Markets for state-contingent claims [PDF]

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 simpli ed versions and in general version, known

as complete markets (komplette markeder) (de nition 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. 1

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 speci c 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

3

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.

4

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. 5

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.

6

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 de ned by S 1 X = a: 1 + r s=1 s

7

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.

8

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, de ned as

A set of transactions which gives us a net gain now, and with certainty no net out ow 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. (In nite demand for some securities, in nite supply of others, not equilibrium.)

9

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 out ow (or in ow) in period one. Similar proof when opposite inequality. In both cases: Need short sales.

10

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?)

11

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).

12

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).

13

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.

14