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Fig. 5.2
American option
price (
top
) and corresponding
exercise boundary (
bottom
)
Proposition 5.4.2
Let
A
satisfy
(i)
and
(ii)
from Proposition
5.4.1
and let (
x
∗
,
λ
∗
)
be the unique solution of
(
5.13
),
then the primal dual active set method converges
superlinearly to (
x
∗
,
λ
∗
)
,
provided that
x
0
x
∗
2
λ
0
λ
∗
2
−
2
+
−
2
is sufficiently small
.
A proof of this result can be found in [84, Theorem 3.1].
Example 5.4.3
Consider an American put with strike
K
=
100 and maturity
T
=
1.
For
σ
T
)aswellasthe
free boundary
s
∗
(t)
. The results are shown in Fig.
5.2
, where we also plot the option
price of the corresponding European option, which is, due to the single exercise
right, lower than the price of the American contract.
=
0
.
3,
r
=
0
.
1, we compute the price of the option (at
t
=
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