FORM A
OMS January 2004
Solvay Business School
Options et Marchés Spéculatifs
A. Farber
FINAL EXAM
January 21, 2004
Question 1
The Belgian Swimming Pool Company (BSPC) plans to invest €50 million in 6 months
from now for 3 months. BSPC plans to buy a zero coupon with a maturity of 3 months after the
date of issuance (it will mature in 9 months from today). The rate on this transaction will depend
on the level of the 3-month interest rate prevailing in 6 months. Sarah Lee, the finance director of
BSPC wishes to lock in the interest rate that she will receive. She is considering how to achieve
this objective.
The Euribor yield curve prevailing today is as follows:
Maturity
Yield
(month)
(With continuous
compounding)
6
2.20%
9
2.60%
(1.1) Calculate the 3-month forward rate (with simple compounding) for a transaction starting in
6 months. Briefly explain the logic underlying your calculation
(1.2) Consider buying forward in 6 months a 3-month zero coupon (9 months to maturity as of
today) for a price (set today) of €50 million. What would be the face value of this zero
coupon?
(1.3) Consider next taking a position on a 6x9 FRA? What would be the fixed rate on the FRA?
What position should BSPC take (long or short)? Explain.
(1.4) Consider finally buying or selling 3-month Euribor futures contracts traded on Liffe. The
size of one contract is € 1 m. The quoted price is 97.10. What position should BSPC take?
Compare the use of IRF contracts with the use of FRAs.
Question 2
The current date is December 16, 2003. Your boss has asked you to provide information on the
June 2004 Euro-Bund Futures Contract that will start trading soon. The underlying asset is a
notional bond with a face value of €100 million and a coupon of 6%. The deliverable bonds are
the following:
Code
Maturity
Coupon
Quoted price
A
B
16.03.2012
16.09.2013
5
3
114.06
99.61
Concordance factor
(June 2004)
0.9392
0.7915
The interest rate (with continuous compounding) is 3%
(2.1) A client wishes to buy bond B forward at the end of June 2004. Calculate the cash forward
price for this transaction.
(2.2) Suppose that the forward price (cash price) quoted on the market for a June 2004 forward
contract on bond B is 110. Set up an arbitrage to take advantage of this market quotation.
(2.3) Why are the concordance factors different? Explain.
(2.4) If the quoted futures price is 122, what would be the cheapest to deliver bond?
Question 3
Two years ago, a firm has bought a 5 years interest rate swap for which it receives floating
payments and pays fixed payments once a year on December 16. The notional amount of the
FORM A
OMS January 2004
swap is € 100 million and the swap rate is 3.50 percent. The last payment took place yesterday
and the swap has a remaining life of 3 years.
The current term structure of interest rate (with continuous compounding) is:
Maturity
(year)
1
2
3
Yield on zero-coupon
(with continuous compounding)
2.80%
3.50%
4.00%
(3.1) Calculate the value of this swap based on a decomposition into two bonds.
(3.2) Explain why this swap can be viewed as a portfolio of FRAs.
(3.3) Show how to value the swap using the current forward interest rates.
(3.4) Calculate the current 3-year swap rate prevailing on the market. Show how to use this rate to
value the “old” swap above.
Question 4
Your firm has sold an option on the XYZ stock. You want to use the Black-Scholes-Merton
model to create the option synthetically. In other words, you want to hold the replicating
portfolio. You have gathered the following market data.
Stock price: 101
Risk-less interest rate (with continuous compounding) : 4%
Type of option sold: European call
Option maturity: 3 months
Volatility: 0.30
(An option is for 1 share).
Strike
100
Black-Scholes value
6.98
Option delta
0.58
Option gamma
0.0260
(4.1) What position should you take in XYZ stock ? Explain.
(4.2) How much should you borrow or lend at the riskless interest rate ?
(4.3) If you hold this replicating portfolio and tomorrow the stock price falls to 98, will you need
to buy stock, sell stock or keep the same position to remain delta neutral ?
(4.4) If the stock turns out to be more volatile than you expected, are you happy, unhappy or
indifferent ?
Question 5
A financial institution has sold a European put option with three month to maturity on a stock that
will not pay a dividend over the life of the option. The current stock price is €20, the volatility is
40 percent and the continuously compounded risk-less rate of interest is 4 percent. The strike
price of the option is €18.
To answer the following question, use a binomial model with 1 step per month (u = 1.1224)
(5.1) Calculate the risk neutral probability of an up movement.
(5.2) Calculate the value of the European put that has been sold.
(5.3) Would the value of the option be different if it were American instead of European?
(5.4) Describe the hedging policy that should be implemented now in order to lock in the result
on this transaction. Check its effectiveness if the stock price goes down.