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CHAPTER 16
Futures
Options
Practice Questions
Problem 16.8.
Suppose you buy a put option contract
on October gold futures with a strike price of
$$1,800
per ounce. Each contract is for
the delivery of 100 ounces. What happens if you
exercise
when the October futures price
is $$1,760?
You gain (1,
800 ?1,760)×100
= $$4,000. This gain is made up of a) a short
futures contract in
October gold and b)
a cash payoff you receive which is 100 times the
excess of $$1,800 over
the previous
settlement price.
The short
futures position is marked to market in the usual
way
until you choose to close it out.
Problem 16.9.
Suppose you sell a call option contract
on April live cattle futures with a strike price
of 90
cents per pound. Each contract is
for the delivery of 40,000 pounds. What happens if
the
contract is exercised when the
futures price is 95 cents?
In this case, you lose
(0
?
95
?
0
?
90)
?
40
?
000
?
$$
2
?
000
. The loss is made up of a) a cash payoff
you have to make equal to 40,000 times
the excess of the previous settlement price over
the
previous settlement price and b) a
short April futures contract.
Problem 16.10.
Consider a two-month call futures
option with a strike price of 40 when the risk-
free interest
rate is 10% per annum.
The current futures price is 47. What is a lower
bound for the value
of the futures
option if it is (a) European and (b)
American?
Lower
bound if option is European is
(
F
0
?
K
)
e
?
rT
?
(47
?
40)
e
?
0
?
1
?
2
?
12
?
6
?
88
Lower bound if option is American is
F
0
?
K
?
7
Problem 16.11.
Consider a four-month put futures
option with a strike price of 50 when the risk-
free interest
rate is 10% per annum.
The current futures price is 47. What is a lower
bound for the value
of the futures
option if it is (a) European and (b)
American?
Lower
bound if option is European is
(
K
?
F
0
)
e
?
rT
?
(50
?
47)
e
?
0
?
1
?
4
?
12
?
2
?
90
Lower bound if option is American is
K
?
F
0
?
3
Problem 16.12.
A futures price is currently 60 and its
volatility is 30%. The risk-free interest rate is
8% per
annum. Use a two-step binomial
tree to calculate the value of a six-month
European call
option on the futures
with a strike price of 60? If the call were
American, would it ever be
worth
exercising it early?
In this case
u
?
e
0
.
3
?
1
/
4
?
1
.
1618
;
d
=
1/
u
= 0.8607; and
1
?
0
?
8607
p
?<
/p>
?
0
?
4626
1
?
1618
?
0
?<
/p>
8607
In the tree shown in
Figure S16.1 the middle number at each node is the
price of the
European option and the
lower number is the price of the American option.
The tree shows
that the value of the
European option is 4.3155 and the value of the
American option is
4.4026. The American
option should sometimes be exercised early.
Figure
S16.1
Tree to
evaluate European and American call options in
Problem 16.12
Problem
16.13.
In Problem 16.12 what value does
the binomial tree give for a six-month European
put option
on futures with a strike
price of 60? If the put were American, would it
ever be worth
exercising it early?
Verify that the call prices calculated in Problem
16.12 and the put prices
calculated
here satisfy put
–
call parity
relationships.
The parameters
u
,
d
, and
p
are the same as in Problem
16.12. The tree in Figure S16.2 shows
that the prices of the European and
American put options are the same as those
calculated for
call options in Problem
16.12. This illustrates a symmetry that exists for
at-the-money futures
options. The
American option should sometimes be exercised
early. Because
K
?
F
0
and
c
?
p
,
the European put
–
call parity
result holds.
c
?
Ke
?<
/p>
rT
?
p
?
F
0
e
?
rT
Also because
C
?
P
, <
/p>
F
0
e
?
rT
?
K
, and
Ke
?
rT
?
F
0
the
result in equation (16.2) holds. (The
first expression in equation (16.2) is
negative; the middle expression is zero, and the
last
expression is positive.)
Figure S16.2
Tree to
evaluate European and American put options in
Problem 16.13
Problem
16.14.
A futures price is currently 25,
its volatility is 30% per annum, and the risk-free
interest rate
is 10% per annum. What is
the value of a nine-month European call on the
futures with a
strike price of
26?
In this case,
F
0<
/p>
?
25
,
K
?
26
,
?
?
0
?
3
,
r
?
0
?
1
,
T
?
0
?
75
ln(
F
0
?
K
)
?
?
2
T
?
2
d
1
?
?
?
0
?
< br>0211
?
T
ln(
F
0
?
K
)
?
?
2
T
?
2
d
2
?
?
?<
/p>
0
?
2809
?
T
c
?
e
?
0
?
075
[25
N
(
?
0
?
0211)
?
26
N
(<
/p>
?
0
?
2809
)]
?
e
?
0
?
075
[
25
?
0
?
4
916
?
26
?
0
?
3894]
?
< br>2
?
01
Problem 16.15.
A futures
price is currently 70, its volatility is 20% per
annum, and the risk-free interest rate
is 6% per annum. What is the value of a
five-month European put on the futures with a
strike
price of 65?
In this case
F
0
?
70
,
K
?
65
,
?
?
0
?
2
,
r
?
0
?
06
,
T
?
0
?
4167
ln(
F
0
?
K
)
?
?
2
T
?
2
< br>d
1
?
?
0
?
6386
?
T
ln(
F
0
?
K
)
?
?
2
T
?
2
d
2
?
?
< br>0
?
5095
?
T
p
?
e
?
0
?
025
[65
N
(
?
0
?
5095)
?
70<
/p>
N
(
?
0
?
6386)]
?
e
?
0
?
025
[65
?
0
?
3052
?
70
?
0
?
2
615]
?
1
?
495
Problem 16.16.
Suppose that a one-year futures price
is currently 35. A one-year European call option
and a
one-year European put option on
the futures with a strike price of 34 are both
priced at 2 in
the market. The risk-
free interest rate is 10% per annum. Identify an
arbitrage opportunity.
In this case
c
?
Ke<
/p>
?
rT
?
2
?
34
e
?
0
?
1
?
1
?
32
?
76
p
?
F
0
e
?
rT
?
2
?
35
e
?
0
?
1
?<
/p>
1
?
33
?
67
Put-call parity
shows that we should buy one call, short one put
and short a futures contract.
This
costs nothing up front. In one year, either we
exercise the call or the put is exercised
against us. In either case, we buy the
asset for 34 and close out the futures position.
The gain
on the short futures position
is
35
?
34
?
1
.
Problem 16.17.
“The price of an at
-the-
money European call futures option always equals
the price of a
similar at-
the-
money European put futures option.”
Explain why this statement is true.
The put price
is
e
?
rT
[
KN
(
?
d
2
)<
/p>
?
F
0
N
(
?
d
1
)]
Because
N
(
?
x
)<
/p>
?
1
?
N
(
x
)
for all
x
the put price can also be written
e
?
rT<
/p>
[
K
?
KN
(
d
2
)
?
F
0
?
F
0
N
(
< br>d
1
)]
Because
F
0
?
K
this is
the same as the call price:
e
?
rT<
/p>
[
F
0
N
(
d
1
)
?
KN
(
d
2
)]
This
result can also be proved from
put
–
call parity showing that
it is not model dependent.
Problem 16.18.
Suppose that
a futures price is currently 30. The risk-free
interest rate is 5% per annum. A
three-
month American call futures option with a strike
price of 28 is worth 4. Calculate
bounds for the price of a three-month
American put futures option with a strike price of
28.
From equation (16.2),
C
?
P
must lie between
30
e<
/p>
?
0
?
05
?
3
?
12
?
28
?
1
?
63
and
30
?
28
e
?
0
?
05
?
3
?
12
?
2
?
35
Because
C
?
4
we must have
1
?
63
?
4
?
P
?
2
?
35
or
1
?
65
?
P
?
2
?
37
Problem 16.19.
Show that if
C
is the price of an American call option
on a futures contract when the strike
price is
K
and the maturity is
T
, and
P
is the price of
an American put on the same
futures
contract with the same strike price and exercise
date,
F
< br>0
e
?
rT
?
K
?
C
?
P
?
F
0<
/p>
?
Ke
?
rT<
/p>
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