Civil Engineering Reference
In-Depth Information
state
n
.Byusing(
12
), the current value of the AO is regarded as the conditional
expectation under the state
n
in
N
t
of all possible outcomes, holding and exercising,
at the next stage.
At any state
n
of time
t
0
, we obtain
T
t
η
n
=
∑
∑
m
∈N
t
w
n
by deriving (
8
)and
=
t
0
+
1
T
t
w
n
(
10
) from
T
backward to
t
0
. This implies 1
=
∑
η
n
.
In order to describe
∑
m
∈N
t
=
t
0
+
1
the new probability measure, we denote a new symbol
T
n
,
n
∈ N
t
0
defined as
T
n
≡{
m
∈ N
t
|
there exists a path
P
connecting
n
and
m
,
but
a
(
n
)
∈
P
,
t
≥
t
0
},
that is,
T
n
is a set of nodes in the subtree of the scenario tree with the root
n
.For
example, the set
T
0
contains all nodes on the scenario tree and
T
t
=
1
∑
w
n
=
∑
n
η
0
=
w
n
.
(13)
n
∈N
t
∈T
0
w
n
η
−
1
0
We s e t
y
n
=
and claim that the collection
Q
=
{
y
n
}
n
∈T
0
is a probability
measure.
Theorem 3.
Let
(
w
,
η
)
be an optimal solution of the feasibility problem, then the
collection Q
=
{
y
n
}
n
∈T
0
is a probability measure.
w
n
Proof.
By (
13
), we have 1
=
∑
n
∈T
0
η
0
=
∑
n
∈T
0
y
n
and
η
0
≥
w
n
≥
p
n
for all
n
∈ T
0
.
So the collection
Q
is a probability measure.
Imposing (
8
)into(
10
) and deriving (
10
) from
T
backward to 0, we have the
following equation,
β
0
C
0
=
m
∈T
0
β
m
F
m
w
n
η
0
=
m
∈T
0
β
m
F
m
y
m
.
(14)
Thus, we conclude our discussion by the following theorem:
Theorem 4.
If the market exists as no arbitrage opportunity then there exists a
probability measure such that the initial value of the AO equals to the expectation
of all the possible future payoff. In fact, the discounted value of the AO at each state
n
∈ N
t
satisfies the following formula:
β
n
C
n
=
m
∈T
n
β
m
F
m
w
n
η
n
=
∑
m
∈T
0
β
m
F
m
y
m
∑
m
∈T
0
y
m
.
4
Conclusion
We have provided an arbitrage model which involves trading the AO and the assets.
In our arbitrage model, the investor can reallocate the position of the AO at each
state. Therefore, the arbitrage model is then constructed by including a crucial
Search WWH ::
Custom Search