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In-Depth Information
This is the Black-Scholes formula. In the expressway investment decision problem
the final condition is:
V
exp
=
Max
(
S
-
E
, 0) =
Max
(
B
-
B
max
, 0)
(10)
V
aban
=
Max
(
E
-
S
, 0) =
Max
(
B
min
-
B
, 0)
(11)
3.4 The Expanded Cost-Benefit Analysis
Instead of the traditional discounted Cash flow (DCF), we can conclude that if a
project satieties
(12)
BV
++ −>
V
C
0
exp
aban
The investment is acceptable.
4 Numerical Studies
4.1 The Solution to the Black-Scholes Equation
We will transform the Black-Scholes equation into constant coefficient diffusion
equation. First, we write
−−
rT t
(
)
VBt
(,)
=
e
UBt
(,)
(13)
τ =−
Tt
(14)
1
2
y
=
log
B
+
(
r
−
στ
)
2
(15)
This takes the differential equation to:
2
(16)
∂
U
1
2
∂
U
y
=
σ
2
∂
τ
∂
2
which is a constant coefficient diffusion equation. Then we can solve this equation,
V
exp
=
−−
rT t
(
)
(17)
BNd
()
−
Ee
Nd
( )
1
2
−−
−−+
rT t
(
)
V
aban
=
BN
(
d
)
Ee
N
(
−
d
)
(18)
1
2
Where
1
2
log(
BE
/
)
++
(
r
σ
)(
T t
−
)
(19)
2
d
=
1
σ
−
Tt
1
log(
BE
/
)
+−
(
r
σ
2
)(
T t
−
)
(20)
2
d
=
2
σ
Tt
−
=−
d
σ
T
−
t
1
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