Stat/Actsc 446/846: Mathematical Models in Finance – Fall 2011
Finding stateprice vectors when
M > N
There are two main approaches for finding stateprice vectors when
M > N
.
Approach 1:
If we’re given a specific matrix
S
(1
,
Ω) and vector of initial prices
~
S
(0), we can write out
the system of
N
equations, parameterize the solution and see if we can find solutions that are positive.
Approach 2:
This is the approach used to prove that an arbitragefree market admits at least one
stateprice vector, when
M > N
. The main idea is to add ArrowDebreu (AD) securities in a way that
preserves the noarbitrage property.
We illustrate the two approaches with an example.
Suppose
~
S
(0) = (1
,
1) and
S
(1
,
Ω) =
1
0
1
1
1
0
1
3
.
Approach 1:
Let
~
ψ
= (
ψ
1
, . . . , ψ
4
). Then we need to see if we can solve
ψ
1
+
ψ
2
+
ψ
3
+
ψ
4
= 1
ψ
2
+ 3
ψ
4
= 1
using a positive vector
~
ψ
. Let
s
=
ψ
3
, then
ψ
4
= (1

s
)
/
3 and if
t
=
ψ
1
, then
ψ
2
= 1

s

t

(1

s
)
/
3 =
2
/
3

(2
/
3)
s

t
. So as long as
s <
1 and
t <
(2
/
3)(1

s
), then
ψ
j
>
0 for all
j
. So for instance if
s
=
t
= 1
/
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 Spring '09
 Adam
 How to Solve It, Ψ, stateprice vector, AD security

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