SK460 Finance Theory, Diderik Lund, 23 January 2001
Markets for state-contingent claims
(Markeder for tilstandsbetingede krav)
Theoretically useful framework for markets under uncertainty.
Used both in simplied versions and in general version, known
as complete markets (komplette markeder) (denition later).
Direct extension of standard general equilibrium and welfare
theory.
Developed by Kenneth Arrow and Gerard Debreu about 1960.
First and second welfare theorem will hold under some assumptions.
Not very realistic.
Description of one-period uncertainty:
A number of dierent states (tilstander) may occur, numbered
s = 1; : : : ; S .
Here: S is a nite number.
Exactly one of these will be realized.
All stochastic variables depend on this state only: As soon
as the state has become known, the outcome of all stochastic
variables are also known.
\Knowing probability distributions" means knowing the outcomes of stochastic variables in each state.
When S is nite, prob. distn.s cannot be continuous.
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Securities with known state-contingent outcomes
Consider n securities (verdipapirer) numbered j = 1; : : : ; n.
May think of as shares of stock (aksjer).
Value of one unit of security j will be pjs if state s occurs.
These values are known.
Buying numbers Xj of security j today, for j = 1; : : : ; n, will
give total outcomes in the S states as follows:
2
66
66
64
p11
pn1
p1S
pnS
..
..
3
7
7
7
7
7
5
2
6
6
6
6
6
4
X1
3
7
7
7
7
7
5
.. =
Xn
2 P
pj 1 Xj
6
6
6
6
6
4P
..
pjS Xj
3
7
7
7
7
7
5
If prices today (period zero) are p10 ; : : : ; pn0, this portfolio (porteflje) costs
3
2
[p10 pn0 ] 6
6
6
6
6
4
X1
..
Xn
2
7
7
7
7
7
5
X
= pj 0Xj
SK460 Finance Theory, Diderik Lund, 23 January 2001
Constructing a chosen state-contingent vector
If we wish some specic vector of values (in the S states), can any
such vector be obtained?
Suppose we wish
2
3
Y
6
1 77
6
.
6
6 . 7
7
6
4
7
5
YS
Can be obtained if there exist S securities with linearly
inde-
pendent (linert uavhengige) price vectors, i.e. vectors of the
type
2
6
6
6
6
6
4
3
pj 1 77
7
7
7
5
..
pjS
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Complete markets
Suppose S such securities exist, numbered j = 1; : : : ; S , where
S n. A portfolio of these may obtain the right values:
2
66
66
64
p11
p1S
..
3
pS 1 77
7
7
7
5
..
pSS
2
6
6
6
6
6
4
X1
..
XS
3
7
7
7
7
7
5
=
2
6
6
6
6
6
4
Y1
..
YS
3
7
7
7
7
7
5
since we may solve this equation for the portfolio composition
2
66
66
64
X1
..
XS
3
77
77
75
=
2
66
66
64
p11
pS 1
p1S
pSS
..
..
3,1
7
7
7
7
7
5
2
6
6
6
6
6
4
Y1
..
YS
3
7
7
7
7
7
5
If there are not as many as S \linearly independent securities," the
system cannot be solved in general.
If S lin. indep. securities exist, the securities market is called complete.
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Remarks on solution: Short sales
Solution likely to contain some negative quantities, Xj < 0.
Known as short sales.
Buying a negative number of a security means selling it.
Starting from nothing, selling requires borrowing the security
rst.
Will have to hand it back in period one.
Must then rst buy it in market in period one.
Short sale raises cash in period zero, but requires outlay in
period one. (Opposite of buying a security.)
Short-seller interested in falling security prices, pjs < pj0.
Remarks on complete markets
To get any realism in description: S must be very large.
But then, to obtain complete markets, n must also be very
large.
Three objections to realism:
{ Knowledge of all state-contingent outcomes.
{ Large number of securities needed.
{ Security price vectors linearly dependent.
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Arrow-Debreu securities
Securities with the value of one money unit in one state,
but zero in all other states.
Also called elementary state-contingent claims, (elementre
tilstandsbetingede krav), or pure securities.
Possibly exist S dierent A-D securities.
If exist: Linearly independent. Thus complete markets.
If not exist, but markets are complete: May construct A-D
securities from existing securities.
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
0 77
.. 77
7
3,1
2
3
2
7
7
0
7
66 p11 pS 1 7
66 X1 77
7
7
.
7
66 .
66 . 77
7
. 75 1 7777
64 . 75 = 64 .
0 777
p1S pSS
XS
.. 777
5
0
with the 1 appearing as element number s in the column vector on
the right-hand side.
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SK460 Finance Theory, Diderik Lund, 23 January 2001
State prices
The state price for state number s is the amount you must pay
today to obtain one money unit if state s occurs, but zero otherwise.
Solve for state prices:
2 3
0 77
6
6
6 .. 7
7
6
7
2
3,1 6
6
7
6
7
p
p
0
6
11
S
1
7
6
7
6
7
6
7
.
.
6
7
6
7
6
7
6
7
as = [p10 pS 0 ] 6 .
.
1
7
6
7
6
7
4
5
6
7
p1S pSS
0
6
7
6
7
6
7
.
6
7
.
6
4 7
5
0
State prices are today's prices of A-D securities, if those exist. In
Copeland and Weston, p. 113f, state prices are called ps, but their
notation is ambiguous, so we use as.
Risk-free interest rate
To get one money unit available in all possible states, need to buy
one of each A-D security. Like risk-free bond. Risk-free interest
rate r is dened by
S
1
X
= a:
1 + r s=1 s
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Pricing and decision making in complete markets
All you need is the state prices. If an asset has state-contingent
values
2
3
Y1 77
6
6
6
6 .. 7
7
6
4
then its price today is simply
2
6
6
6
6
6
4
YS
Y1
[a1 aS ] ..
YS
7
5
3
7
7
7
7
7
5
=
S
X
s=1
asYs:
Can show this must be true for all traded securities.
For small potential projects: Also (approximately) true. Exception for large projects which change (all) equilibrium prices.
Typical investment project: Investment outlay today, uncertain
future value. Accept project if outlay less than valuation (by
means of state prices) of uncertain future value.
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Absence-of-arbitrage proof for pricing rule
If some asset with future value vector
2
6
6
6
6
6
4
Y1
..
YS
is traded for a dierent price than
3
7
7
7
7
7
5
2
6
6
6
6
6
4
Y1
[a1 aS ] ..
YS
3
7
7
7
7
;
7
5
then one can construct a riskless arbitrage, dened as
A set of transactions which gives us a net gain now,
and with certainty no net outow at any future date.
A riskless arbitrage cannot exist in equilibrium when people have
the same beliefs, since if it did, everyone would demand it. (Innite
demand for some securities, innite supply of others, not equilibrium.)
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Proof contd., exploiting the arbitrage
Assume that a claim to
2
6
6
6
6
6
4
is traded for a price
Y1
..
YS
3
7
7
7
7
7
5
pY < [a1 aS ] 2
6
6
6
6
6
4
Y1
..
YS
3
7
7
7
7
:
7
5
\Buy the cheaper, sell the more expensive!"
Here: Pay pY to get claim to Y vector, shortsell A-D securities in
amounts fY1 ; : : : ; YS g, cash in a net amount
2
6
6
6
6
6
4
Y1
[a1 aS ] ..
YS
3
7
7
7
7
7
5
, pY > 0:
Whichever state occurs: The Ys from the claim you bought is exactly enough to pay o the short sale of a number Ys of A-D securities for that state. Thus no net outow (or inow) in period
one.
Similar proof when opposite inequality. In both cases: Need short
sales.
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Separation principle for complete markets
As long as rm is small enough | its decisions do not aect
market prices | all its owners will agree on how to decide on
investment opportunities: Use state prices.
Everyone agree, irrespective of preferences and wealth.
Also irrespective of probability beliefs | may believe in dierent probabilities for the states to occur.
Exception: All must believe that the same S states have strictly
positive probabilities. (Why?)
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Individual utility maximization with complete markets
Assume for simplicity that A-D securities exist. Consider individual
who wants consumption today, C , and in each state next period,
Qs. Budget constraint:
W0 =
X
s
asQs + C:
Let s Pr(state s). Assume separable utility function
u(C ) + E [U (Qs )]:
(Possibly u() 6= U (), maybe because of time preference.)
X
has f.o.c.
X
max[u(C ) + s sU (Qs ) s.t. W0 = s asQs + C
sU 0(Qs)
= as for all s:
u0(C )
(and the budget constraint).
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Remarks on rst-order conditions
sU 0(Qs)
= as for all s:
u0(C )
Taking a1 ; : : : ; aS as exogenous: For any given C , consider how
distribute budget across states. Higher s ) lower U 0(Qs) )
higher Qs. Higher probability attracts higher consumption.
Consider now whole securities market. Assume total consumption
in each future state
X
Q s =
Qs
individuals
is given. Assume also everyone believes in same 1; : : : ; S . If
some s increases, everyone wants own Qs to increase. Impossible.
Equilibrium restored through higher as.
Assume now Q s increases. Generally people's U 0(Qs) will decrease.
Equilibrium restored through decreasing as (less scarcity).
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SK460 Finance Theory, Diderik Lund, 23 January 2001
Complete markets and welfare economic theory
(Not an important topic in this course.)
Complete markets for state-contingent claims represent exten-
sion of general equilibrium theory to uncertainty.
Also versions with more than one uncertain period.
Same physical good in dierent states and/or at dierent points
in time must be considered as dierent goods.
General equilibrium theory (existence, uniqueness, etc.) works
as under certainty.
First and second welfare theorem also work.
In particular, everyone has same MRS between \consumptions"
in any pair of states.
But: With fewer than S linearly independent securities, markets are called \incomplete," and these results (generally) do
not hold.
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