The
is matrix multiplication. The notation
is kth component.
Definition
Given the market structure
the assumption of "no acceptable opportunities" is the requirement that the
set
is empty. Note the absence of stress measures in the present definition (see
the last remark).
We are going to drop the stress measures from further considerations of this
chapter.
Notation
We introduce the following
notation
Hence,
Condition
(Basic coherence condition) We always
assume the
following:
Proposition
(Basic existence of
incomplete market pricing) There are no acceptable opportunities iff there
exists a pricing vector.
In other words
iff there exists a vector
s.t.
We explain the geometrical meaning of the proposition
(
Basic existence of
incomplete market pricing
) using the pictures
(
Incomplete Market
1
),(
Incomplete Market 2
).
If the projections of vectors
on the plane
are pointing in all directions then
may be replicated by a positive linear combination of
.
Otherwise, if the projections of
are in the same halfplane with respect to any subdivision to halfplanes in
then such replication is not possible because then
would have a nonzero projection on the plane.
It is now clear why we need the condition
(
Basic coherence condition
): If all
the
lie in the plane then the replication is never possible.
Note that the vector
is defined as a normal vector to some separating hyperplane. Set
and consider the consequence of
:
If we assume existence of a riskless asset then there is no choice but to
conclude
.
Hence, the
represents some convex combination of the original measures and the
equality
means that
is a risk neutral measure.
Definition
Given the set of available instruments
we introduce the set of risk neutral probability
measures:
Corollary
If there are no acceptable
opportunities and there is a riskless asset
then
Here
refers to the convex hull across various ways to model the market (across the
index
).
Proof
Even though the corollary above clearly follows from considerations of the
present section, such considerations are not easily extensible to more complex
situations. Hence, we give a more generic proof.
For any portfolio
with the property
we introduce the
set
Clearly, by assumption of no acceptable
opportunities
The next task is to prove
that
We argue by contradiction. Assume
and
introduce
By the assumption
we have
By the theorem (
Separating
hyperplane theorem
) there exists
s.t.
From the latter inequality we derive that
.
Then from the former inequality we
conclude
But
is another portfolio with the property
.
Hence, we arrived to contradiction with the assumption of no acceptable
opportunities. We conclude that any finite intersection
is non empty.
We next
want to prove that the intersection
is not empty for any countable collection of
.
We introduce the linear
subspace
The
may be empty. Any x s.t.
may be represented as a
sum
where the
and the y has the property
Also,
Hence, we will restrict further proof to
only (switch to the orthogonal complement of
).
For
the set
always has an interior. Let
be the Lebesgue measure acting in the space of
:
.
By the structure of
the mapping
is continuous with respect to y. We choose some
sequence
where
is the unit ball. By taking a subsequence we
have
for some
.
We
have
Since
always has an interior we also
have
By taking further subsequence and using continuity of
we
have
where we use the notation
for the difference of sets. Note, that
has properties of distance between sets. Hence,
cannot converge to zero (because
and
are
small
and
is at least as large as
).
Consequently,
cannot converge to zero. Hence, we proved that
is not empty for any countable collection
.
To see
that the same statement holds for any collection
we note that there exists a countable everywhere dense set
on
and
is continuous. Hence, any
is infinitely close to some point
and we can always
have
for however small
.
Since
we
conclude
It
remains to note that the structure of the set
is such that the result extends from
to general
.
