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Fig. 6.2
Fixed strike Asian
call and European call option
with same strike in the same
market model
(ii) The
floating strike call
has the two-dimensional payoff
g(s,y)
=
(s
−
y/T )
+
.
We set
q(t)
=
t/T
, introduce a new variable
z
=
q(t)
−
y/(sT )
, and make the
ansatz
V(t,s,y)
=
sH(t,z)
in (
6.5
) to obtain (
6.6
).
Since Theorem 4.3.1 also holds for path-dependent options and both payoffs, the
fixed and the floating strike, satisfy the growth condition (4.10) with respect to the
supremum
M
T
, we can localize the problem (
6.5
) to a bounded domain
(
1
/R
s
,R
s
)
2
in the variables
s
and
y
. This results again in a bounded computational domain
G
=
(
−
R
1
,R
2
)
for
z
in (
6.6
).
Example 6.2.2
Consider a fixed strike Asian option. Set
S
=
100,
T
=
1,
σ
=
0
.
3
and
r
0
.
09. We plot the price of the Asian options and the corresponding European
call price for various strike prices
K
in Fig.
6.2
where we used finite elements for
the discretization. It can be seen that Asian and European option prices are both
decreasing convex functions in the exercise price
K
. We also observe the fact that
when the time-to-maturity is equal to the length of the averaging period (which is
the case here since we set
t
0
=
=
0in(
6.3
)), European option prices are higher than
the corresponding Asian option prices [103].
6.3 Compound Options
Compound options are options on options. Let
V
1
(t, s)
be the option price of a Euro-
pean option with payoff
g
1
(s)
and maturity
T
1
>
0. Then, the value of a compound
option
V
with payoff
g(s)
and maturity 0
<T <T
1
is given by
= E
e
r(t
−
T)
g(V
1
(T , S
T
))
s
.
V(t,s)
|
S
t
=
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