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15.1.2 BNS Model
The stochastic volatility model suggested by Barndorff-Nielsen and Shepard (BNS)
[10] specifies the volatility
σ
as an Ornstein-Uhlenbeck process, driven by a pure
jump subordinator
(for the definition of a subordinator, see Defini-
tion 10.2.1). The construction of structure preserving equivalent martingale mea-
sure
{
L
t
:
t
≥
0
}
is discussed in [128]. In particular, it is shown in [128, Theorem 3.2] that the
dynamics of the pair process
(X
t
,σ
t
)
t
≥
0
under
Q
Q
is given by
1
/
2
σ
t
)
d
t
+
σ
t
d
W
t
+
ρ
d
L
λt
,
d
X
t
=
(r
−
λ
κ
(ρ)
−
(15.8)
d
σ
t
λσ
t
=−
d
t
+
d
L
λt
.
Here,
ρ
≤
0 is a correlation parameter and
λ>
0. The constant
κ
(ρ)
is defined as
e
ρζ
1
w(ζ)k(ζ)
d
ζ,
κ
(ρ)
=
−
ρ <
0
,
(15.9)
R
+
where
w
: R
+
→ R
+
satisfies
w(ζ)
1
2
k(ζ)
d
ζ<
−
∞
,
R
+
.Let
ν
Q
(
d
ζ)
and
k
is the Lévy density of
L
under
P
=
λw(ζ )k(ζ )
d
ζ
be the Lévy
(X, σ
2
)
.From(
15.8
), we readily
measure of
L
under
Q
and let
Z
=
(X, Y )
:=
deduce that this model fits into (
15.1
), with
c
=
0 and coefficients
b, Σ, ς
l
given by
r
,Σ )
√
y
0
,
l
(z, ζ )
ρζ
ζ
.
1
−
λ
κ
(ρ)
−
2
y
b(z)
=
=
=
(15.10)
−
λy
It is shown in that the process
(X
t
,σ
t
)
t
≥
0
is Markovian, so that
(X
t
,σ
t
)
t
≥
0
is also
Markovian (the Markov property is invariant under bijective mappings).
15.2 Pricing Equations
(X, Y
1
,...,Y
n
v
)
be the unique solution of (
15.1
) and let
z
Let
Z
(x, y
1
,
...,y
n
v
)
. As in the previous chapters, we show that the fair value of a European
derivative
=
:=
:= E
e
−
r(T
−
t)
g(e
X
T
)
z
v(t,z)
|
Z
t
=
n
v
+
1
. To this end, we
solves a parabolic partial integro-differential equation in
R
consider a generalization of Proposition 8.1.2.
Proposition 15.2.1
Let the Lévy measure satisfy
(10.11).
Denote by
Q
(z)
:=
(ΣΣ
)(z) and by
A
the infinitesimal generator of Z which is
,
for functions
C
2
(
d
) with bounded derivatives
,
given by
f
∈
R
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