-
CHAPTER 11
Trading
Strategies Involving Options
Practice Questions
Problem 11.8.
Use put
–
call
parity to relate the initial investment for a bull
spread created using calls to the
initial investment for a bull spread
created using puts.
A bull spread using calls
provides a profit pattern with the same general
shape as a bull
spread using puts (see
Figures 11.2 and 11.3 in the text). Define
p
1
and
c
1
as
the prices of
put and call with strike
price
K
1
and
p
2
and
c
2
as the prices of a put and call with
strike
price
K
2
. From put-call
parity
p
1
?
S
?
c
1
?
K
1
e
?
rT
p
2
?
S
?
c
2
?
K
2
e
?
rT
Hence:
p
< br>1
?
p
2
?
c
1
?
c
2
?
(
K
2
?
K
1
)
e
?
rT
This shows that the initial
investment when the spread is created from puts is
less than the
initial investment when
it is created from calls by an amount
(
K
2
?
K
1
)
e
?
rT
. In fact as
mentioned in the text the initial
investment when the bull spread is created from
puts is
negative, while the initial
investment when it is created from calls is
positive.
The profit when
calls are used to create the bull spread is higher
than when puts are used by
(
K
2
?
K
1<
/p>
)(1
?
e
?<
/p>
rT
)
. This reflects
the fact that the call strategy involves an
additional risk-free
investment of
(
K<
/p>
2
?
K
1
)
e
?
rT
over the put strategy. This
earns interest of
(
< br>K
2
?
K
1
)
e
?
r
T
(
e
rT
?
1)
?
(
K<
/p>
2
?
K
1
)(1
?
e
?
rT
)
.
Problem 11.9.
Explain how an
aggressive bear spread can be created using put
options.
An aggressive bull spread using call
options is discussed in the text. Both of the
options used
have relatively high
strike prices. Similarly, an aggressive bear
spread can be created using
put
options. Both of the options should be out of the
money (that is, they should have
relatively low strike prices). The
spread then costs very little to set up because
both of the
puts are worth close to
zero. In most circumstances the spread will
provide zero payoff.
However, there is
a small chance that the stock price will fall fast
so that on expiration both
options will
be in the money. The spread then provides a payoff
equal to the difference
between the two
strike prices,
K
2
?
K
1
.
Problem 11.10.
Suppose that put options on a stock
with strike prices $$30 and $$35 cost $$4 and $$7,
respectively. How can the options be
used to create (a) a bull spread and (b) a bear
spread?
Construct a table that shows
the profit and payoff for both spreads.
A bull spread
is created by buying the $$30 put and selling the
$$35 put. This strategy gives rise
to an
initial cash inflow of $$3. The outcome is as
follows:
Stock Price
S
T
?
35
Payoff
0
Profit
3
30
?
S
T
?
< br>35
S
T
?
30
S
T
?
35
?
5
S
T
?
32
?
2
A bear spread is created by selling the
$$30 put and buying the $$35 put. This strategy
costs $$3
initially. The outcome is as
follows
Stock Price
S
T
?
35
Payoff
0
Profit
?
3
30<
/p>
?
S
T
?
35
35
?
S
T
5
32
?<
/p>
S
T
2
S
T
?
30
Problem 11.11.
Use
put
–
call parity to show that
the cost of a butterfly spread created from
European puts is
identical to the cost
of a butterfly spread created from European
calls.
Define
c
1
,
c
2
, and
c
3
as
the prices of calls with strike prices
K
1
,
K
2
and
K
3
. Define
p
1
,
p
2
and
p
3
as
the prices of puts with strike prices
K
1
,
K
2
and
K
3
. With the
usual
notation
c
1
?
K
1
e
?
rT
?
p
1
?
S
c
2
?
K
2
e
?
rT
?
p
2
?
S
c
3
?
K
3
e
?
rT
?
p
3
?
S
Hence
c
1
?
c
3
?
2
c
< br>2
?
(
K
1
?
K
3
?
2
K
2
)
e
?
rT
?
p
1
?
p
3
?
2
p
2
Because
K
2
?
K
1
?
K
3
?
K
2
, it follows that
K
1
?
K
3
?
2
K
2
?
0
and
c
1
?
c
3
?<
/p>
2
c
2
?
p
1
?
p
3
?
2
p
2
The cost of a
butterfly spread created using European calls is
therefore exactly the same as
the cost
of a butterfly spread created using European puts.
Problem 11.12.
A call with a strike price of $$60 costs
$$6. A put with the same strike price and
expiration date
costs $$4. Construct a
table that shows the profit from a straddle. For
what range of stock
prices would the
straddle lead to a loss?
A straddle is created by
buying both the call and the put. This strategy
costs $$10. The
profit/loss
is shown in the following table:
Stock Price
S
T
?
60
Payoff
Profit
S
T
?
60
60
?
S
T
S
T
?
70
50
?
S
T
S
< br>T
?
60
This shows that the
straddle will lead to a loss if the final stock
price is between $$50 and $$70.
Problem 11.13.
Construct a table showing the payoff
from a bull spread when puts with strike prices
K
1
and
K
2
are
used
(
K
2
?
K
1
)
.
The bull spread is created by buying a
put with strike price
K
1
and
selling a put with strike
price
K
2
. The payoff is
calculated as follows:
Stock Price
S
T
?
K
2
K
1
?
S
T
?
K
2
Payoff from Long
Put
0
0
Payoff from
Short
Put
0
Total
Payoff
0
S
T
?
K
2
< br>S
T
?
K
2
?
(
K
2
?
S
T
)
?
(
K
2
?
K
1
)
S
< br>T
?
K
1
K
1
?
S
T
Problem 11.14.
An investor
believes that there will be a big jump in a stock
price, but is uncertain as to the
direction. Identify six different
strategies the investor can follow and explain the
differences
among them.
Possible
strategies are:
Strangle
Straddle
Strip
Strap
Reverse calendar spread
Reverse butterfly spread
The strategies all provide
positive profits when there are large stock price
moves. A strangle
is less expensive
than a straddle, but requires a bigger move in the
stock price in order to
provide a
positive profit. Strips and straps are more
expensive than straddles but provide
bigger profits in certain
circumstances. A strip will provide a bigger
profit when there is a
large downward
stock price move. A strap will provide a bigger
profit when there is a large
upward
stock price move. In the case of strangles,
straddles, strips and straps, the profit
increases as the size of the stock
price movement increases. By contrast in a reverse
calendar
spread and a reverse butterfly
spread there is a maximum potential profit
regardless of the
size of the stock
price movement.
Problem 11.15.
How can a
forward contract on a stock with a particular
delivery price and delivery date be
created from options?
Suppose that
the delivery price is
K
and
the delivery date is
T
. The
forward contract is
created by buying a
European call and selling a European put when both
options have strike
price
K
and exercise
date
T
. This portfolio
provides a payoff of
S
T
?
K
under
all
circumstances where
S
T
is
the stock price at time
T
.
Suppose that
F
0
is the forward price.
If
K
?
F
0
, the forward contract that is created has zero value. This shows that the price of a
call equals the price of a put when the
strike price is
F
0
.
Problem 11.16.
“A box spread comprises four
options. Two can be combined to create
a long forward
position and two can be
combined to create a short forward position.”
Explain this statement.
A box spread is a bull
spread created using calls and a bear spread
created using puts. With
the notation
in the text it consists of a) a long call with
strike
K
1
, b) a
short call with
strike
K
2
,
c) a long put with
strike
K
2
, and d)
a short put with
strike
K
1
. a) and
d) give a long forward
contract with
delivery price
K
1
;
b) and c) give a short forward contract with
delivery price
K
2
.
The two forward contracts taken
together give the payoff of
K
2
?
K
1
.
Problem 11.17.
What is the result if the strike price
of the put is higher than the strike price of the
call in a
strangle?
The result is
shown in Figure S11.1. The profit pattern from a
long position in a call and a put
is
much the same when a) the put has a higher strike
price than a call and b) when the call has
a higher strike price than the put. But
both the initial investment and the final payoff
are
much higher in the first case.
Figure S11.1
Profit Pattern
in Problem 11.17
Problem
11.18.
One Australian dollar is
currently worth $$0.64. A one-year butterfly spread
is set up using
European call options
with strike prices of $$0.60, $$0.65, and $$0.70. The
risk-free interest
rates in the United
States and Australia are 5% and 4% respectively,
and the volatility of the
exchange rate is 15%. Use the DerivaGem
software to calculate the cost of setting up the
butterfly spread position. Show that
the cost is the same if European put options are
used
instead of European call
options.
To use
DerivaGem select the first worksheet and choose
Currency as the Underlying Type.
Select
Black-Scholes European as the Option Type. Input
exchange rate as 0.64, volatility as
15%, risk-free rate as 5%, foreign
risk-free interest rate as 4%, time to exercise as
1 year, and
exercise price as 0.60.
Select the button corresponding to call. Do not
select the implied
volatility button.
Hit the
Enter
key and click
on calculate. DerivaGem will show the price of
the option as 0.0618. Change the
exercise price to 0.65, hit
Enter
, and click on
calculate again.
DerivaGem will show
the value of the option as 0.0352. Change the
exercise price to 0.70, hit
Enter
, and click on
Calculate
. DerivaGem will
show the value of the option as 0.0181.
Now select the button
corresponding to put and repeat the procedure.
DerivaGem shows the
values of puts with
strike prices 0.60, 0.65, and 0.70 to be 0.0176,
0.0386, and 0.0690,
respectively.
The cost of setting up the
butterfly spread when calls are used is therefore
0
?
061
8
?
0
?
01
81
?
2
?
0
?
0352
?
0
?
0095
The cost of setting up the
butterfly spread when puts are used is
0
?
0176
?
0
?
0690
?
2
?
0
?
0386
?
0
?
0094
Allowing for rounding errors, these two
are the same.
Problem 11.19
An index
provides a dividend yield of 1% and has a
volatility of 20%. The risk-free
interest rate is 4%. How long does a
principal-protected note, created as in Example
11.1,
have to last for it to be
profitable for the bank issuing it? Use DerivaGem.
Assume that the investment
in the index is initially $$100. (This is a scaling
factor that makes
no difference to the
result.) DerivaGem can be used to value an option
on the index with the
index level equal
to 100, the volatility equal to 20%, the risk-free
rate equal to 4%, the
dividend yield
equal to 1%, and the exercise price equal to 100.
For different times to
maturity,
T
, we value a call option
(using Black-Scholes European) and calculate the
funds
available to buy the call option
(=100-100e
-0.04×
T
). Results are as follows:
Time to
Funds Available
Value of Option
maturity, T
1
3.92
9.32
2
7.69
13.79
5
18.13
23.14
10
32.97
33.34
11
35.60
34.91
This table shows that the
answer is between 10 and 11 years. Continuing the
calculations we
find that if the life
of the principal-protected note is 10.35 year or
more, it is profitable for the
bank.
(Excel’s Solver
can be used in conjunction with the DerivaGem
function
s to
facilitate
calculations.)
Further Questions
Problem 11.20
A
trader creates a bear spread by selling a six-
month put option with a $$25 strike price
for $$2.15 and buying a six-month put
option with a $$29 strike price for $$4.75. What is
the initial investment? What is the
total payoff when the stock price in six months is
(a) $$23,
(b) $$28, and (c) $$33.
The initial investment is
$$2.60. (a) $$4, (b) $$1, and (c) 0.
Problem 11.21
A trader sells
a strangle by selling a call option with a strike
price of $$50 for $$3 and
selling a put
option with a strike price of $$40 for $$4. For what
range of prices of the
underlying asset
does the trader make a profit?
The
trader makes a profit if the total payoff is less
than $$7. This happens when the price of
the asset is between $$33 and
$$57.
Problem 11.22.
Three put
options on a stock have the same expiration date
and strike prices of $$55, $$60, and
$$65.
The market prices are $$3, $$5, and $$8,
respectively. Explain how a butterfly spread can
be created. Construct a table showing
the profit from the strategy. For what range of
stock
prices would the butterfly spread
lead to a loss?
A butterfly spread is created by buying
the $$55 put, buying the $$65 put and selling two of
the
$$60 puts. This costs
3
?
8
?
2
?
5
?
$$
1
initially. The
following table shows the profit/loss
from the strategy.
Stock Price
Payoff
Profit
0
?
1
S
T
?
65
60
?
S
T
?
65
55
?
S
T
?
60
65
?
S
T
64
?
S
T
S
T
?
55
The butterfly
spread leads to a loss when the final stock price
is greater than $$64 or less than
$$56.
Problem 11.23.
A diagonal spread is created by buying
a call with strike price
K
2
and
exercise date
T
2
and selling a call with strike price
K
1
and
exercise date
T
1
(
T
2
?
T
1
)
. Draw a diagram
showing the the value of the spread at
time T
1
when (a)
K
2
?
K
1<
/p>
and (b)
K
< br>2
?
K
1
.
There are two alternative profit
patterns for part (a). These are shown in Figures
S11.2 and
S11.3. In Figure S11.2 the
long maturity (high strike price) option is worth
more than the
short maturity (low
strike price) option. In Figure S11.3 the reverse
is true. There is no
ambiguity about
the profit pattern for part (b). This is shown in
Figure S11.4.
S
T
?
55
0
S
T
?
56<
/p>
?
1