The Contingent Claims Model of Speculation
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MICHAEL A. S. GUTH, Ph.D., J.D. |
The Contingent Claims Model of Speculation
For the modern reader, the literature on profitable
destabilizing speculation as a whole appears confusing and contradictory. Many of the early works in the 1950s and
1960s employed models which simply failed to come to grips adequately with a
complex phenomenon like speculation.
For example, speculation can only occur under uncertainty. Yet many of the early works contained models
with no randomness. In a world of
certainty, profits from purchases and sales at a known price illustrate arbitrage
not speculation.
Financial
economists now have a general equilibrium, contingent claims model to
illustrate speculation. This model
solves comparative static issues via standard economic reasoning. The standard format provides individuals
with production functions, preferences, time-distributed endowments, etc., and
specifies the available market range.
Economists then solve for the general equilibrium and pose comparative
static questions, e.g., what happens if preferences change?
In the
Hirshleifer (1975, 1976, 1977) model, investors buy Arrow-Debreu contingent
claims knowing that they will have the opportunity to revise their portfolio
after new information arrives and before the state of the world is
revealed. In presenting Hirshleifer's
model, we benefit from hindsight. A
zero complementarity assumption in the original model led to several
objections. We will introduce the model
without the assumption, show the first-order conditions, and then impose the
assumption to simplify the expressions.
Circa
1975 speculation theorists included, but were not limited to, two competing
groups. The first group maintained in
the Keynes-Hicks tradition that different tolerances for risk largely explained
who would hedge and who would speculate.
In the Keynes-Hicks view, futures markets serve the beneficial social
role of transferring price risks from those more risk averse (hedgers) to those
less risk averse (speculators). The
opposing group, led by the empirical writings of Holbrook Working, contended
that different beliefs (subjective probability values) primarily determined who
chose to speculate. Working regarded
speculation as the means by which market prices come to reflect the best
information among market participants.
Jack
Hirshleifer (1975) initially slanted his work toward providing evidence for the
Working view of speculation:
differences in beliefs account for speculation. For example, his article illustrates how
prices would change if unexpected information arrives and causes beliefs to
change. He also attempted to show that
only a `belief deviant' individual would speculate. But Hirshleifer's (1976, 1989) subsequent replies to comments
expanded on the impact of other significant factors including the
regime/completeness of markets; risk preferences; the nature of information and
endowments; and, due to subsequent work by his son David Hirshleifer (1989a,
1989b, 1990, 1991), demand elasticities and transaction costs.
Hirshleifer's
valuable framework reflects the importance of (1) planning based on the arrival
of new information, (2) multiple trading periods C so that traders have time to buy (sell) and resell (repurchase), (3)
subjective probability distributions over shifts in underlying demand or
supply;[1]
and (4) market incompleteness. As we
describe in Sections 3.4 and 3.5, Hirshleifer's model represents a philosophy
about looking at speculative activity:
identifying the factors most important to speculation, some of which are
missing even from recent attempts to portray speculation. Perhaps this chapter will rekindle interest
his contingent claims model of speculation.
If so, the general theory of speculation will benefit.
3.2
Hirshleifer's (1975) Model Specification
Consider a two-commodity pure exchange economy
with a risky commodity, Z, and a riskless commodity, N. The total quantity of `riskless' commodity N
is fixed and consequently does not depend on the state of the world. The economy begins in period 0 and
ends in period 2, at which time either state A or state B
is revealed. The total quantity of
commodity Z will depend on the state of the world; states A and B
can be thought of as a `good crop' and `bad crop' scenario, respectively. Let some individual believe state A
will occur with probability p and thus state B with probability
1-p. This individual's endowment
vector can then be represented as
, (3.1)
and his expected utility as
. (3.2)
We can treat the riskless commodity, N, as
a numeraire and set its price equal to 1:
. The market regime
is `semicomplete' in that state-contingent claims are available on asset Z but not on asset N. Hence, the available contingent claims have
prices
. If state-contingent
claims
and
were available on N
as well, the market regime would be fully complete.
3.2.1.
The Noninformed Equilibrium
Consider now the following sequence of
events. First, the individual trades
from his endowment, E, to a consumptive optimum,
, based on his preferences, beliefs, and budget
constraint. After the trading has been
completed, state A or state B is revealed. The state contingent claims pay off
according to which state of the world occurs.
Graphically, the sequence of events appears as
Endowment Trading Consumptive Gamble E (period 0) Gamble
C *
Nature's Choice of Consumption
state A or state B (period 2)
(period 1)
In this
pure exchange economy, the individual faces the budget constraint
. (3.3)
When the expected utility function (3.2) is
maximized with respect to
and
and subject to (3.3),
the consumptive optimality conditions are
and
. (3.4)
where
We can eliminate
the
term from (3.4) by
assuming the individual's utility function is additively-separable:
Recall this
assumption appeared in numerous extensions of the Arrow-Debreu model reviewed
in Chapter 2. Additive separability
implies the individual derives no complementarity from consuming the two
goods. If the two goods are food and
gold, the zero complementarity assumption seems reasonable. If the two goods are different kinds of
food, then some complementarity effects would be expected.
With
zero complementarity, conditions (3.4) reduce to
and
. (3.5)
Conditions (3.5) together with the budget
constraint (3.3) determine what quantity of commodities N and Z
the individual will purchase. But the
right-hand side of the equations in (3.5) still contain the market prices,
which any individual will take as given under the competitive assumptions of
this model.
The
prevailing market prices cannot be determined from (3.5), because we do not
know how many other market participants share this individual's endowments,
preferences (utility function), and beliefs (probability assignments). We have arrived at the fundamental issue
confronting all asset pricing theory and the central theme of Chapter 5: how can individual equilibrium conditions be
translated into an aggregate, market-clearing equilibrium model? Hirshleifer (1975) used the heuristic device
of a `representative individual,' i.e., all the people in the market are so
similar that any one individual's beliefs and preferences could be assumed to
reflect the market's as well. As
Hirshleifer noted,
In a
world of pure exchange the equilibrium prices are then governed by the
necessity of `sustaining' the representative endowment vector; since a
representative individual can find no one to trade with in a closed economy,
his endowment vector must be his consumptive optimum....Without a
representative individual, equilibrium prices would of course reflect some average
measure of the various individuals' differing probability beliefs and some average
measure of their respective marginal utilities. Hirshleifer (1975), p. 527.
If
denotes the market's
probability assigned to state A occurring and
denotes the market's
collective utility function, then market clearing prices can be determined from
the optimality conditions
and
. (3.6)
Assuming
zero complementarity and the arrival of unexpected information that changes
the representative probability beliefs from (
) to (
), the new market clearing prices
and
will adjust according
to the following ratios:
and
. (3.7)
Note that (3.7) contains no marginal utility
functions. Hirshleifer observed that
condition (3.7) will hold even if the `representative individual' assumption is
relaxed. In its place, he assumes
individuals have homogeneous or `concordant beliefs.' His logic then holds that if all of the individual probability
assignments, p, equaled
before the new
information and now equal
, then condition (3.7) still holds. Of course, while beliefs change, we must assume that preferences
and endowments are held constant.
3.2.2 The
Informed Equilibrium
Suppose the following sequence of events now
occurs. Individuals trade from their
endowment position to a consumptive optimum, knowing that new information will
arrive that will change probability beliefs.
After the information event, individuals can retrade their contingent claims.
Following the second round of trading, the state of the world will be revealed
as either state A or state B.
Traders will then consume their commodities. Graphically, the new sequence of events prior to the state of the
world being revealed is
Endowment Prior
Trading Speculative
Gamble E Round Gamble T *
(period
0)
Arrival of Posterior
Consumptive
Information Trading
Round Gamble C **
(period 1) (period
2)
The
complexity of the individual's decision problem has now increased substantially,
because in selecting his prior trading consumptive optimum, he must not only
take into account the prices that prevail in the market at that time, but also
the prices he anticipates will prevail in the posterior trading round as well. To simplify the analysis further,
Hirshleifer assumes that the information that arrives will be decisive, so that
every individual will know with certainty what the state of the world will
be. Thus in the posterior trading
round, the probability assigned to either state A or state B will
drop to zero. Consequently, the prices
of contingent claims for that state will drop to zero as well. Thus the budget constraint in the prior
trading round will contain both state A and state B contingent
prices, whereas the posterior trading round budget constraint will only contain
the actual state's prices.
In the
prior trading round, the individual faces the following dynamic optimization
problem:
(3.8)
subject to
![]()
(3.9)
![]()
The single primed notation reflects the
information that the state of the world will be A, and the double primed
notation applies to state B. The
first two budget constraints indicate that the individual's wealth in period 2
(when the state is revealed to be either state A or B) must equal
the value of his prior round trading position. The third budget constraint shows that the value of the prior
trading round (with superscript t) must equal the value of the
individual's endowed wealth (with superscript e).
One
prominent feature emerging from this informed environment relates to prices in
the uninformed environment:
and
. The prior round
prices in the informed environment match the prevailing prices in the
uninformed environment. Hirshleifer
(1976) shows that this key result does not depend on the zero complementarity
assumption. However, individuals must
be able to compute future prices, a subject we will return to in Section 3.2.4.
3.2.3
Speculation Defined
Hirshleifer defined `speculation' as any
portfolio position T * in the informed environment that
differed from the consumptive optimum C * in the noninformed
environment. However, in a regime of
incomplete markets, essentially everyone would speculate once the zero complementarity
assumption is relaxed.
In
earlier work, Hirshleifer and Rubinstein (1973) examined the operation of a
single commodity market with a state-complete regime as well as for information
event-state-complete markets.
Obviously, with a single commodity, no complementarity effects will
complicate the first-order conditions.
For this model, Hirshleifer and Rubinstein defined a speculator as
someone who plans for differential contingent consumptions over information
events.
Speculation
in this earlier model does not depend on portfolio revision but does require a
capital gains motive or plan. If
information unexpectedly arrived in the market, prices would normally adjust
and some traders would revise their portfolios. However, these transactions would not amount to speculation,
since potential capital gains did not motivate any prior behavior. The only individuals who met the
nonspeculator definition had ratios of conditional posterior probabilities
equal to the corresponding ratio of conditional posterior prices. The nonspeculator definition in this earlier
model thus depends only on beliefs and prices, not on resources or preferences.
Similarly,
Rubinstein (1975) found if all individuals have homogeneous beliefs in a single
commodity contingent claims economy, then all would have `nonspeculative'
beliefs. In a heterogeneous beliefs
version of the same economy, those individuals who happened to match the
consensus beliefs would be `nonspeculators.'
These consensus beliefs are more complicated than just the average of
individual beliefs, because additional information is imparted in computing the
average. Joseph Stiglitz has similarly
encountered problems in defining speculation.
Stiglitz
(1970) also identifies speculation with portfolio revision.... Stiglitz introduces uncertainty by assuming
the date t=1 price of long-term bonds is uncertain at date t=0. Unfortunately, since Stiglitz admits no
purely risky securities into his model, in an equilibrium context his economy
is riskless and the injected uncertainty disappears together with the purpose
of his model. Rubinstein (1975),
p.815n.
Recall
from Chapter 1 that a semantic controversy arose over how to define a
`nonspeculator' in the speculation and stability literature. If `nonspeculators'
could be defined, then everyone else left in the model would presumably be a
speculator. Friedman and Telser
maintained that `nonspeculators' could not base their investment decisions on
past prices or price trends. Baumol and
others countered that people would be irrational to ignore past prices.
Against
this backdrop, Hirshleifer (1975) introduced his contingent claims model of
speculation, and the resolution appeared clear and simple. Traders who utilize the informed environment
to alter their choice of a consumption lottery from what it would have been in
a noninformed environment are speculators.
Those who choose the same lottery in the two environments are
nonspeculators. Yet this result
depended on the particular regime of incomplete markets selected and the zero
complementarity assumption. When the
latter assumption was relaxed, then utility-deviant traders as well as
belief-deviant traders were found to be speculators. The only nonspeculators in the semicomplete market regime were
those who exhibited zero complementarity in their preferences for goods. As a result, the current generation of
speculation theorists seem disillusioned with the general equilibrium
contingent claims framework and favor instead noise trader, partial
equilibrium, and so-called rational expectations models of speculation.
Returning
now to the Keynes-Hicks tradition that risk preferring individuals speculate,
the Hirshleifer (1977) model offers some support for this position. `[G]iven differences of belief so that
speculation occurs, the extent of an individual's speculative commitments will
vary with his risk-tolerance.' But
without trading rounds before and after the known arrival of information,
traders will have no capital gains motive.
The new information can certainly change equilibrium beliefs. Preferences and degrees of risk aversion
normally will not change in a model just with the arrival of information; the
theorist afterall has fixed the utility functions as a specification of the
model. These facts suggest that
differences in beliefs have a greater impact than differences in risk aversion
in defining who will speculate.
Ironically,
we have come full circle. The extent to
which speculators profitably destabilize markets remains as an empirical
question. The same can be said for
whether differences in beliefs or risk aversion are more important determinants
of speculation.
3.2.4.
Common Knowledge Beliefs and Price Computability
Even under the zero complementarity assumption,
the relationship between prices and probability assignments will not be valid
with merely a `concordant beliefs' assumption.
Although individual market traders may agree in their probability
beliefs, they have no way of knowing a priori they agree. Chapter 4 is devoted entirely to this topic,
and Chapter 5 applies this topic to asset pricing theories. Unless the traders' beliefs are both agreed
and common knowledge, equation (3.7) will not hold once the representative individual
assumption is relaxed.[2] To say that traders' beliefs are common
knowledge means that all the traders know their own and everyone else's
beliefs, know everyone knows everyone else's beliefs, knows everyone knows
everyone knows everyone else's beliefs, and so on.
Hirshleifer
(1976) was aware of this point when he stated `If all have concordant beliefs
as to the likelihood of the possible states of the world, and all know this,
there will be no problem in assigning probability weights.' (p.693, emphasis
added) However, we face an additional
difficulty, which is precisely the same problem Nagatani (1975) identified for
the Arrow-Debreu securities market in Chapter 2. The future spot market prices to which probability measures are
to be assigned cannot be computed from the information available in the prior
trading round.[3]
Hirshleifer
avoided this difficulty by establishing a martingale process on prices:
If
everyone believes that the martingale proposition is valid, and acts
accordingly, then (in a zero-complementarity world of concordant beliefs) the
proposition will be valid!
But...equation (3.7) does not in general hold true if the
zero-complementarity assumption is relaxed.
Absent independence in demand, if everyone believes in martingales, and
acts accordingly, then everyone will then be wrong! So without zero complementarity, the conditional posterior prices
that are necessary to form a correct probability distribution for guiding
decisions in the prior round of trading cannot simply be computed from publicly
available data (the agreed probabilities) available in the prior round. In general, detailed information as to
endowments and utility functions would be needed to compute the conditional
posterior prices properly, knowledge of which is needed to guide prior-round
decisions properly. (Hirshleifer 1976,
p.693)
The other alternative is to assume that future
spot market prices are common knowledge.
This assumption would limit the robustness of the martingale results,
but it would enable individuals to solve the maximization problem set forth in
(3.8) and (3.9).
3.2.5.
Zero Complementarity Revisited
Two of Hirshleifer's results in one application
of his framework C speculation
only by a belief-deviant individual and the martingale property of prices (
and
) C hinge on
the zero complementarity assumption.
One critic complained that zero complementarity was unrealistic for a
consumptive lottery between a left and a right shoe or for different grades of
wheat. But clearly Hirshleifer's
(1975) Semicomplete Market model intended to portray consumption among more
varied commodities.
Readers
may be left with the unfortunate impression that most economists find the zero
complementarity assumption objectionable.
Any theoretical model abstracts from reality. Useful models include the most important elements of reality and
exclude the rest. For many commodities,
zero complementarity is not only a close but a perfect approximation to
reality.
David
Hirshleifer (1991) uses the zero complementarity assumption to establish
futures prices follow a martingale, i.e., the futures price exhibits no upward
or downward bias. However, futures
prices do reflect seasonal variations with harvesting. David Hirshleifer (1990) also shows that
with positive (negative) complementarity, the futures prices is a downward
(upward) biased predictor of the future spot price. Furthermore when transactions costs are introduced, if demand is
inelastic/elastic, the futures price is a downward/upward biased predictor of
the future spot price. However, David
Hirshleifer's results on the effect of demand elasticity depend on the extent
of consumer participation. David
Hirshleifer (1990) finds that nonparticipation by consumers causes the
direction of the bias to correlate to the hedging positions of producers. If consumers more actively trade futures,
then the bias results could be reversed.
Zero complementarity tends to support the martingale property and offset
bias effects in contingent claims or futures prices.
3.3
Market Incompleteness and Speculation
Commenting on Hirshleifer's article, Feiger
(1976) noted that with complete contingent claims markets, no one has any
desire to speculate. With incomplete
markets, everyone will form contracts with the intent of revising them.[4] A reader may question why only
belief-deviant individuals were found to speculate in Hirshleifer (1975), and
whether the zero complementarity assumption is so strong that it negates the
general proposition that everyone will speculate. We will therefore employ another variant of Hirshleifer's (1977)
framework to explore how speculation and incomplete markets relate.
This
section will show that in a general equilibrium contingent claims model,
utility maximizing individuals will generally speculate even if zero
complementarity is assumed. The
particular form of the contingent claims model we employ resembles a stock
market economy. It turns out that the
zero complementarity assumption only eliminated speculation for a so-called
Semicomplete Market (with claims for
and
), which is the application covered in Section 3.2. First we will define a Fully Complete
Market, an Unconditional (incomplete) Market, and speculation.
Again we
have a pure exchange economy with riskless good N, risky good Z,
and states A and B. All
individuals receive the same endowment for good N, but society's total
endowment for good Z differs across states:
and ![]()
Definition 1: A Fully Complete Market
contains separate contingent claims for
and ![]()
A fully
complete regime only requires enough contingent claims to span the space of
potential consumption vectors; separate state-commodity contracts shown in
Definition 1 are sufficient but not necessary.
Definition 2 (Hirshleifer 1977): An Unconditional
Market contains claims to N and
, where
represents a 1:1
entitlement package of ![]()
The
unconditional market regime corresponds to a claim for a numeraire, such as
money, and a stock market security that entitles the purchaser of
to one share of Z
for every possible state of the world.
Definition 3: Speculation occurs in
any market regime when individuals trade to intermediate consumption lotteries
in an informed environment that differ from their consumption lotteries in a
non-informed situation.
Proposition: Assuming zero complementarity
between N and Z, a utility-maximizing individual will speculate
in an Unconditional Market regime.
Proof: The Noninformed Environment. Let some individual assign probability
to state A occurring
and (1-
) to state B. The
individual receives the same endowment for the riskless asset N in
either state; moreover, the individual cannot trade claims for N
conditioned on each state of the world so that
. The individual maximizes
expected utility
![]()
subject to the budget constraints
![]()
where
represents the 1:1
sandwich price for
and the superscripts e denote the
individual's endowment.
Let
. Feasibility in
the pure exchange economy requires
and
Because
= 0 by the zero
complementarity assumption, the optimality conditions emerging from this
expected utility maximizing problem reduce to
![]()
As a further consequence of zero complementarity,
the individual's marginal rate of substitution for N across states will
equal unity:
= 1. (3.10)
The Informed Environment. The
individual trades first to an intermediate contingent claim bundle, denoted
with superscript t.
In period 1,
conclusive information on the state emerges. In period 2, the contingent claim
market reopens, and the individual retrades to his final period 2
consumptive optimum. The conclusive
information collapses the 1:1 sandwich across states to simply a claim for Z
in the known state.
The
individual now maximizes expected utility
(3.11)
subject to the budget constraints
(3.12)
(3.13)
(3.14)
and
represent the
implicit prices for good Z in states A and B, had such
claims existed. In particular, since
represents a 1:1
package of
,
The primes denote
the conditional prices and quantities that would prevail in each state: one prime denotes state A, two primes
denote state B. Furthermore ![]()
Let ![]()