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Therefore, it is sufficient to consider the prices of knock-out barrier contracts. For
notational simplicity, we omit the subscripts.
Let
r
be the first
hitting time of
B
by the process
S
. In real price, the value of the down-and-out
option
V
is then given by
V
do
(t, s)
= E
e
r(t
−
T)
g(S
T
)
1
≥
0 the constant interest rate and let
τ
B
=
inf
{
t
≥
0
|
S
t
=
B
}
T<τ
B
}
|
S
t
=
s
,
{
where
g
is the payoff of the option. The barrier option price
V
is again the solution
of the deterministic BS equation but with different boundary conditions.
C
1
,
2
(J
C
0
(J
Theorem 6.1.1
Let V
∈
×[
B,
∞
))
∩
×[
B,
∞
)) with bounded deriva-
tives in s beasolutionof
∂
t
V
+
A
V
−
rV
=
0
in J
×
(B,
∞
),
V (T , s)
=
g(s)
in (B,
∞
),
and boundary conditions
V(t,s)
=
0
in J
×[
0
,B
]
,
where
A
denotes the Black-Scholes generator
(4.5).
Then
,
V(t,s) can be repre-
sented as
= E
e
r(t
−
T)
g(S
T
)
1
s
.
V(t,s)
T<τ
B
}
|
S
t
=
{
Proof
We show the result only for
t
=
0. As in the proof of Theorem 4.1.4, we know
e
−
rt
V(t,S
t
)
is a martingale for 0
that the process
M
t
:=
≤
t
≤
τ
B
, since
∂
t
V
+
A
V
−
rV
=
0in
J
×[
B,
∞
)
. Thus,
V(
0
,s)
= E[
M
0
|
S
0
=
s
]
= E[
M
τ
B
∧
T
|
S
0
=
s
]
= E
e
−
rτ
B
∧
T
V(τ
B
∧
s
T,X
τ
B
∧
T
)
|
S
0
=
= E
e
−
rT
V(T,S
T
)
1
{
T<τ
B
}
|
s
S
0
=
+ E
e
−
rτ
B
V(τ
B
,S
τ
B
)
1
{
T
≥
τ
B
}
|
s
S
0
=
= E
e
−
rT
g(S
T
)
1
s
.
T<τ
B
}
|
S
0
=
{
Changing to log-price and time-to-maturity, we obtain the weak formulation:
Find
u
∈
L
2
(J
;
V)
∩
H
1
(J
;
L
2
)
such that
(∂
t
u, v)
+
a
BS
(u, v)
=
0
,
∀
v
∈
V,
a.e. in
J,
(6.2)
u(
0
)
=
u
0
,
g(e
x
)
and
V
H
1
((
log
(B),
where
u
0
=
={
u
∈
∞
))
:
u(
log
(B))
=
0
}
. As in Sect. 4.3,
we can localize the problem to a bounded domain
G
=
(
log
(B), R)
.
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