CHAPTER 22
Exotic Options and Other Nonstandard Products
Practice Questions
Problem 22.8.
Describe the payoff from a portfolio consisting of a floating lookback call and a floating
lookback put with the same maturity.
A floating lookback call provides a payoff of
minT
SS−
. A floating lookback put provides a
payoff of
max T
SS−
. A combination of the lookback call and the lookback put therefore
provides a payoff of
max min
SS−
.
Problem 22.9.
Consider a chooser option where the holder has the right to choose between a European call
and a European put at any time during a two-year period. The maturity dates and strike
prices for the calls and puts are the same regardless of when the choice is made. Is it ever
optimal to make the choice before the end of the two-year period? Explain your answer.
No, it is never optimal to choose early. The resulting cash flows are the same regardless of
when the choice is made. There is no point in the holder making a commitment earlier than
necessary. This argument also applies when the holder chooses between two American
options providing the options cannot be exercised before the two-year point. If the early
exercise period starts as soon as the choice is made, the argument does not hold. For example,
if the stock price fell to almost nothing in the first six months, the holder would choose a put
option at this time and exercise it immediately.
Problem 22.10.
Suppose that
1
c
and
1
p
are the prices of a European average price call and a European
average price put with strike price
K
and maturity
T
,
2
c
and
2
p
are the prices of a
European average strike call and European average strike put with maturity
T
, and
3
c
and
3
p
are the prices of a regular European call and a regular European put with strike price
K
and maturity
T
. Show that
1 2 3 1 2 3
c c c p p p+ − = + −
The payoffs are as follows:
1
c
ave
max( 0)SK−
2
c
ave
max( 0)
T
SS−
3
c
max( 0)
T
SK−
1
p
ave
max( 0)KS−
2
p
ave
max( 0)
T
SS−
3
p
max( 0)
T
KS−
The payoff from
11
cp−
is always
ave
SK−
; The payoff from
22
cp−
is always
aveT
SS−
; The
payoff from
33
cp−
is always
T
SK−
; It follows that
1 1 2 2 3 3
c p c p c p− + − = −
or
1 2 3 1 2 3
c c c p p p+ − = + −
Problem 22.11.
The text derives a decomposition of a particular type of chooser option into a call maturing
at time
2
T
and a put maturing at time
1
T
. By using put–call parity to obtain an expression for
c
instead of
p
, derive an alternative decomposition into a call maturing at time
1
T
and a put
maturing at time
2
T
.
Substituting for
c
, put-call parity gives
2 1 2 1
( ) ( )
1
max( ) max q T T r T T
c p p p S e Ke
− − − −
= + −
2 1 2 1
( ) ( )
1
max 0 q T T r T T
p S e Ke
− − − −
= + −
( ) ( )( )
q T T r q T T
− − − − −
