Clearly, no investmentη0 ≡ (η1, η2) satisfies (18) since Zη =
µ (5η1+ 2η2)
− (5η1+ 2η2)
¶
(2.21) cannot be nonnegative unless it is identically zero. However, the investmentη0 = (2, −5) is an arbitrage opportunity satisfying (19) since
10η = −3, Zη = µ 0
0
¶
. (2.22)
An economy with arbitrage opportunities only of the first type is
Z=
à 1 0
−1 −1
1 1
!
. (2.23)
The payoff vector for any investmentη = (η1, η2) is (η1, −η1− η2, η1 + η2). Clearly, this is nonnegative only ifη1 + η2 = 0. Thus, any arbitrage portfolio (ω, −ω) with ω > 0 satisfies (18), creating an arbitrage of the first type; however, there is no way to set up an arbitrage of the second type with a current negative commitment of funds.
A sufficient condition for an arbitrage opportunity of the second type to guarantee the existence of an arbitrage opportunity of the first type is the existence of a (positive invest-ment) portfolio with a semipositive return; that is, Zw > 0. (For example, an asset with limited liability.) To demonstrate this proposition, assume thatη is an arbitrage opportunity of the second type with 10η = −a < 0, and that w is the portfolio with semipositive return and positive return in states: z0s·w = b > 0. Then η + aw is an arbitrage opportunity of the first type since
10(η + aw) = −a + a = 0,
Z(η + aw) > 0 with z0s·(η + aw) > az0s·w= ab > 0. (2.24) If any asset has limited liability (and its payoff is not identically zero), then this condition is met. Since this will typically be true, we will be concerned primarily with (the existence of) arbitrage opportunities of the second type, which then guarantees (the existence of) arbitrage opportunities of the first type.
2.6 Pricing in the Absence of Arbitrage
Whenever arbitrage opportunities are available, then as was previously demonstrated, no investor will be satisfied with any unbounded portfolio, and no equilibrium can obtain. On the other hand, if there are no arbitrage opportunities in the set of returns, then these returns can be supported in some equilibrium. This proposition will not be proved until later. Here we prove the important preliminary result that in the absence of arbitrage, there is a linear pricing relation among the returns.
Definition 1 A vector p is said to be a pricing vector supporting the economy Z if Z0p= 1.
The more usual way of writing this result is in terms of payoffs and current values: Y0p= v.
Note that this is the same pricing vector; the former result is transformed to the latter by premultiplying by diag(v).]
Typically some supporting pricing vector will exist. Often it will not be unique We will be concerned with the conditions under which a positive pricing vector exists. The reason for this concern is illustrated in the following example.
Consider the economy
Z=
µ z11 z12 0 z22
¶
. (2.25)
Clearly, only pricing vectors withp1 = 1/z11can support this economy. So ifp1is negative, z11must be as well. In this case just shorting the first asset is an arbitrage (of the second type). In a simple situation like this p1 is the cost today of getting one dollar in state 1 at the end of the period (and no dollars in every other state). Therefore, a negative (zero) price indicates an arbitrage opportunity of the second (first) type. Similarly, a portfolio investing the fraction w1 = z12/(z12− z11) in the first asset would have a nonzero return only in the second state. Its return in state 2 is−z22z11/(z12− z11), so p2 = (z11− z12)/z22z11
also must be positive, or this portfolio is an arbitrage opportunity, The connection between positive state prices and the lack of arbitrage is valid in general.
Theorem 2 There exists a nonnegative pricing vector p which supports the returns tableau of an economy if and only if there are no arbitrage opportunities of the second type.
Proof. Consider the linear programming problem
Minimize 10η Subject to Z0η = 0. (2.26) Clearly, the objective can be reduced at least to zero sinceη = 0 accomplishes this. If there are no arbitrage opportunities of the second type, then by (19) zero is the minimum objective possible as well. Now from the theorem of duality (see Mathematical Background) a finite objective minimum for a primal linear programming problem guarantees that the dual is also feasible. The dual of (26) is
Maximize 00p
Subject to Z0p= 1 p = 0. (2.27)
p 0 The two constraints form the linear pricing result desired, so the sufficiency of no arbitrage is proved.
Conversely, if there is a nonnegative pricing support vector, then the dual problem in (27) is feasible. By necessity its objective is zero. Again by the theorem of duality, the primal problem in (26) has a minimal objective of zero. Thus, no arbitrage opportunities exist. Q.E.D.
This theorem requires the existence of some nonnegative pricing vector. It does not prohibit the existence of other pricing vectors with some negative prices. For example, for the economy described in (3) the pricing support equations are
p1+ 2p2+ 3p3 = 1,
3p1 + p2+ 2p3 = 1. (2.28)
with solutions
p1 = .2 − .2p3,
p2 = .4 − 1.4p3. (2.29)
p1 is positive ifp3 < 1, and p2 is positive forp3 < 27. Thus, any value ofp3 between 0 and
2
7 results in all three state prices being positive [e.g.,p3 = .1 and p0 = (.18, .26, .1)] and
2.6 Pricing in the Absence of Arbitrage 31
shows that no arbitrage opportunities (of the second type) are available. Nevertheless, the pricing vector(.1, −.3, .5) with a negative component also supports the economy.
As a second example consider the economy in (15) where state 1 is insurable. The pricing support equations are
3p1+ p2+ 2p3 = 1,
2p1 + 2p2+ 4p3 = 1. (2.30)
with solutions
p1 = .25,
p2 = .25 − 2p3. (2.31)
Thus,p1 is fixed, andp2 > 0 for p3 < 18. Again there is no arbitrage.
For the economy in (20) the pricing support equations are 5p1 − 5p2 = 1,
2p1 − 2p2 = 1. (2.32)
There are obviously no solutions to this set of equations and, a fortiori, no positive so-lutions. Therefore, we know that arbitrage of the second type is possible as previously demonstrated.
For the economy
Z=
à 5 2 3 12 1
!
. (2.33)
the solutions to the pricing support equations are p1 = 2 − p3,
p2 = −3 + p3. (2.34)
In this case there are solutions for the pricing vector, but clearlyp1 andp2 cannot both be nonnegative. In this case, also, arbitrage of the second type is possible.
In some cases there is a nonnegative pricing support vector, but one or more its com-ponents must be zero. For example, consider the economy in (23). For the two assets the pricing support equations are
p1− p2+ p3 = 1,
−p2+ p3 = 1. (2.35)
Clearly, the only solutions have p1 = 0. As we have already seen, this economy has an arbitrage opportunity of the first type. The next theorem shows that to ensure that the pricing vector is strictly positive we must assume that there are no arbitrage opportunities of the second or first types.
Theorem 3 There exists a positive pricing vector p which supports the returns tableau of an economy if and only if there are no arbitrage opportunities of the first or second types.
Proof. (Only the proof of necessity is given here. Sufficiency is demonstrated in the next chapter.) Assume thatη is an arbitrage opportunity of the first type. Then Zη > 0 [condi-tion (18b)] and p> 0 imply
p0Zη > 0. (2.36)
But, by assumption, p is a pricing support vector, so p0Z= 10 and (36) then imply
10η > 0, (2.37)
which contradicts (18a). Thus,η cannot be an arbitrage opportunity as was assumed.
Now assume that η is an arbitrage opportunity of the second type. Then Zη = 0 [condition (19b)] and p> 0 imply
p0Zη > 0. (2.38)
As before, p is a pricing support vector, so p0Z= 10and (38) then imply
10η > 0, (2.39)
which violates (19a). So again η cannot be an arbitrage opportunity as was assumed.
Q.E.D.
The two important features of these two theorems are the positivity of the pricing vector and the linearity of the pricing result. Positive state prices assure the absence of arbitrage while linearity guarantees the absence of any monopoly power in the financial market.
The first statement has already been discussed. ps is the price today of one dollar next period if state s occurs. If any of these prices are zero or negative, then this valuable claim can be obtained at no (or negative) cost.
The linearity of the pricing result guarantees that a payment of two dollars in any state is worth twice as much as a payment of one dollar; that is, there are no scale effects. In addition, the incremental value of one dollar in state s does not depend on the other states’
payoffs; that is, there are no scope effects. If the pricing relation were not linear, then it would be profitable to combine assets into particular packages worth more than the parts or to split assets into parts worth more than the whole.