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10.6.1 Finite Difference Discretization
As in the Black-Scholes case, we replace the domain J × G by discrete grid points
(t m ,x i ) and approximate the partial derivatives in ( 10.24 ) by difference quotients at
the grid points. Let the grid points in the log-price coordinate x be given by
x i =− R + ih,
i =
0 , 1 ,...,N +
1 ,h :=
2 R/(N +
1 ) = x,
(10.26)
which are equidistant with mesh width h , and the time levels by
t m =
mt, m
=
0 , 1 ,...,M, t
:=
T/M.
We define the weights for j
=
0 , 1 , 2 ,... ,
(j
+
1 )h
ν j
k ( 2 ) (z) d z
k ( 3 ) ((j
k ( 3 ) (jh),
=
=
+
1 )h)
jh
jh
ν j
k ( 2 ) (z) d z
k ( 3 ) (
k ( 3 ) (
=
=
jh)
(j
+
1 )h),
(j
+
1 )h
which satisfy j = 0 j
=
ν j )
k ( 2 ) (z) d z<
+
. Then, we can discretize the
R
C 0 (G) at the mesh point x i , i
generator ( 10.24 )for f
=
1 ,...,N ,by
N i
R x i
z)k ( 2 ) (z) d z
jh)ν j
xx f(x i +
=
xx f(x i +
+ O
(h)
0
j =
0
N
i
xx f) i + j ν j
(h 2 ),
=
+ O
+ O
(h)
j
=
0
i
1
0
xx f(x i + z)k ( 2 ) (z) d z =
xx f) i j ν j
+ O (h).
R + x i
j
=
0
Using the θ -scheme for the time discretization, we obtain
Find u m + 1
N
∈ R
such that for m
=
0 ,...,M
1 ,
I
θt G J u m + 1
= I
θ)t G J u m ,
+
( 1
(10.27)
u 0
= u 0 ,
with G J
N
×
N
∈ R
defined for i
=
1 ,...,N ,
h 2 2 0
ν 1 ,
1
G i,i =
ν 0 )
ν 1
+
h 2 2 ν 1
ν 0 ,
1
G i,i + 1 =
ν 2
ν 0
h 2 2 ν 1
ν 0 ,
1
G i,i 1 =
ν 2
ν 0
h 2 2 ν j
ν j 1 ,j
1
G i,i + j =
ν j + 1
=
2 ,...,N
i,
h 2 2 ν j
ν j 1 ,j
1
G i,i j =
ν j + 1
=
2 ,...,i
1 .
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