Information Technology Reference
In-Depth Information
1
2
y∂
xx
f(z)
+
(r
−
λ
κ
(ρ)
−
y/
2
)∂
x
f(z)
−
λy∂
y
f(z)
S
f )(z)
:=
(
A
f(x
f(z)
ν
Q
(
d
ζ),
+
+
ρζ,y
+
ζ)
−
R
+
with
ν
Q
(
d
ζ)
λw(ζ )k(ζ )
d
ζ
. Note that there is no diffusion component in the sec-
ond coordinate direction
∂
yy
f(z)
.
=
S
is the generator of the process
Z
=
(X, Y )
in (
15.8
)
under a structure preserving EMM. However, one can consider the pricing of options
under the minimal entropy martingale measure (MEMM). For
ρ
=
Remark 15.2.3
The operator
A
0, this is done
by Benth and Meyer-Brandis [17], see also [16]. In particular, they derive that
S
A
under the MEMM is given by
1
2
y∂
xx
f(z)
+
(r
−
y/
2
)∂
x
f(z)
−
λy∂
y
f(z)
S
(t)f )(z)
:=
(
A
λ
f(x,y
f(z)
H(t,y
ζ)
H(t,y)
+
+
+
ζ)
−
ν(
d
ζ),
(15.14)
R
+
where
H
:[
0
,T
]×R
+
→ R
is solution of the PIDE
∂
t
H
−
λy∂
y
H
−
r(y)H
+
λ
H(t,y
+
ζ)
−
H(t,y)
ν(
d
ζ)
=
0in
[
0
,T)
× R
+
,
R
+
H(T,y)
=
1in
R
+
.
1
2
(μy
−
1
/
2
βy
1
/
2
)
2
, where
μ, β
Herewith,
r(y)
=
+
∈ R
.
15.3 Variational Formulation
We derive the variational formulation of the pricing PIDE (
15.12
) exemplarily for
the Bates model with
n
v
=
1,
λ
1
=
0 and the BNS model.
B
Consider the operator
A
in (
15.13
) and let
n
v
=
1,
λ
1
=
0. For this choice of
parameters, we can write
B
f )(z)
H
f )(z)
(
A
(
A
=
(
A
+
f )(z),
is the generator of the Heston model (9.12) and the operator
A
H
where
A
is given
by
λ
0
f(x
f(z)
ν
0
(
d
ζ).
(
A
f )(z)
:= −
λ
0
κ∂
x
f(z)
+
+
ζ,y)
−
R
+
To prepare the variational formulation, we perform in the pricing equation (
15.12
)
the variable transformation (9.13), i.e. we set
y
2
)
. The oper-
v(t,x,
y)
:=
v(t,x,
1
/
4
B
changes accordingly to
A
=
A
+
A
B
H
ator
A
, where the transformed Heston oper-
ator
A
H
is given in (9.15). Additionally, we consider the change of variables (9.17)
Search WWH ::
Custom Search