Information Technology Reference
In-Depth Information
Table 11.2
Algorithm to
compute sensitivities with
respect to arguments of
solution
Choose
η
0
∈
S
η
.
Calculate the matrices
M
,
A
and
D
h
.
Let
u
0
be the coefficient vector of
u
0
N
in the basis of
V
N
.
For
j
=
0
,
1
,...,M
−
1
solve
M
−
θ)t
A
)u
0
u
1
←
+
θt
A
,(
M
−
(
1
Set
u
0
:=
u
1
.
Next
j
Set
v
D
h
u
1
.
:=
of order 2 in one dimension
D
h
=
1
Q
h
=
h
−
2
diag
0
(
1
,
−
T
2
,
1
)
. In the multidi-
−
mensional case where
V
N
is given by (8.19), the matrix
D
h
is given by
T
γ
d
Q
n
h
Q
n
h
D
h
=
C
γ,
n
T
γ
1
⊗···⊗
⊗···⊗
.
γ,
|
n
|=
n
In Table
11.2
, the algorithm how to obtain an approximation to the derivative
n
u(T , x)
at maturity
T
is illustrated. The vector
v
N
D
∈ R
is the coefficient vector
h
u
N
in the basis of
V
N
.
We have the following convergence result for the approximation of sensitivities
with respect to solution arguments.
n
of
D
;
V
∗
) and for
0
θ<
2
C
1
(J
C
3
(J
Theorem 11.2.7
Assume u
∈
;
V
)
∩
≤
also
(3.30).
Assume that the approximation ∂
h
u
0
N
is quasi-optimal in L
2
(G) for all β
≤
α
.
As-
n
h
n
sume further that
D
approximates
D
in the sense of Definition
11.2.5
.
Then
,
M
−
1
n
u
M
h
u
N
n
2
n
u
m
+
θ
h
u
m
+
θ
n
2
V
D
−
D
L
2
(G
0
)
+
t
0
D
−
D
N
m
=
C
(t)
2
T
Ch
2
(s
−
r)
2
∗
T
¨
u(τ )
d
τ,
θ
∈[
0
,
1
]
0
2
≤
+
˙
u(τ )
H
s
−
r
(G)
d
τ
(t)
4
0
..
u(τ )
2
∗
1
2
d
τ,
θ
=
0
Ch
2
(s
−
r)
2
+
max
0
T
u(t)
H
s
(G)
.
≤
t
≤
11.3 Numerical Examples
In this section, we compute various sensitivities for different models. We choose
models where the price is known in closed form so that we are able to compute the
errors between the exact price/sensitivities and their approximations. We measure
the
L
∞
-norm of the error on a subset
G
0
⊂
T
. In all computations,
we discretize by linear finite elements and the Crank-Nicolson scheme where the
time steps are chosen sufficiently small.
G
at maturity
t
=
Search WWH ::
Custom Search