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BFC 5915 Options, futures and risk management
Practice questions for revision
These are questions from past (written closed book) examinations that I have been involved
with over the years. They will provide you with some practice for exam questions and
revision. You should try to solve them before looking at the solutions.

Question 1

The current NZD/USD exchange rate is 0.7420 (1 NZD = 0.7420 USD). Pessimo Ltd is a New
Zealand importer of goods from the US and is expecting a large shipment of merchandise in 6
months time for which Pessimo will pay the supplier $2 million USD. Pessimo wants to hedge
against a depreciation in the New Zealand dollar. The risk free rate in NZ is 3.5 percent per
annum and in the US the risk free rate is 1.5 percent per annum. Assume a flat yield curve.
Pessimo’s treasurer Joanne uses the Garman Kohlhagen model to estimate the price of 6-month
options with a strike price of 0.7400, with the output given in the following table.

GARMAN KOHLHAGEN MODEL

spot rate 0.7420
strike rate 0.7400
interest rate base currency ("foreign") 0.035
interest rate terms currency ("domestic") 0.015
volatility 0.14
time to maturity 0.5
d1 -0.02425
N(d1) 0.490325
d2 -0.12325
N(d2) 0.450955
Gkcall price 0.026296
N(-d1) 0.509675
N(-d2) 0.549045
Gkput price 0.031639

(a) Describe exactly what option contracts Joanne enters to hedge Pessimo against a
depreciation in the New Zealand dollar. (2 marks)




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(b) Describe exactly how the option contract(s) work as a hedge for Pessimo. (2 marks)










(c) How much does Pessimo pay for the option contract? (1 mark)








(d) Suppose the exchange rate in 6 months time is 0.6500, when the option matures and
Pessimo takes delivery of the goods from the supplier. Calculate the profit/loss on the
hedge set up, by showing the exact cash flows on the option in 6 months time.
(2 marks)








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(e) Calculate the effective exchange rate achieved by Pessimo when it pays $1 million
USD for the goods delivered, under the assumption that the exchange rate is 0.6500
on the delivery date. (Hint: you must take into account the premium paid for the
option) (3 marks).







Question 2

The current GBP/USD exchange rate is 0.5197 (1USD = 0.5197GBP) with a volatility of
15% per annum. Interest rates in the UK are 5% per annum and in the US interest rates are
4% per annum. Nifty (a UK company) is an exporter of widgets to the US and is expecting
payment of USD 1,000,000 for goods in 90 days time. In order to hedge its exposure to
movements in the exchange rate, Nifty decides to purchase a put option on USD at a strike of
0.5500.

The Garman Kohlhagen model is used to price currency options, and you are given the
following correct information in order to price the option.

GARMAN KOHLHAGEN MODEL
days to maturity 90
spot rate 0.5197
strike rate 0.55
interest rate base currency ("foreign") 0.04
interest rate terms currency ("domestic") 0.05
volatility 0.15
time to maturity(years) 0.246575
d1 -0.69044
N(d1) 0.24496
d2 -0.76492
N(d2) 0.222159
Gkcall price 0.005366
N(-d1) 0.75504
N(-d2) 0.777841
Gkput price 0.034027

(a) How much does Nifty pay in GBP for a put option on a notional principal of USD 1
million at a strike of 0.5500?
(2 marks)

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(b) If the exchange rate at maturity is 0.5000 is the option in or out-of-the-money?
Explain your answer. Calculate the pay-off and the profit in GBP (ignoring the time
value of money) on the option to Nifty.
(3 marks)

Question 3
Indicate whether the statement is true/false or uncertain and provide a one-two sentence
explanation. (No explanation, no marks)

1. If the storage costs associated with an asset increase, the futures price for that asset
will increase.

2. The effective interest cost to company X from (a) borrowing for 5 years on a floating
rate basis with interest reset semi-annually at LIBOR+2% p.a., and (b) entering a 5
year interest rate swap, with semi-annual settlements, receiving LIBOR and paying a
fixed rate of 7%, will be less than (c) borrowing at a fixed rate of 8%.

3. The number of futures contracts on the share price index required to hedge an equity
portfolio will increase with the beta of the portfolio.

4. The fixed rate currently quoted in the market for entering an interest rate swap
(against a floating rate of LIBOR) must be such that the swap has a zero Net Present
Value.

5. If the zero interest rates (continuously compounded) are 0.06 for 1 year, 0.055 for 2
years, and 0.05 for 3 years, the one year forward interest rate (for the period from one
year hence to two years hence) is 0.055.

6. It is not optimal to exercise an American call option on stock early, except
immediately prior to the stock trading ex-dividend.

7. The payoff on a call option in a one period binomial model can be replicated with a
long position in delta units of stock and a short position in a riskless bond.

8. A European call option with exercise price of $5.00 currently has a premium of $0.20,
when the stock price is $5.00. If the stock price increases to $5.20 the call option price
will increase to exactly $0.40.

9. A non-dividend paying stock is currently trading at $20.00. The risk-free rate of
interest (continuously compounded) is 5.6% p.a. The price of a European call option
on this stock with a strike price of $19.50 and expiring in one month’s time must be
greater than 50 cents.

10. A stock is currently trading at $5.00. European call options on the stock with an
exercise price of $5.20 and a maturity date in 3 months are trading at $0.14. The risk-
free rate of interest (continuously compounded is 5.0%). European put options with
the same strike and maturity date on the same share must trade at $0.35 to the nearest
cent.

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11. An unexpected cash dividend will cause a European call option price to increase and a
European put option price to decrease.

12. A portfolio contains only short (sold) put options. The gamma of the portfolio is
negative.

13. The time premium for a European put option is always positive.

14. A position created by buying a call option and a put option at the same strike price
will make money when the underlying stock price is volatile.

Question 4

Provide a BRIEF comment (i.e. no more than TEN lines) on EACH of the following
statements.

(a) "A covered call, created by buying the stock and writing a call option on the stock can
be a risk-reducing strategy".

(c) "The Black-Scholes model relies on risk-neutral pricing".

(c) "Implied volatility is the market’s best guess of the volatility of the stock over the
remaining life of the option".

(d) "A company which anticipates having to borrow in three months, and which is
concerned about a possible increase in interest rates, can hedge itself against that risk
by using a forward rate agreement (FRA)".

(e) "Basis risk is often a problem in using futures contracts for hedging purposes".


Question 5

Shares on XYZ are currently trading at $5.00. European options on XYZ exist with 60 days
until maturity. The following table gives information based on the Black-Scholes option
pricing model.


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1 2 3 4 5 6
days to maturity 60 60 60 60 60
stock price 5.00 5.00 5.00 5.00 5.00
strike price 4.60 4.80 5.00 5.20 5.40
interest rate 0.060 0.060 0.060 0.060 0.060
volatility 0.200 0.200 0.200 0.200 0.200
time to maturity (years) 0.164 0.164 0.164 0.164 0.164
d1 1.190 0.666 0.162 -0.322 -0.787
N(d1) 0.883 0.747 0.564 0.374 0.216
d2 1.109 0.585 0.081 -0.403 -0.868
N(d2) 0.866 0.721 0.532 0.344 0.193
bscall 0.4692 0.3111 0.1866 0.1003 0.0480
N(-d1) 0.1169 0.2528 0.4356 0.6261 0.7843
N(-d2) 0.1336 0.2794 0.4677 0.6564 0.8073
bsput 0.0240 0.0640 0.1376 0.2492 0.3950
delta(Call) 0.8831 0.7472 0.5644 0.3739 0.2157
delta(Put) -0.1169 -0.2528 -0.4356 -0.6261 -0.7843
gamma 0.4845 0.7885 0.9711 0.9344 0.7220
vega(answer in cents) 0.3982 0.6480 0.7982 0.7680 0.5934
theta call -0.4790 -0.5997 -0.6437 -0.5734 -0.4228
theta put -0.2057 -0.3145 -0.3466 -0.2644 -0.1020
rho(call)(answer in cents) 0.6487 0.5630 0.4332 0.2908 0.1694
rho(put)(answer in cents) -0.1001 -0.2183 -0.3806 -0.5556 -0.7096

(a) The Black-Scholes model has a number of assumptions underlying it. List the
assumptions and discuss why the model has been adopted so widely when in the real
world most of the assumptions are violated.

(b) Why is the delta for the call option (K = 5.00) greater than the delta for the call option
(K=5.40)? Explain fully.

(c) Using the delta estimate the value of the call option (K=5.00) if the share price
increases to $5.20.

(d) Will your estimate in (c) underestimate or overestimate the new value of the call
option? Explain your answer with the aid of a diagram.

(e) Construct a delta neutral, gamma neutral portfolio containing calls at a strike price of
$5.40 and puts at a strike price of $4.80, (and shares if necessary). Assume the share
price is $5.00.


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