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d
S
t
=
+
rS
t
d
t
σS
t
d
W
t
,
d
Y
t
=
S
t
d
t.
We need the two-dimensional version of Proposition 4.1.1.
AS
A
Proposition 6.2.1
Let
denote the differential operator which is
,
for functions
C
2
,
1
(
f
∈
R × R
) with bounded derivatives
,
given by
1
2
σ
2
s
2
∂
ss
f(s,y)
AS
f )(x)
(
A
=
+
rs∂
s
f(s,y)
+
s∂
y
f(s,y).
(6.4)
Then
,
the process M
t
:=
f(S
t
,Y
t
)
−
0
(
A
f )(S
τ
,Y
τ
)
d
τ is a martingale with respect
to the filtration of W
.
Proof
We apply the two-dimensional Itô formula
∂
2
f
∂x
2
(X
t
,Y
t
)
∂f
∂x
(X
t
,Y
t
)
d
X
t
+
∂f
∂y
(X
t
,Y
t
)
d
Y
t
+
1
2
(
d
X
t
)
2
d
f(X
t
,Y
t
)
=
·
∂
2
f
∂x∂y
(X
t
,Y
t
)
∂
2
f
∂y
2
(X
t
,Y
t
)
1
2
(
d
Y
t
)
2
,
+
·
d
X
t
·
d
Y
t
+
·
to the vector process
(S
t
,Y
t
)
∈ R
2
+
and obtain, using d
t
·
d
t
=
d
t
·
d
W
=
0,
AS
f )(S
t
,Y
t
)
d
t
+
σS
t
∂f
d
f(S
t
,Y
t
)
=
(
A
∂x
f(S
t
,Y
t
)
d
W
t
.
The result follows since it is shown in Proposition 4.1.1 that the stochastic integral
0
σS
τ
∂
x
f(S
τ
,Y
τ
)
d
W
τ
is a martingale with respect to the filtration of
W
.
Similar arguments as in Theorem 4.1.4 lead to the following PDE for
V(t,s,y)
,
AS
V
∂
t
V
+
A
−
rV
=
0
.
(6.5)
Equation (
6.5
) is a PDE in two variables,
s
and
y
. But it can be reduced to a univari-
ate one,
1
2
σ
2
(q(t)
z)
2
∂
zz
H
∂
t
H
+
−
+
r(q(t)
−
z)∂
z
H
=
0in
J
× R
,
(6.6)
with the terminal condition
H(T,z)
(z)
+
. The function
q(t)
depends on the op-
tion type. We consider both cases, fixed and floating strike calls:
=
(i) The
fixed strike call
has the two-dimensional payoff
g(s,y)
=
(y/T
−
K)
+
.We
t
set
q(t)
=
1
−
T
, introduce a new variable
z
=
(y/T
−
K)/s
+
q(t)
, and insert
the ansatz
V(t,s,y)
=
sH(t,z)
into (
6.5
) to obtain (
6.6
).
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