Civil Engineering Reference
In-Depth Information
The alive AO always has a market price at each state and can be traded in the
practical market. In the practical market, the investors always hold more than one
unit of the AO and can reallocate the position of the AO at each state. Therefore, our
arbitrage model is constructed by including a variable which indicated the position
of the AOs. The objective function in the arbitrage model is defined as the expected
arbitrage profit. When the market exists as an arbitrage opportunity, the solution of
our model provides a trading strategy to maximize the expected arbitrage profit.
On the other hand, the optimal value of the arbitrage model is equal to zero when
there is no arbitrage opportunity in the market. We analyze the arbitrage model
by using the mathematical technique of duality theory. The dual problem turns out
to require the existence of the probability measure. Furthermore, we show that the
value of the AO at each state is equal to the conditional expectation of all the possible
future payoff under the synthesis probability measures.
2
Notations and Assumptions
Following the notation of
King
(
2002
), we denote
F
is called a filtration. The true state of the world is revealed to the investors at time
t
by the atoms of
F =
{F
t
|
t
=
0
,
1
,
2
,...,
T
}
and
F
t
.Forany
t
, let the set
N
t
be a partition of
Ω
which generates
the algebra
F
t
. Thus,
N
t
+
1
will be a refiner of
N
t
,forall
t
. It is convenient to
t
model the relation of
0
by a
T
-depth scenario tree, in which each atom in
N
t
corresponds to a unique node at the depth
t
in the tree. The leaf nodes
n
{N
t
}
=
∈ N
T
of the scenario tree are one-to-one correspondence to the elements
ω
in the sample
space
Ω
. The unique parent node of node
n
∈N
t
for 1
≤
t
≤
T
is denoted as
a
(
n
)
∈
N
t
−
1
, and the set of child nodes of node
n
∈ N
t
for 0
≤
t
≤
T
−
1 is denoted as
c
(
n
)
∈ N
t
+
1
. Let the probability of each leaf node
n
∈ N
T
is weighted as
p
n
with
p
n
=
1
.
Hence, the probability of the internal nodes
n
∈N
t
,
t
=
1
,
2
,...,
T
−
1
∑
n
∈N
T
is obtained recursively by
p
n
=
∑
p
m
and the conditional probability of the
state
n
occurred under the information of state
a
m
∈
c
(
n
)
(
n
)
is defined by
p
n
/
p
a
(
n
)
.
Denoted
P
=
{
p
n
}
n
∈N
t
for all
t
, the triple
(
Ω
,F,
P
)
becomes a probability
t
space. Suppose that
{
X
t
}
0
is a stochastic process adapted by the filtration
F
,then
=
the condition expectation of
X
t
+
1
under the information of state
n
∈ N
t
and the
p
m
probability
P
is defined as E
P
[
X
t
+
1
|N
t
]
≡
∑
X
m
p
n
,
which is a random variable
m
∈
c
(
m
)
of states in
N
t
.
In the market, there are
K
+
1 traded securities indexed by
k
=
0
,
1
,...,
K
with
the vector price process
S
n
;
n
∈ N
t
are adapted by the filtration
F
,thatis,
S
n
is
S
n
)
where security 0 is chosen to be a numeraire which is essentially a bank account
process. We introduce the discounted processes denoted by
S
n
,
S
n
,...,
measurable in the algebra
F
t
. The vector
S
n
can be represented as
(
1
S
n
. The discounted
β
n
=
security prices relative to the numeraire are denoted by
Z
n
=
β
n
S
n
for
k
=
0
,
1
,...,
K
.
The price
Z
n
of the numeraire will be exactly one in any state
n
.
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