant cousin of a well known author) manages for Mr Zeprof a welldiversified portfolio of European stocks. The current market value of this portoflio is
14 million euros. The portfolio follows closely the evolution of the DJ Euro Stoxx 50
index. She is wondering whether she could increase the return by selling call options.
The DJ Euro Stoxx 50 currently stands at 3,500 and has a volatility of 20% per
annum. The risk-free interest rate is 4% (with continuous compounding) and no
dividend will be paid during the next 6 months. Each call option is for € 100 times the
value of the index.
Anita Hull has calculated the values and the “greeks” for six-month calls as a function
of the strike price.
Strike
3,100
3,300
3,500
3,700
3,900
Value
500
351
232
144
84
Delta
0.858
0.735
0.584
0.428
0.290
Gama
0.00045
0.00066
0.00079
0.00079
0.00069
Theta
-211.46
-250.97
-265.55
-248.47
-206.73
Suppose that she sells 20 6-month call options with a strike price of 3,500
(4.1) Calculate the maximum return on her portfolio. Under what condition will the
sale of the call option increase the return on the portfolio?
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(4.2) Suppose that the DJ Euro Stoxx 50 drops abrutly by 5%. Calculate the impact of
this variation on the value of her portfolio.
(4.3) What does gamma represent? Explain why gamma varies with the strike price?
(4.4) What does theta represent? Explain why theta is negative.
QUESTION 5 - Interest Rate Options
The Bowtie Company will receive 100,000 euros in 6 month from now. The company
plans to invest that money for 1-year following a rollover strategy of 2 successive
investments at the 6-month Euro Libor interest rate.
The current term structure of the Euro Libor interest rate is as follow:
Maturity (year)
0
0.5
1
1.5
2
Discountfactor
1
0.9841
0.9656
0.9446
0.9250
In the figure below you are given the lattice of six-month Euro Libor (a simple interest
rate), which has been derived using the Ho and Lee model. The time interval used in
this lattice is six months. The risk neutral probability of an up movement is 0.5.
0
0.5
1
5.61%
4.41%
3.23%
4.45%
3.26%
3.30%
Various techniques are considered to hedge the interest rate risk.
Let us first analyse possible hedges for the interest risk of the first 6-month period
(starting in 6 month). Two solutions are considered: a FRA and a call on interest rate.
(5.1) Consider first a 6  12 FRA on 6-month Euro Libor (the settlement date is in 6
months). Calculate the fixed rate, and indicate the position (long or short) that
Bowtie Company should take.
Consider next a European call option on the 6-month Euro Libor. The maturity of the
option is 6 month and the fixed rate is equal to the fixed rate of the FRA calculated
previously (a simple interest).
(5.2) Compare the payoff of the call to the payoff of the FRA. Calculate the value of
the call option.
(5.3) Show how this option can be valued as an option on a zero-coupon.
Let us now analyse a possible hedge for the two successive future 6-month periods
with a swaption.
(5.4) Consider a swaption with 6 month to maturity on a 1-year swap (with payments
every 6 months) and a fixed rate of 4.00%. Define a swaption and explain how
to value it. What would be the value of this swaption?