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1
2 Q 11 x 1 x 1 + Q 12 x 1 x 2 +
1
2 Q 22 x 2 x 2 μ 1 x 1 μ 2 x 2 .
BS
A
=
BS by finite difference quotients, we define, for N 1 ,N 2 ∈ N
To discretize
A
,agrid
G
R,R) 2
on G
=
(
by
G := {
(x 1 ,i 1 ,x 2 ,i 2 )
|
1
i 1
N 1 , 1
i 2
N 2 }
,
where the grid points (x 1 ,i 1 ,x 2 ,i 2 ) are given by
(x 1 ,i 1 ,x 2 ,i 2 ) = ( R + i 1 h 1 , R + i 2 h 2 ),
h k :=
2 R/(N k +
1 ),
1
i k
N k ,k
=
1 , 2 .
C 4 (G) ,weset f i 1 ,i 2 :=
For f
f(x 1 ,i 1 ,x 2 ,i 2 ) for the function value of f at the grid
point (x 1 ,i 1 ,x 2 ,i 2 )
G
and consider the difference quotients
h 2
1
(h 1 )
x 1 x 1 f(x 1 ,i 1 ,x 2 ,i 2 )
=
(f i 1 1 ,i 2
2 f i 1 ,i 2 +
f i 1 + 1 ,i 2 )
+ O
x 1 x 1 f) i 1 ,i 2 + O
(h 1 ),
=:
x 2 x 2 f(x 1 ,i 1 ,x 2 ,i 2 ) = h 2
2 f i 1 ,i 2 + f i 1 ,i 2 + 1 ) + O (h 2 )
(f i 1 ,i 2 1
2
x 2 x 2 f) i 1 ,i 2 + O
(h 2 ).
=:
Furthermore,
( 2 h 2 ) 1 (f i 1 ,i 2 + 1
(h 2 )
x 1 x 2 f(x 1 ,i 1 ,x 2 ,i 2 )
=
x 1 [
f i 1 ,i 2 1 )
+ O
]
( 4 h 1 h 2 ) 1 f i 1 + 1 ,i 2 + 1
f i 1 1 ,i 2 1
=
f i 1 1 ,i 2 + 1
f i 1 + 1 ,i 2 1 +
+ O (h 1 ) + O (h 2 )
x 1 x 2 f) i 1 ,i 2 + O
(h 1 )
(h 2 ),
=:
+ O
as well as
( 2 h 1 ) 1 (f i 1 + 1 ,i 2
(h 1 )
(h 1 ),
x 1 f(x 1 ,i 1 ,x 2 ,i 2 )
=
f i 1 1 ,i 2 )
+ O
=:
x 1 f) i 1 ,i 2 + O
( 2 h 2 ) 1 (f i 1 ,i 2 + 1
(h 2 )
(h 2 ).
x 2 f(x 1 ,i 1 ,x 2 ,i 2 )
=
f i 1 ,i 2 1 )
+ O
=:
x 2 f) i 1 ,i 2 + O
Denote by v i 1 ,i 2
v(t m ,x 1 ,i 1 ,x 2 ,i 2 ) an approximation of the option value v on time
level t m at the grid point (x 1 ,i 1 ,x 2 ,i 2 )
BS v
G
. We now replace the PDE, t v
A
+
rv
=
0, by the finite difference equations
m
E
i 1 ,i 2 =
0 ,
1
i k
N k ,k
=
1 , 2 ,m
=
0 ,...,M
1 ,
(8.15)
g(e x i 1 ,e x i 2 ) and homogeneous boundary condi-
with the initial condition v i 1 ,i 2 =
tions v 0 ,i 2 =
v i 1 , 0 =
m
i 1 ,i 2
0, i k =
0 ,...,N k +
=
E
1, m
0 ,...,M .In( 8.15 ),
is the
finite difference operator given by
k 1 v m + 1
i 1 ,i 2
v i 1 ,i 2 θ(
v) i 1 ,i 2 +
m
v) m + 1
i 1 ,i 2
rθv m + 1
i 1 ,i 2
E
i 1 ,i 2 :=
F
+
( 1
θ)(
F
θ)v i 1 ,i 2 ,
+
r( 1
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