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9.3 Variational Formulation
We consider the weak formulation of the pricing equation (
9.11
). For the ensuing
numerical analysis, it is convenient to multiply the value of the option
v
in (
9.11
)
with an exponentially decaying factor, i.e. we consider
ve
−
η
,
w
:=
(9.17)
where
η
∈
C
2
(
R
1
)
is assumed to be at most polynomially growing at infinity.
The function
w
solves
∂
t
w
n
v
+
G
Y
,
−
(
A
+
A
η
)w
+
rw
=
0 n
J
× R ×
(9.18)
g(e
x
)e
−
η
G
Y
,
w(
0
,z)
=
w
0
:=
in
R ×
where the operator
A
η
is given by
2
tr
Q
D
2
η
+
1
1
2
(Σ
η
+
A
η
:=
Σ
η
∇+
2
b)
∇
η.
(9.19)
n
v
+
1
The coefficient
Σ
η
∈ R
appearing in (
9.19
) is defined by
n
v
+
1
(Σ
η
,...,Σ
n
v
+
1
)
,Σ
η
:=
1
Q
ik
∂
x
i
η.
Consider now the pricing equation of the (transformed) Heston model (
9.14
). For
notational simplicity, we drop “
Σ
η
=
η
i
=
v
and
A
”in
. Consider the change of variables
1
2
κy
2
,
κ>
0. By the definition of
(
9.17
) with
η
=
η(x,y)
:=
A
η
(
9.19
), it follows
that the pricing equation for
w
:=
(v
−
v
0
)e
−
η
in the Heston model becomes
H
f
κ
∂
t
w
−
A
κ
w
+
rw
=
in
J
× R × R
≥
0
,
(9.20)
w(
0
,x,y)
=
0in
R × R
≥
0
,
e
−
κ/
2
y
2
(
where
f
κ
H
v
0
−
:=
A
rv
0
)
and
1
8
y
2
∂
xx
f(x,y)
1
2
ρβy∂
xy
f(x,y)
1
2
β
2
∂
yy
f(x,y)
H
(
A
κ
f )(z)
:=
+
+
r
2
βκρy
2
∂
x
f(x,y)
1
8
y
2
1
+
−
+
(
2
β
2
κ
∂
y
f(x,y)
β
2
1
2
4
αm
−
+
−
α)y
+
y
1
2
y
2
κ(β
2
κ
2
ακm
f(x,y).
+
−
α)
+
(9.21)
)
the
L
2
(G)
-inner product, i.e.
(ϕ, φ)
Let
G
:= R × R
≥
0
and denote by
(
·
,
·
=
G
ϕφ
d
x
d
y
. We associate to
H
κ
r
the bilinear form
a
κ
−
A
+
(
·
,
·
)
via
:=
(
r)ϕ,φ
,ϕ,φ
a
κ
H
κ
C
0
(ϕ, φ)
−
A
+
∈
(G).
Integration by parts yields
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