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=
where it is assumed that the market prices of volatility risk
γ
i
(y)
,
i
1
,...,n
v
,are
smooth bounded functions of
y
(y
1
,...,y
n
v
)
only (compare with [68, Chap. 2],
[67]). We introduce the combined market prices of volatility risk
Λ
i
defined by
=
i
−
ρ
i
(μ
r)
=
+
=
Λ
i
(y)
γ
k
(y)L
i
+
1
,k
+
1
,i
1
,...,n
v
.
(9.7)
ξ(y)
k
=
1
(S, Y
1
,...,Y
n
v
)
Under a risk neutral probability measure, the dynamics of
Z
:=
is as in (
9.3
), with coefficients
b
,
Σ
given by
⎛
⎝
⎞
⎠
rs
g
1
(y
1
)Λ
1
(y)
.
c
p
(y
p
)
−
g
p
(y
p
)Λ
p
(y)
c
1
(y
1
)
−
b(z)
=
,
(9.8)
diag
sξ(y), g
1
(y
1
),...,g
2
(y
2
)
L,
Σ(z)
=
(9.9)
1
. In the next section, we derive the pricing equa-
tion for European options under these SV models.
n
v
+
where
z
:=
(s, y
1
,...,y
n
v
)
∈ R
9.2 Pricing Equation
As in the BS model, we switch in (
9.1
) to the log-price process
X
=
ln
(S)
.
We therefore consider instead of the process
(S, Y
1
,...,Y
n
v
)
:=
the process
Z
(X, Y
1
,...,Y
n
v
)
. Assuming constant interest rate
r
≥
0 and setting
z
:=
(x, y
1
,
n
v
+
1
, we are interested in calculating the option value
...,y
n
v
)
∈ R
:= E
e
−
r(T
−
t)
g(e
X
T
)
z
.
V(t,z)
|
Z
t
=
(9.10)
Note that the option price is a function of
z
=
(x, y
1
,...,y
n
v
)
and not just
x
, since
we have to condition on
Z
t
=
x
. The reason for this is that the
process
Z
is Markovian (the unique strong solution to the SDE (
9.3
)isaMarkov
process, see, e.g. [3, Theorem 6.4.5]), but the process
X
is not. Proceeding as in
Sect. 8.1, we find that the function
v(t,z)
z
and not just on
X
t
=
:=
V(T
−
t,z)
is a solution of the PDE
G
Y
,
∂
t
v
−
A
v
+
rv
=
0 n
J
× R ×
(9.11)
g(e
x
)
G
Y
,
v(
0
,z)
=
v
0
(z)
:=
in
R ×
[
Q
(z)D
2
f(z)
]+
b(z)
∇
f(z)
is the infinitesimal generator
of the process
Z
and
G
Y
1
where
(
A
f )(z)
:=
2
tr
n
v
denotes the state space of the process
Y
.
Consider now the Heston model with coefficients (
9.4
)-(
9.5
) (recall that we set
⊆ R
λ
0). Since the coefficient
Σ
in (
9.5
) is not Lipschitz continuous, we can only
formally conclude that the infinitesimal generator
=
H
appearing in the pricing
A
=:
A
equation (
9.11
) is given in log-price by
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