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Using the results for the American options (5.1.1), the function
u
(p)
in time-to-
maturity and log-price is the solution of the variational inequality
∂
t
u
(p)
BS
u
(p)
ru
(p)
−
A
+
≥
0
in
J
× R
,
u
(p)
≥
ϕ
(p)
in
J
× R
,
(∂
t
u
(p)
BS
u
(p)
ru
(p)
)(u
(p)
ϕ
(p)
)
−
A
+
−
=
0
in
J
× R
,
u
(p)
(
0
,x)
ϕ
(p)
(
0
,x)
=
in
R
,
with
p
th payoff function
g(T
t,e
x
)
w
(p)
(t, x)
−
+
∈[
if
t
δ,T ),
ϕ
(p)
(t, x)
=
t,e
x
)
g(T
−
if
t
∈[
0
,δ).
This payoff function involves a European option price
w
(p)
which, according to
Theorem 4.1.4, satisfies the partial differential equation
∂
τ
w
(p)
BS
w
(p)
rw
(p)
+
A
+
=
0
in
(t
−
δ,t)
× R
,
u
(p
−
1
)
(t
w
(p)
(t
−
δ,x)
=
−
δ,x)
in
R
.
Similar as for the compound option the pricing problem amounts to pricing itera-
tively an American option on a European option. Therefore, using the localization
arguments from Theorem
6.3.1
, we can again truncate the domain to
G
=
−
(
R,R)
.
As in (5.7), we have the weak formulation
Find
u
(p)
L
2
(J
H
0
(G))
H
1
(J
L
2
(G))
such that
u
(p)
(t,
∈
;
∩
;
·
)
∈
K
0
,R
and
(∂
t
u
(p)
,v
u
(p)
)
a
BS
(u
(p)
,v
u
(p)
)
a
BS
(ϕ
(p)
,v
u
(p)
),
−
+
−
≥−
−
∀
∈
K
0
,R
,
v
u
(p)
(
0
)
=
0
,
where the European option price
w
(p)
is also the weak solution of
Find
w
(p)
L
2
((t
H
0
(G))
H
1
((t
L
2
(G))
such that
∈
−
δ,t)
;
∩
−
δ,t)
;
(∂
τ
w
(p)
,v)
a
BS
(w
(p)
,v)
H
0
(G),
a.e. in
(t
+
=
0
,
∀
v
∈
−
δ,t),
w
(p)
(
0
)
u
(p
−
1
)
(t
=
−
δ).
Example 6.4.2
Consider a swing option. Set
K
=
100,
T
=
1,
σ
=
0
.
3,
r
=
0
.
05,
δ
=
5. We plot the price of the swing options and the corresponding
exercise boundary in Fig.
6.4
where we used finite elements for the discretization.
It is not surprising that swing and American put option values are similar in appear-
ance. The influence of the refraction period is not visible on the computed swing
prices but is clearly seen on the exercise regions. Moreover, it is observed that for
p,p
∈ N
0
.
1 and
p
=
p
with
p
≥
the exercise boundaries satisfies the monotonicity
s
p
(t)
s
p
(t),
≥
∀
∈[
]
t
0
,T
,
and are strictly increasing on the time interval
[
(p
−
1
)δ, T
]
.
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