The Fundamental Theorem of Asset Pricing

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In this section we shall link up the Radner equilibrium with the existence of "no arbitrage" asset prices. This, of course, is analogous to what we derived earlier in our heuristic two-period, no uncertainty case. Obviously, with multiple states, assets, etc. we have to redefine our concepts somewhat. Arbitrage opportunities can be defined (adopting the terminology of Ingersoll, 1987: 52-3) as follows:

Arbitrage Opportunity I: there is an "arbitrage opportunity of the first kind" if there is a portfolio ah RF such that qah = ・/font> f qfafh 0 and Vah > 0.

Arbitrage Opportunity II: there is an "arbitrage opportunity of the second kind" if there is an ah such that qah = ・/font> f qf afh < 0 and Vah 0.

(as our typesetting does not allow us to be more precise about notation, we should note here that Vah > 0 does not mean that all returns are positive in all states, i.e. it allows for some zero values, while Vah 0 allows for the possibility that all returns are zero).

These definitions are no more complicated than our previous ones. If qfafh < 0, then an agent is selling asset f F short; if qfafh 0, then an agent is either selling asset f short or not buying it. Summing up over assets, then qah = ・/font> f Fafh < 0 implies the agent is selling short on net whereas if qah = ・/font> f Fafh 0 merely implies that he is either selling short on net or not making any net purchases of assets. In either case, there is "no commitment", or basically, on net, he uses none of his endowment at time t = 0 to purchase assets. Similarly, Vah 0 implies that the agent is making no net losses in the future, while Vah > 0 in our first definition implies that at least one return in some state must be positive. Following our previous section, we can define the condition of "no arbitrage" or "arbitrage free" asset prices q as follows (again using the definition of arbitrage of the first kind):

Arbitrage-Free: A set of asset prices q RF is "arbitrage free" if there is no portfolio ah RF such that qah 0 and Vah > 0.

In other words, there is no portfolio with non-positive commitment (qah 0) that yields a non-negative return in every state and a strictly positive return in some states. The intuitive meaning of these conditions have already been given earlier in our simple diagrams.

The mathematics of our earlier case generalizes easily to the multi-good/asset/state case. However, for simplicity, we shall assume that all payoffs are merely in terms of one good (i.e. "gold") which has a spot price of 1 in every state. This, as we shall see later, is a somewhat necessary simplifying assumption. Thus, in terms of analyzing asset market equilibrium, we have F assets, S +1 states but only one good. With this, we now have S + 1 elements in the net trade vector, t = {t 0, t 1, ..., t S} where t s = ps(xsh - esh). Thus our net trade space is S+1 dimensional. Thus, we can define W as the hyperplane or linear subspace:

W = {t RS+1 | t 0 = - qah, t s = Vsah for s = 1, .., S with q, ah RF}

Where Vs = [rs1, rs2, .., rsf]. Consequently, for "no-arbitrage" we need that W R+S+1 = {0}, i.e. the subspace W only shares the origin with the positive orthant R+S+1. Then, the orthogonal subspace is:

W^ = {y RS+1 yW = 0}

where yW = 0 implies that yt = 0 for all t W. Thus, as before, we can find a m W^ such that m = [1, m 1, m 2, ..., m S] where m s is the "present value" of a unit of income in state s S (notice that the first element remains 1 for normalization). Thus, in general, m W = 0 implies that m ｷ[-q, V] = 0 or simply, that there is an S-dimensional vector m -0 = [m 1, .., m S] such that q = m -0V. Thus, for any particular asset f F, then qf = m -0Vf = ・/font> s S m srfs, i.e. the price of an asset is some linear combination of its return in each of the states rf = (rf1, ...,rfS) weighted by the state prices (m 1, .., m S).

Let us now turn to what has been termed the fundamental theorem of asset pricing, which has its roots in Stanley Fischer (1972) and Stephen Ross (1976). This comes in two parts: the first part establishes the equivalence of arbitrage-free asset prices q and the existence of non-negative state prices m ; the second part establishes that the existence of utility-maximizing agents with monotonic preferences implies that asset prices are arbitrage free. Let us to prove them in turn:

Theorem: (First Fundamental Theorem) an asset price vector q RF is "arbitrage free" if and only if there is a semi-positive vector of state prices, m = (1, m 1, ..., m S) 0 where m ｷ[-q, V] = 0 (i.e. q = m -0V).

Proof: (i) (ii): Assume all assets have non-negative prices and returns, thus qf 0 for all f F and V 0. Then we can define W = {t RS+1 t 0 = -qah and t s = Vsah for all s = 1, .., S, ah RF} which is a convex subspace in RS+1. Then, define the simplex D = {t R+S+1 | ・/font> s S+1 t s = 1}. This is a convex set in the non-negative orthant, thus D R+S+1. Note also that 0 D by the definition of the simplex, so D R+S+1\{0}. Now, the "no-arbitrage" condition implies that W R+S+1\{0} = , thus W D = . Since both W and D are disjoint, convex sets then we can apply the separating hyperplane theorem, i.e. there is a vector r RS+1 where p 0 and a scalar b such that:

r t b for all t W

and:

r t b for all t D

As 0 W and 0 D , then r t 0 for all t W and r t > 0 for all t D . Also, because t W implies -t W, then r t = 0 for all t W, thus r W^ . All that remains to show is that r 0. Suppose not. Suppose there is a state s such that r s < 0. But then consider a t D where t s = 1 and t s = 0 for all s s . But then, r t < 0. A contradiction. Thus, there is no s where r s < 0, i.e. r s 0 for all s S+1, thus r = (r 0, r 1, ..., r S) 0. Finally, define m s = r s/r 0. As r 0, then m = (1, r 1/r 0, ...., r S/r 0) = (1, m 1, .., m S) 0. Furthermore, m W^ as W^ is a linear subspace and so m = (1/r 0)r W^ . Thus, there exists a m = (1, m 1, .., m s) 0 such that m t = 0 for all t W. As [-q, V] W, then m ｷ[-q, V] = 0 or, defining m -0 = (m 1, .., m S), then q = m -0V.

(ii) (i): Suppose m [-q, V] = 0 but nonetheless arbitrage opportunities remain, i.e. there is an ah such that qah 0 and Vah > 0. Now, as q = m -0V, then post-multiplying by the arbitrage portfolio ah, then qah = m -0Vah. As m -0 0, then as qah 0 by hypothesis, then the only possibility that could make this true is that Vah 0. But then that contradicts the fact that ah is an arbitrage opportunity.

In short, the first part of the "Fundamental Theorem" effectively implies a consistent linear pricing rule, so that the price of asset f is some linear combination of returns. The second part of the "Fundamental Theorem" associates the absence of arbitrage with the existence of utility-maximizing agents with monotonic preferences:

Theorem: (Second Fundamental Theorem) Under the assumption that utility functions are continuous and strongly monotonic (i.e. uh: Rn(S+1)+ R is continuous and for any x x, where x x, then uh(x ) > uh(x)) then the optimization problem has a solution if and only if there are no arbitrage opportunities on the financial markets, i.e. the following two are equivalent:

(i) for every h H, consumption plan xh* and portfolio ah* solves the following:

max uh[x0h, x1h, ...., xSh]

s.t. p0x0h + qah p0e0h

psxsh Vsah + psesh for all s S

(ii) There is no portfolio ah RF such that qah 0 and Vah 0 (but Vah 0).

Proof: (i) (ii): Let the pair (xh*, ah*) be a solution to the optimization problem, then p0(x0h* - e0h) = -qah* and ps(xsh* - esh) = Vsah* for all s S. Now, suppose q allows for arbitrage opportunities. Then there is a feasible arbitrage portfolio ah RF such that qah 0 and Vah > 0, thus -qah + Vah > 0. Thus, there is an xh = [x0h, x1h, .., xSh] such that p0(x0h - e0h) = -qah - qah* and ps(xsh - esh) = Vsah + Vsah*. As a result,

p0(x0h - e0h) > p0(x0h* - e0h)

ps(xsh - esh) > ps(xsh* - esh) for all s S.

This implies that xh > xh*, or assuming strong monotonicity, then uh(xh) > uh(xh*), a contradiction of the individual optimality of x*.

(ii) (i): We start by defining a pair of sets, BA and BR. For the first:

BA = {yh R+n(S+1) | p0(y0h - e0h) + ・/font> s S m sps(ysh - esh) 0}

which is a closed and bounded (and hence compact) subset of R+n(S+1) (notice also that this is the budget constraint of an Arrow-Debreu problem); we also define

BR = {yh R+n(S+1) | p0(y0h - e0h) -qah, and ps(ysh - esh) Vsah for all s S, ah RF}

which is a closed set (notice here that this is the budget set of a Radner problem). Now, if q is arbitrage-free, then there is no arbitrage portfolio ah RF such that qah 0 and Vah > 0. As we saw earlier by the First Fundamental Theorem, this implies there is a semi-positive vector m = (1, m 1, .., m S) RS+1+ such that q = m -0V. We can use these to define our set BA. However, if q = m -0V, then we see immediately that if xh BR, then it must be true that there exists an ah such that p0(x0h - e0h) -qah and ps(xsh - esh) Vsah which, if there is no arbitrage so q = m -0V, consequently implies that p0(y0h - e0h) + ・/font> s S m sps(ysh - esh) 0, and so xh BA. Thus, BR BA. As BA is compact, then BR is a closed subset of a compact set BA and thus is itself compact. However, notice that BR merely summarizes the set of budget constraints for a Radner optimization problem. As BR is a compact set and uh is a continuous function, then by the Weierstrass theorem, it achieves a maximum over BR. Thus a solution to the Radner optimization problem is defined.

Thus, the second half of the Fundamental Theorem of Asset Pricing connects the concept of an "arbitrage-free" q with the solution to the optimization problem. The meaning of this theorem can be gauged by examining the multi-state, multi-asset analogue to our earlier two-dimensional Lagrangian solution. Recall that if q is arbitrage-free, then there is a semi-positive vector m -0 = (m 1, ..., m S) 0 such that q = m -0V implying that for any f F:

qf = ・/font> s S m srfs

Thus, as we saw earlier, can assign multiplier values m s to returns in different states so that the price of a unit of asset f F is equal to the weighted sum of the value of returns across states. Thus, we can think of m s as shadow price of the state-contingent commodity that pays a unit of account if state s occurs and nothing otherwise. To illustrate this more clearly, let us assume that our agent has a differentiable von Neumann-Morgenstern utility function ・/font> s S+1p shush(xsh), where p sh is the probability (subjective/objective) of a particular state and ush is the elementary utility function that emerges in a particular state. Thus, our optimization problem for agent h H becomes:

max ・/font> s S+1 p sh ush(xsh)

s.t.

p0x0h + qah p0e0h

psxsh Vsah + psesh for all s S

and consequently the Lagrangian would looks like:

L = ・/font> s S+1 p shush(xsh) - l 0h[qah + p0(x0h - e0h)] + l 1h[V1ah - p1(x1h - e1h)] + ..... + l Sh[VSah - pS(xSh - eSh)]

where we have S+1 Lagrangian multipliers, l 0h, ..., l Sh. Let us substitute our utility-function ush(xsh) with an indirect utility function, y sh(ps, msh) where income is defined as msh = psesh + Vsah* = psesh + ・/font> f F rfsafh*, the income from the sale of endowments in state s (psesh) and the returns from the chosen portfolio in state s (Vsah*). Consequently, the Lagrangian can be rewritten as:

L = ・/font> s S+1 p sh y s h(ps, msh) - l 0h[qah]

where all that remains is the choice of the optimal portfolio, ah*. The first order conditions for a maximum implies that for any asset f:

dL/dafh = ・/font> s S+1 p sh( y sh/ msh)( msh/ afh) - l 0h qf = 0

which holds true for all assets f = 1, ..., F. Recognizing that msh/ afh = rfs, we can rewrite this as:

(1/l 0h)・/font> s S+1 p sh( y sh/ msh)rfs = qf for f = 1, ..., F.

so the expected marginal utilities of F assets are proportional to the vector of asset prices, q. Letting m sh = (p sh /l 0h)( y sh/ msh), then this reduces to:

・/font> s S+1 m sh rfs = qf for f = 1, ..., F.

We can think of m sh as the marginal utility of state s for agent h H weighted by the personal proportionality factor (p sh/l 0h) which obviously depends on the subjective probability of state s (p sh) and the personal marginal utility of wealth at t = 0 (l 0h). Thus, m sh is the ratio of expected utility of one extra unit of wealth at state t = 1 and state s (i.e. p sh( y sh/dmsh)) to the actual utility of an extra unit of wealth at t = 0 (l 0h).

It might be useful to notice the analogy between this last rule and the characterizations familiar to us from the Arrow-Debreu model. Notice first that our result implies that for any two assets f, g F that:

・/font> s S+1 m sh rfs/qf = ・/font> s S+1 m sh rgs/qg for all f, g F

which can be seen as a financial analogue of the individual optimum of the state-preference approach. It effectively states that at the individual optimum, with given asset prices, qf and qg, the individual will adjust his asset holdings until the expected marginal utility per dollar in each asset is equal. There is also an analogue to the fundamental theorem of risk-bearing in Arrow-Debreu, namely for any two individuals h, k H and for any pair of assets f, g F:

[・/font> s S+1 m sh rfs]/[・/font> s S+1 m sh rgs] = qf/qg = [・/font> s S+1 m sk rfs]/[・/font> s S+1 m skrgs]

Notice that the term on the left, agent h's ratio of expected marginal utilities of assets f and g, can be conceived of as the marginal rate of substitution between assets f and g. Thus, this states that the agent h's marginal rate of substitution is equal to agent k's marginal rate of substitution - the familiar Pareto-optimality condition, now in a financial context.

Notice that qf and rfs are given in this result and thus are exogenous to the consumer. As a result, it must be true that this holds true for any household h H. This implies that m sh = m s for all households h H, so that ・/font> s S+1 m s rsf = qf for f = 1, ..., F. Thus, marginal utilities of states will be equal across agents at each state at the rate m s. However, it must be stressed here that this relies on the existence of complete asset markets, i.e. that there are at least S linearly-independent assets.

To see why, notice that the linear pricing rule ・/font> s S+1 m sh rsf = qf can be rewritten as m -0hVf = qf where m -0h = [m 1h, m 2h, .., m Sh] and Vf is the fth column of the V matrix. Consequently, completing the matrix m -0hV = q which is a simple linear equation. As a result, as V and q are given, we are trying to solve for m -0h, then there are three possible results: either there is no solution, there is a unique solution or there is an infinite number of solutions. The first case of no solution can be ruled out as rank(V, q) = rank(V) because, by the linear pricing rule we obtained from no-arbitrage, q is a linear combination of the vectors in V.

Now, for a unique solution, we require that rank(V) = S, so that we need it that the number of linearly-independent assets equal the number of states. Notice that this implies that V is a square matrix (or can be made one by dropping the linearly dependent assets) and is non-singular (|V| 0), thus inverting, m -0h = V-1q, exactly as we predicted earlier. Because the solution is necessarily unique, then any other solution m -0k = V-1q will necessarily imply that m -0k = m -0h, thus the marginal values of the states are equal across all agents k, h H.

Can we have an infinite number of solutions? We will if the rank(V) < S, or the number of linearly independent assets fall below the number of states. We refer to such a situation as one of incomplete asset markets. Notice that this implies that it is no longer necessarily true that we have m -0k = m -0h in equilibrium Some commentators have noted that m -0h m -0k can be regarded as the "basic property of incomplete markets" (Magill and Quinzii, 1996: p.97). We shall return to incomplete markets later, but, for now, we make note of the fact that this "basic property" immediately implies that we do not have Pareto-optimality as the marginal rates of substitution are no longer equated to each other across agents, i.e. we will not have an "optimal risk-bearing allocation."

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