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4.2 Algorithm Development
To resolve the stochastic partial differential equation, we develop an algorithm based
on finite-difference methods. Let's introduce some notation. The time step will be
δ
t
δ
B
and the benefit step
, both of which are constant. Thus the finite-difference grid is
made up of point at benefit
BiB
=
δ
(21)
and times
tTkt
=−
δ
(22)
where
0
≤≤
iI
and
0
≤≤
kK
. We write the real option value at each of the
finite-difference grid point as
k
i
VVi
=
(
δ
BTkt
,
−
δ
)
(23)
so that the superscript is the time variable and the subscript the benefit variable. We
write (9) in a more general form as
2
∂
V
∂
V
∂
V
(24)
+
aBt
(,)
+
bBt
(,)
+
cBtV
(,)
=
0
2
∂
t
∂
B
∂
B
Furthermore, take the approximation to the derivatives, and put them into this
equation:
k
k
+
1
k
k
k
k
k
VV
−
V
−
2
VV
+
V V
−
(25)
k
k
k
k
2
i
i
+
a
i
+
1
i
i
-1
+
b
i
+
1
i
-1
+
c V
=
O
(,
δδ
t
S
)
i
i
i
i
δ
t
δ
B
2
2
δ
B
Rearrange this difference equation to put all of the
k
+1 term on the left-hand side:
k
+
1
k
k
k
k
k
k
= + + +
(26)
For the Black-Scholes equation the coefficient above simplify to
V
V
(1
BVCV
)
i
i
i
−
1
i
i
i
i
+
1
1
1
A
k
=
(
σ
22
i
−
ri
)
δ
t
B
k
= −
(
σ
22
i
+
r
)
δ
t
C
k
=
(
σ
22
i
+
ri
)
δ
t
(27)
i
i
i
2
2
V
+
. This make us prescribe a relationship
between the real option value at an end point and interior value.
The boundary conditions for expansion option is
k
k
1
V
If we know
for all
i
then (26) tell us
k
V
=
0
(28)
0
I
VI
k
=
δ
B
-
e
δ
-
rk
t
(29)
The boundary conditions for abandon option is
k
e
δ
-
rk
t
V
=
(30)
0
k
V
=
0
(31)
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