Chapter 22
Exotic Options: II
Question 22.1.
With a premium of
P
paid at maturity if
S
T
>K
, the COD will have the same value (which will
initially be set to zero) as a regular call minus
P
cash or nothing call options. That is,
0
=
BSCall(S
0
,K,σ,r,T,δ)
−
P
×
CashCall(S
0
,K,σ,r,T,δ).
(1)
a)
Given the inputs and pricing the above options,
P
, must satisfy
0
=
10
.
45
−
P (
0
.
5323
)
(2)
which implies
P
=
10
.
45
/.
5323
=
19
.
632.
b)
The delta of the COD is
0
.
637
−
19
.
632
×
.
01875
=
.
2689
(3)
and the gamma of the COD is
.
019
−
19
.
632
(
−
.
00033
)
=
2
.
55%.
(4)
c)
As the option approaches maturity, the gamma will explode close to the money making delta
hedging difFcult.
Question 22.2.
In the same way as the COD, the paylater is priced initially using
0
=
BSPut(S
0
−
P
×
DR(S
0
,K,σ,r,T,δ,H).
Thus, the amount to be paid if the barrier is hit is
P
=
0
0
,K,σ,r,T,δ,H)
=
2
.
3101
0
.
7590
=
3
.
0436
.
At subsequent times prior to hitting the barrier, the value of the paylater put is
0
,K,σ,r,T
−
t,δ)
−
P
×
0
−
t,δ,H)
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