2 Linear Pricing

2.1 Introduction In analyzing security , two concepts are central: linearity and positivity. Lin- earity of pricing, treated in this chapter, is a consequence of the law of one . The law of one price says that portfolios that have the same payoff must have the same price. It holds in a securities equilibrium under weak restrictions on agents’ preferences. Positivity of pricing is treated in the next chapter.

2.2 The Law of One Price The law of one price says that all portfolios with the same payoff have the same price. That is, if hX = h X, then ph = ph , (2.1) for any two portfolios h and h . If there are no redundant securities, only one portfolio generates any given payoff, and thus the law of one price is trivially satisfied. A necessary and sufficient condition for the law of one price to hold is that every portfolio with zero payoff has zero price. If the law of one price does not hold, then every payoff in the asset span can be purchased at any price. To see this, note first that the zero payoff can be purchased at any price because any multiple of a portfolio with zero payoff is also a portfolio with zero payoff. If the zero payoff can be purchased at any price, then any payoff can be purchased at any price.

2.3 The Payoff Pricing Functional For any security prices p we define a mapping q : M → R that assigns to each payoff the price(s) of the portfolio(s) that generate(s) that payoff. Formally, q(z) ≡{w : w = ph for some h such that z = hX}. (2.2)

15

Downloaded from Cambridge Books Online by IP 140.117.120.241 on Sat Dec 18 07:43:29 GMT 2010. http://dx.doi.org/10.1017/CBO9780511753787.004 Cambridge Books Online © Cambridge University Press, 2010 16 Linear Pricing In general, the mapping q is a correspondence rather than a single-valued function. If the law of one price holds, then q is single-valued. Further, it is a linear functional:

Theorem 2.3.1 The law of one price holds iff q is a linear functional on the asset span M.

Proof: If the law of one price holds, then, as just noted, q is single valued. To prove linearity, consider payoffs z, z ∈ M such that z = hX and z = h X for some portfolios h and h . For arbitrary λ, µ ∈ R, the payoff λz +µz can be generated by the portfolio λh +µh with price λph+µph . Because q is single valued, definition 2.2 implies that

q(λz + µz ) = λph + µph . (2.3)

The right-hand side of Eq. (2.3) equals λq(z) + µq(z ), and thus q is linear. Conversely, if q is a functional, then the law of one price holds by definition. 

Whenever the law of one price holds, we call q the payoff pricing functional. The payoff pricing functional q is one of three operators that are related in a triangular fashion. Each portfolio is a J-dimensional vector of holdings of all securities. The set of all portfolios, RJ , is termed the portfolio space. A vector of security prices p can be interpreted as the linear functional (portfolio pricing functional) from the portfolio space RJ to the reals,

p : RJ → R, (2.4) assigning price ph to each portfolio h. Note that we are using p to denote either the functional or the price vector as the context requires. Similarly, payoff matrix X can be interpreted as a linear operator (payoff operator) from the portfolio space RJ to the asset span M,

X : RJ → M, (2.5) assigning payoff hX to each portfolio h. Assuming that q is a functional, we have

p = q ◦ X, (2.6) or, more explicitly,

ph = q(hX), (2.7) for every portfolio h.

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2.4 Linear Equilibrium Pricing The payoff pricing functional associated with equilibrium security prices is the equi- librium payoff pricing functional. If the law of one price holds in equilibrium, then, by Theorem 2.3.1, the equilibrium payoff pricing functional is a linear functional on the asset span M.Wehave

Theorem 2.4.1 If agents’ functions are strictly increasing at date 0, then the law of one price holds in an equilibrium, and the equilibrium payoff pricing functional is linear.

Proof: If the law of one price does not hold at equilibrium prices p, then there is a portfolio h0 with zero payoff, h0 X = 0, and nonzero price. We can assume that ph0 < 0. For every budget-feasible portfolio h and consumption plan (c0, c1), portfolio h + h0 and consumption plan (c0 − ph0, c1) are budget feasible and strictly preferred. Therefore, an optimal consumption and portfolio choice for any agent cannot exist. 

Note that Theorem 2.4.1 holds whether or not consumption is restricted to be positive. We will see in Chapter 4 that the law of one price may fail in the presence of restrictions on portfolio holdings. If date-0 consumption does not enter the agents’ utility functions, the strict mono- tonicity condition for Theorem 2.4.1 fails. In that case the law of one price is satisfied under the conditions established in the following:

Theorem 2.4.2 If agents’ utility functions are strictly increasing at date 1 and there exists a portfolio with positive and nonzero payoff, then the law of one price holds in an equilibrium, and the equilibrium payoff pricing functional is linear.

Proof: If the law of one price does not hold, then, as in the proof of Theorem 2.4.1, we consider portfolio h0 with zero payoff and nonzero price, and an arbitrary budget-feasible date-1 consumption plan c1 and portfolio h. Let hˆ be a portfolio with positive and nonzero payoff. There exists a number α such that αph0 = phˆ . But then portfolio h + hˆ − αh0 and date-1 consumption plan c1 + hXˆ are budget

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The following examples illustrate the possibility of failure of the law of one price in equilibrium if the conditions of Theorems 2.4.1 and 2.4.2 are not satisfied.

Example 2.4.3 Suppose that there are two states and three securities with payoffs x1 = (1, 0), x2 = (0, 1), and x3 = (1, 1). The utility function of the representative agent is given by

2 2 2 u(c0, c1, c2) =−(c0 − 1) − (c1 − 1) − (c2 − 2) . (2.9) His or her endowment is 1 at date 0 and (1, 2) at date 1. Because the endowment is a satiation point, any prices p1, p2, and p3 of the securities are equilibrium prices. When p1 + p2 = p3, the law of one price does not hold. Here the condition of strictly increasing utility functions is not satisfied. 

Example 2.4.4 Suppose that there are two states and two securities with payoffs x1 = (1, −1) and x2 = (2, −2). The utility function of the representative agent depends only on date-1 consumption and is given by

u(c1, c2) = ln(c1) + ln(c2), (2.10) for (c1, c2)  0. His or her endowment is 0 at date 0 and (1, 1) at date 1. Let the security prices be p1 = p2 = 1. The agent’s optimal portfolio at these prices is the zero portfolio. Therefore, these prices are equilibrium prices even though the law of one price does not hold. Here the condition of strictly increasing utility functions at date 1 is satisfied, but there is no portfolio with positive and nonzero payoff. 

2.5 in Complete Markets S Let es denote the sth basis vector in the space R of contingent claims, with 1 in the sth place and zeros elsewhere. Vector es is the state claim or the Arrow security of state s. It is the claim to one unit of consumption contingent on the occurrence of state s. If markets are complete and if the law of one price holds, then the payoff pricing functional assigns a unique price to each state claim. Let

qs ≡ q(es) (2.11) denote the price of the state claim of state s. We call qs the state price of state s.

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Because any linear functional on RS can be identified by its values on the basis vectors of RS, the payoff pricing functional q can be represented as q(z) = qz (2.12) for every z ∈ RS, where q on the right-hand side of Eq. (2.12) is an S-dimensional vector of state prices. Observe that we use the same notation for the functional and the vector that represents it. Because the price of each security equals the of its payoff under the payoff pricing functional, we have

p j = qxj , (2.13) or, in matrix notation, p = Xq. (2.14) Equation (2.14) is a system of linear equations that associates state prices with given security prices. When the left inverse of the payoff matrix is used, it follows that q = Lp. (2.15) The results of this section depend on the assumption of market completeness because otherwise state claim es may not be in the asset span M, and thus q(es) may not be defined. In Chapter 5 we will introduce state prices in incomplete markets.

2.6 Recasting the Optimization Problem When the law of one price is satisfied, the payoff pricing functional provides a convenient way of representing the agent’s consumption–portfolio choice problem. Substituting z = hX and q(z) = ph, the problem (1.4)–(1.6) can be written as

max u(c0, c1) (2.16) c0,c1,z subject to

c0 ≤ w0 − q(z) (2.17)

c1 ≤ w1 + z (2.18) z ∈ M. (2.19) This formulation makes clear that the agent’s consumption choice in security mar- kets depends only on the asset span and the payoff pricing functional. Any two sets

Downloaded from Cambridge Books Online by IP 140.117.120.241 on Sat Dec 18 07:43:29 GMT 2010. http://dx.doi.org/10.1017/CBO9780511753787.004 Cambridge Books Online © Cambridge University Press, 2010 20 Linear Pricing of security payoffs and prices that generate the same asset span and the same payoff pricing functional induce the same consumption choice. If markets are complete, restriction (2.19) is vacuous. Further, we can use state prices in place of the payoff pricing functional. The problem (2.16)–(2.19) then simplifies to

max u(c0, c1) (2.20) c0,c1,z subject to

c0 ≤ w0 − qz (2.21)

c1 ≤ w1 + z. (2.22) This problem can be interpreted as the consumption–portfolio choice problem with Arrow securities. The first-order conditions for the problem (2.20) (at an interior solution) imply that ∂ u q = 1 . (2.23) ∂0u Thus, state prices are equal to marginal rates of substitution. Security prices can be obtained from state prices using Eq. (2.14). Equation (2.23) can also be obtained by premultiplying Eq. (1.13) by L and using Eq. (2.15). The following example illustrates the use of state prices for determining equilib- rium security prices in complete markets.

Example 2.6.1 Suppose that there are two states and two securities with payoffs x1 = (1, 1) and x2 = (2, 0). The representative agent’s utility function is given by 1 1 u(c , c , c ) = ln(c ) + ln(c ) + ln(c ), (2.24) 0 1 2 0 2 1 2 2 for (c0, c1, c2)  0. His or her endowment is 1 at date 0 and (1, 2) at date 1. Equilibrium security prices are such that the agent’s optimal portfolio is the zero portfolio. Through simple substitution of variables, the agent’s problem (1.4)–(1.6) can be written 1 1 max ln(1 − p1h1 − p2h2) + ln(1 + h1 + 2h2) + ln(2 + h1). (2.25) h1,h2 2 2

The first-order condition for problem (2.25) evaluated at h1 = h2 = 0 yields equi- librium security prices p1 = 3/4 and p2 = 1. The same prices can be calculated by using the payoff pricing functional. Because markets are complete, the payoff pricing functional is given by the state prices

Downloaded from Cambridge Books Online by IP 140.117.120.241 on Sat Dec 18 07:43:29 GMT 2010. http://dx.doi.org/10.1017/CBO9780511753787.004 Cambridge Books Online © Cambridge University Press, 2010 Bibliography 21 which, by Eq. (2.23), are equal to the marginal rates of substitution at the equilibrium consumption plan. The equilibrium consumption plan is (1, 1, 2), and the marginal are 1 for date-0 consumption, 1/2 for state-1 consumption, and 1/4 for state-2 consumption. Marginal rates of substitution are (1/2, 1/4); hence,

1 1 q = , . (2.26) 2 4

Equilibrium security prices are p1 = qx1 = 3/4 and p2 = qx2 = 1. 

2.7 Notes As an inspection of the proof of Theorem 2.4.1 reveals, linear equilibrium pricing obtains under nonsatiation of agents’ utility functions at equilibrium consumption plans. Nonsatiation is a weaker restriction than strict monotonicity. The linearity of payoff pricing is a very important result. It is much discussed in elementary finance texts under the name value additivity. One implication of value additivity is the Miller–Modigliani theorem (Miller and Modigliani [3]), which says that two firms that generate the same future profits have the same market value regardless of their debt–equity structure. Another implication is that corporate managers have no motive to diversify into unrelated activities: if a firm pays market value for an acquisition, then the value of the two cash flows together is the sum of their values separately, and no more. Thus, acquisitions do not create value by making the firm more attractive to stockholders via, say, reduced cash-flow volatility. It remains true, though, that if the summed cash flows increase owing to reduced costs or “synergies” of management, then value is created. Other important implications of the law of one price are parity relations such as rate parity, put-call parity, and others. For articles emphasizing the role of state prices in analysis of security pricing, see Hirshleifer [1], [2].

Bibliography [1] Hirshleifer, J. Investment decision under : Choice theoretic approaches. Quarterly Journal of Economics, 79:509–36, 1965. [2] Hirshleifer, J. Investment decision under uncertainty: Application of the state approach. Quarterly Journal of Economics, 80:252–77, 1966. [3] Miller, M. and Modigliani, F. The cost of capital, corporation finance and the theory of investment. American Economic Review, 48:261–97, 1958.

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