Financial Derivatives
Problem Set 4
Spring 2016
Futures Options
1.
The following prices are observed:
§ Futures price is $102 for expiration in 90 days
§ Call option with $100 strike price and 90 days to expiration has premium of $4
§ Put option with $100 strike price and 90 days to expiration has premium of $1.75
§ T-bill rate is 10%
Is this an equilibrium situation? Can you formulate a profitable arbitrage strategy?
Problems 2 through 5 use some options on futures quoted on January 29. Determine the
profit from the strategy for each of the following futures prices at expiration: 470, 475,
485, 490, 495. Graph the results. Determine the breakeven futures price at expiration.
Use a multiplier of 500.
2.
The June 485 call premium is 4.875. Construct a simple long call position.
3.
The June 485 put premium is 6.75. Construct a simple long put position.
4.
On January 29, the June futures price is 483.10. Using June call data from problem
2, construct a covered call position.
5.
Using the data in problems 3 and 4, construct a protective put with the June 485 put.
The remaining questions are for class discussion
6.
Explain why American call options on futures could be exercised early when call
options on the spot are not. For simplicity, assume no dividends.
7.
Consider the Black model for pricing futures options. What problems arise when
applying it to options on Eurodollar futures?
8.
Explain why the futures option-pricing model is simply a pricing model for options
on instruments with a zero cost of carry.
9.
Evaluate the following statement: “Dividends are relevant to the pricing of European
call options but not to the pricing of European call options on futures. Amazingly
enough, this is true even though the two calls have the same price.”
Prof. Kensinger
page 1
Financial Derivatives
1.
Solutions: Problem Set 4
We’ll need to make an assumption
about the expiration of the futures
contract that underlies the options. If its
expiration is the same as the options,
then this is not an equilibrium situation.
Someone could sell a call and buy a put,
receiving $2.25. This would create an
obligation to sell a futures contract with
futures price of $100. At the same time,
one could purchase a listed futures
contract with futures price of $102 at no
immediate cost. Doing these things
would result in an inflow of $2.25
immediately, with an obligation to pay
$2 at the expiration date (this arises
from the obligation to buy the
underlying commodity for $102 and sell
it for $100).
2.
Actual premium paid for the contract is
$2,437.50 (4.875 times 500). This is
the maximum loss. If the price at
expiration were 485 or less, the optionholder would incur the maximum loss.
Breakeven futures price is 485 + 4.875
= 489.875. If the price at expiration
were 490, there would be a profit of
$62.50. If the price at expiration were
495, there would be a profit of
$2562.50
3.
Actual premium paid for the contract is
$3,375 (6.75 times 500). This is the
maximum loss. If the price at
expiration were 485 or higher, the
option-holder would incur the
maximum loss. If the price at expiration
were 480, there would be a loss of
$875. Breakeven futures price is 485 –
6.75 = 478.25. If the price at expiration
were 475, there would be a profit of
$1625. If the price at expiration were
470, there would be a profit of $4125.
4.
To construct a covered call, buy the
underlying futures contract and write a
call. The underlying long futures
contract has zero value on the date of
origination. The sale of the call would
produce income of $2437.50 (4.875
times 500). If the futures price at
expiration were less than 485, the call
would expire worthless and the futures
position would dominate the story. At
470, the loss would be 2437.50 –
500(483.10 – 470) = –4112.50. At 475,
the loss would be 2437.50 – 500(483.10
– 475) = –1612.50. Breakeven futures
price at expiration is 478.225. At 480,
the profit would be 2437.50 –
500(483.10 – 480) = 887.50. At 485,
the profit would be 2437.50 + 500(485–
483.10) = 3387.50. This is the
maximum profit, because the option
would be exercised if the futures price
were above 485.
If the futures prices were 102.31, this
situation would be in equilibrium.
Suppose that the expiration of the
underlying futures contract were 30
days following the expiration of the
options. Someone could repeat the
transactions described above. At option
expiration there would be two steps in
the settlement process. First, the long
102 futures contract would be settled
for the spot price minus 102. Then the
options package would be settled for
100 minus the 30-day futures price.
The net obligation at option expiration
then would be the basis for the 30-day
futures contract plus $2.
Prof. Kensinger
Spring 2016
Financial Derivatives
5.
Solutions: Problem Set 4
To construct a protective put, buy the
underlying futures contract and buy a
put. The underlying long futures
contract has zero value on the date of
origination. The purchase of the put
would cost $3,375 (6.75 times 500). If
the futures price at expiration were
higher than 485, the put would expire
worthless and the futures position
would dominate the story. At 495, the
profit would be 500(495–483.10)– 3375
= 2575. At 490, the profit would be
500(490–483.10)– 3375 = 75.
Breakeven futures price at expiration is
489.85. At 485, the loss would be
500(485–483.10)– 3375 = –2425. This
is the maximum loss, because the
option would be exercised if the futures
price were below 485.
6. through 9. For class discussion.
Prof. Kensinger
Spring 2016