Pricing by Arbitrage

Magister Ludi



It might be worthwhile to make a few more comments on the overlap between Radner equilibrium and financial theory a little further. One of the more significant results so far has been the fundamental theorem of asset pricing which yielded q = m -0V as a result. For a specific asset, recall, this is written as:

qf = ・/font> s S ms rfs

so that the price of asset f is linearly related to the variety of rates of return it obtains, with the weights representing the marginal value of different states or "state prices". This relationship is a fundamental result with many applications to financial theory. In this section, we will only consider a few points of contact heuristically. More details can be found in general textbooks on financial economics: standard references include Ingersoll (1987), Huang and Litzenberger (1988) and, especially, Duffie (1992).

One of the basic consequences of the fundamental theorem is the following "linear pricing rule": if the returns on an asset are linearly related to the returns on any other set of assets, then the price of that asset is also linearly related to the prices of the other assets. To see this, suppose we have nothing but Arrow securities in our economy and that markets are complete (thus we have F Arrow securities). Recall that for an Arrow security which pays a unit of the numeraire good in state s and nothing otherwise (so we can refer to it as the "sth" Arrow security), we have a return vector rs = [0, 0, ..0, 1, 0, .., 0] . As a consequence, by the fundamental theorem, the value of that Arrow security is merely qs = m s. Now, suppose a new bond is introduced which pays a more complicated return, e.g. rf = [r1 , r2, ..., rs,..., rS]. It is obvious that the returns on this bond can be replicated by a series of Arrow securities, specifically:

rf = ・/font> s S a fsrs

where rs is the return of an Arrow security which pays a unit in state s S and a fs can be thought of as the number of Arrow securities of the type which pay in state s. For instance, with a three-state world, a bond with return structure rf = [3, 4, 2] can be replicated by taking three Arrow securities which pay in state 1, four Arrow securities that pay in state 2 and two Arrow securities that pay in state 3 (thus a f1 = 3, a f2 = 4, a f3 = 2). Consequently, the linear pricing rule states that the price of this bond is equal to the price of a bundle of nine Arrow securities (three of the type which pay in state 1, four of which pay in state 2, etc.). Letting qf be the price of this bond and letting qs be the price of an Arrow security which pays in state s, then:

qf = ・/font> s S afs qs

so the price of the bond is a linear combination of the prices of the Arrow securities using the same coefficients a fs. The basic consequence of the linear pricing rule, then, is that as long as the number of Arrow securities equals the number of states covered by any asset, then the return of that asset can be replicated by a portfolio of Arrow securities and the price of the resulting asset is merely the price of that portfolio.

This implication is more general: namely, that if there is any set of assets whose returns are linearly independent and cover all states, then the return on any new asset can be replicated by taking a linear combination of the old assets. Consequently, the price of any new asset qf can be determined as a linear combination of these "fundamental" assets. We state this as a proposition:

Theorem: (Pricing by Arbitrage) Let there be n fundamental assets (i.e. n linearly independent assets which span the space of state returns) and let there be unlimited short-sales. Then (i) the return on any given asset f with payoffs can be expressed as a linear combination of returns on the n fundamental assets, i.e. rf = ・/font> i=1n a iri and (ii) in equilibrium, the price of asset f is the same linear combination of the prices of the n fundamental assets, qf = ・/font> i=1n a iqi.

Proof: Part (i) is evident from our previous discussion: as long as the new asset does not offer returns in "new" states which the fundamental assets did not cover, then we can always construct a portfolio of fundamental assets a = {a i}i=1n such that the return on this portfolio replicates the return on the new asset exactly, i.e. rf = ・/font> i=`n a iri. For (ii), we shall prove this for the simple case when n = 2, which then generalizes to higher n by construction. Suppose we have two fundamental assets i = 1, 2, have and a new asset f. Let a = [a 1, a 2] be the portfolio of fundamental assets which replicates the returns on the new asset, thus rf = a 1r1 + a 2r2, where rf, r2 and r2 are all vectors of returns over different states. Then we want to show that qf = a 1q1 + a 2q2. To see this, suppose not. Suppose qf > a 1q1 + a 2q2. Then consider a portfolio a* which has -(a 1q1 + a 2q2) units of asset f, a 1qf units of asset 1 and b 2qf units of asset 2 (thus we are short-selling asset f and purchasing assets 1 and 2). Thus, a* = [-(a 1q1 + a 2q2), a 1qf, a 2qf] . Thus, letting q = [qf, q1, q2], this implies:

qキa* = -(a 1q1 + a 2q2)qf + a 1qfqg + a 2qfqf = 0

so that the cost of the portfolio a* is zero. Now, for any state s, the returns on the portfolio a* are:

Vsa* = ・/font> f F rfsaf = -rfs(a1q1 + a2q2) + r1s a1qf + r2sa2qf

but as rf =a 1r1 + a 2r2, this is reduced to Vsa* = -rfs(a 1q1 + a 2q2) + rfsqf or simply:

Vsa* = rfs(qf - a1q1 - a2q2)

Let us assume, without loss of generality, that rf is semi-positive. Since we hypothesized that qf > a 1q1 + a 2q2, then this implies that Vsa* 0 for every s (strictly for some s). Thus, portfolio a* yields positive returns in the future period. However, recall that the cost of the portfolio is zero, i.e. qキa* = 0. Thus, if we allow unlimited short-sales, then we can always add a* to any portfolio without violating any budget constraint - and, as Vsa* 0, we can thus always increase future returns costlessly, i.e. we have an arbitrage opportunity. Thus, with strictly monotonic preferences, agents can increase their utility by adding portfolio a*, which violates equilibrium. Thus, it must be that qf a 1q1 + a 2q2. By reverse reasoning, we can construct a portfolio a which costs nothing that we can subtract from any other portfolio and increases utility. Thus, it must be that qf = a1q1 + a2q2.

This linear pricing rule forms the heart of the "pricing by arbitrage" logic that underlies much of finance theory. Notice that to obtain the linear pricing rule, we required unlimited short-sales - a proposition which directly goes against Radner's (1972) assumption of a lower bound to ensure existence of a Radner equilibrium. This incongruity is one of the touchier points in attempts to connect Radner equilibria to standard financial theory and shall be passed over in silence here. Instead, we shall use the power of "pricing by arbitrage" to illustrate a few results in financial economics which have relevance to Radner economies.

Consider first the risk-neutral/martingale representation of Cox and Ross (1976) and Harrison and Kreps (1979). The basic notion is that a state price, m s, can be interpreted as a risk-neutral (martingale) probability p s discounted at a riskless rate of return r, i.e. ms = ps/(1+r). Recall that our fundamental theorem claims that qf = ・/font> s S ms rfs. Applying the risk-neutral representation, this becomes:

qf = (1+r)-1 ・/font>s S ps rfs

so the value of asset f is the expected value of its returns evaluated under the risk-neutral probabilities p 1, p 2, .., p S and discounted at the riskless rate of return. Notice that the no-arbitrage condition, which guaranteed the existence of positive state prices m 1, m 2, .., m S also guarantees the existence of positive risk-neutral probabilities and an associated riskless rate of return (shadow or real). We can let E* denote the expectation operator associated with risk-neutral probabilities, thus, E*(rf) = ・/font>sS ps rfs.

We can apply this immediately to return/risk structure of the Arbitrage Pricing Theory (APT) of Stephen Ross (1976). To do this, we must change our notation a little bit. So far, we have been referring to rfs as the return to asset f in state s by which we meant the value of the payoff of that asset in state s. Let us now define the financial return of asset f in state s, Rfs, as the value of the payoff divided by the purchasing price minus 1, i.e. Rfs = rfs/qf - 1. Thus, the fundamental theorem relation q = ・/font> sS ms rfs becomes 1 = ・/font> s S ms(1+Rfs). Note that if Rfs = r for all s S (i.e. asset f pays off a riskless rate of return r), then this implies that ・/font> s S ms = 1/(1+r).

Ross's APT suggests that we consider the following exact return on a dollar invested in asset f:

Rf = Ef + bf11 + bf22 ... + bfkk

where Ef is the expected return on f, the terms (1, 2, .., k) are the exogenous factors and the coefficients (b f1, b f2, .., b fk) are the factor-loading coefficients. The term ・/font> i=1n b fi i can be thought of as the unexpected part of an asset's return. Notice that as there is exact factor structure here, then we are omitting idiosyncratic risk. Now, by the fundamental theorem, we know that:

1 = ・/font> s S ms(1+Rfs)

or, as ms = ps/(1+r) by the risk-neutral martingale representation rule:

(1+r) = ・/font> s S ps(1+ Rfs) = E*(1+Rf)

Now, E*(1+Rf) = E*(1 + Ef + ・/font> ib fi i) = E*(1+ Ef) + ・/font>if E*( i), or, as E*(1+Ef) = 1+Ef, then:

(1+r) = 1 + Ef + ・/font>i bfiE*(i)


Ef - r = ・/font> ibfi(-E*(i))

or, using Ross's notation, we obtain the resulting expression of the (exact factor) APT:

Ef - r = ・/font> ibfi li

where li = -E*(i). Thus, the difference between the expected return on asset f and the risk-free return r (i.e. the risk premium of asset f) is a linear combination of some set of terms l i which have commonly been termed the "factor risk premia", where li = -・/font>s ps i = -(1+r)・/font>s ms i. The question is how to interpret this last term. A useful way to view this is to think of ( 1, .., k) as the unexpected returns on a set of Arrow securities. An Arrow security, recall, pays a return in a particular state and nothing otherwise, so we can think of the return to an Arrow security as a one-factor model with factor loading coefficients b ii = 1 and b ij = 0 for all i j. Thus, the return on the Arrow security is Ri = Ei + i where Ei is the expected return and i is the unexpected part. Consequently, going through the same motions as before, we obtain as a result Ei - r = -E*( i) = l i. Thus, we can think of l i as the risk premium on an Arrow security. Thinking of the factor loading coefficients b fi as the analogues of the coefficients a i we had before, then we can say that the risk premium on any asset is some linear combination of the risk premia on the fundamental assets (such as Arrow securities) that compose it - which fits in precisely with the "pricing by arbitrage" intuition we had before.

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