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 difficult.
Question 22.2.
In the same way as the COD, the paylater is priced initially using
0
=
BSP ut(S
0
, K, σ, r, T, δ)
−
P
×
DR(S
0
, K, σ, r, T, δ, H).
Thus, the amount to be paid if the barrier is hit is
P
=
BSP ut(S
0
, K, σ, r, T, δ)
DR(S
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
BSP ut(S
0
, K, σ, r, T
−
t, δ)
−
P
×
DR(S
0
, K, σ, r, T
−
t, δ, H)
271
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Part 5 Advanced Pricing Theory
The paylater premium has the potential to be much lower than the COD premium. Compare a
paylater with
H
=
K
to a COD. You’ll find in many cases that the COD premium is approximately
twice as great. This is a consequence of the reflection principle—once you have hit the barrier, there
is approximately a 50% chance that the option will move out of the money, which means that half
the time, you’ll pay the premium without the option paying off. Thus, the premium is half that of
the COD, where the premium is always paid when and only when the option is in the money.
The initial delta will be
−
.
1903
−
3
.
0436
(
−
.
0439
)
= −
0
.
0567. The
DR
has a gamma very close
to zero hence, initially, there is little difference between the paylater’s gamma and a regular put
option’s gamma.
As time evolves the behavior of delta and gamma becomes similar to the COD, since in each case
a small move can trigger a discrete payment; the main difference being that the discrete payment
is likely to occur before expiration when
S
t
gets close to the barrier.
Question 22.3.
If
S
=
H
,
d
6
=
d
8
implying
N (d
6
)
=
N (d
8
)
. This makes the probability given in equation (22.7)
N (d
2
)
=
P (S
T
> K)
which satisfies condition 1. If at time
T
,
S
T
≤
H
and
S
T
≥
K
the probability
will be
N (d
2
)
=
1 since the barrier has been hit and the option is in the money; hence condition
2 is satisfied. For the last condition we have to examine the probability at time
T
when
S
T
> H
or
S
T
< K
and verify it will be zero. If
S
T
< K
the barrier has been hit since
K < H
and the
probability will be
N (d
2
)
which will equal zero. When
S
T
> H
,
S
T
must be greater than both
H
and
K
. This implies
N (d
2
)
=
1,
N (d
6
)
=
1 and
N (d
8
)
=
0. This implies equation (22.7) will
equal zero (i.e. condition 3 is satisfied).
Question 22.4.
We must show the formula is a solution to
e
−
r(T
−
t)
P
(
S
T
≥
H
and
S
T
< K
)
where
P
stands for
risk neutral probability.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '06
 Adam
 Advanced Pricing Theory

Click to edit the document details