Asset Pricing John H. Cochrane June 12, 2000

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price, p(x) > 0. CHAPTER 4 THE DISCOUNT FACTOR No-arbitrage says that you can’t get for free a portfolio that might pay off positively, but will certainly never cost you anything. This definition is different from the colloquial use of the word “arbitrage.” Most people use “arbitrage” to mean a violation of the law of one price – a riskless way of buying something cheap and selling it for a higher price. “Arbitrages” here might pay off, but then again they might not. The word “arbitrage” is also widely abused. “Risk arbitrage” is a Wall Street oxymoron that means making specific kinds of bets. An equivalent statement is that if one payoff dominates another, then its price must be higher – if x ≥ y, then p(x) ≥ p(y) (Or, a bit more carefully but more long-windedly, if x ≥ y almost surely and x>ywith positive probability, then p(x) >p(y). You can’t forget that x and y are random variables.) m>0 ⇒No-arbitrage The absence of arbitrage opportunities is clearly a consequence of a positive discount factor, and a positive discount factor naturally results from any sort of utility maximization. Recall, m(s) =β u0 [c(s)] u 0 (c) It is a sensible characterization of preferences that marginal utility is always positive. Few people are so satiated that they will throw away money. Therefore, the marginal rate of substitution is positive. The marginal rate of substitution is a random variable, so “positive” means “positive in every state of nature” or “in every possible realization.” Now, if contingent claims prices are all positive, a bundle of positive amounts of contingent claims must also have a positive price, even in incomplete markets. A little more formally, > 0. Theorem: p = E(mx) and m(s) > 0 imply no-arbitrage. Proof: m>0; x ≥ 0 and there are some states where x>0. Thus, in some states mx > 0 and in other states mx =0. Therefore E(mx) > 0. ¤ No arbitrage ⇒ m>0 Now we turn the observation around, which is again the hard and interesting part. As the law of one price property guaranteed the existence of a discount factor m, no-arbitrage guarantees the existence of a positive m. The basic idea is pretty simple. No-arbitrage means that the prices of any payoff in the positive orthant (except zero, but including the axes) must be strictly positive. The price = 70
SECTION 4.2 NO-ARBITRAGE AND POSITIVE DISCOUNT FACTORS 0 plane divides the region of positive prices from the region of negative prices. Thus, if the region of negative prices is not to intersect the positive orthant, the iso-price lines must march up and to the right, and the discount factor m, must point up and to the right. This is how we have graphed it all along, most recently in figure 9. Figure 11 illustrates the case that is ruled out: a whole region of negative price payoffs lies in the positive orthant. For example, the payoff x is strictly positive, but has a negative price. As a result, the (unique, since this market is complete) discount factor m is negative in the y-axis state. x * , m x p = -1 p = 0 p = +1 Figure 11. Counter-example for no-arbitrage ⇒ m>0 theorem. The payoff x is positive, but has negative price. The discount factor is not strictly positive The theorem is easy to prove in complete markets. There is only one m, x ∗ .Ifitisn’t positive in some state, then the contingent claim in that state has a positive payoff and a negative price, which violates no arbitrage. More formally, Theorem: In complete markets, no-arbitrage implies that there exists a unique m> 0 such that p = E(mx). Proof: No-arbitrage implies the law of one price, so there is an x ∗ such that p = E(x ∗ x), and in a complete market this is the unique discount factor. Suppose that x ∗ ≤ 0 for some states. Then, form a payoff x that is 1 in those states, and zero elsewhere. This payoff is strictly positive, but its price, P s:x ∗ (s)
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Asset Pricing John H. Cochrane June
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Contents Acknowledgments 2 Preface
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9.2 Intertemporal Capital Asset Pri
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19.2 Yield curve and expectations h
Page 9 and 10:
Asset pricing problems are solved b
Page 11 and 12:
try to focus on the basic ideas and
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Chapter 1. Consumption-based model
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SECTION 1.1 BASIC PRICING EQUATION
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discount factors, SECTION 1.3 PRICE
Page 19 and 20: SECTION 1.4 CLASSIC ISSUES IN FINAN
Page 23 and 24: equation (1.2) as SECTION 1.4 CLASS
Page 25 and 26: ate! In equations, if then SECTION
Page 33 and 34: SECTION 1.5 DISCOUNT FACTORS IN CON
Page 39 and 40: SECTION 1.6 PROBLEMS function with
Page 41 and 42: Chapter 2. Applying the basic model
Page 43 and 44: SECTION 2.2 GENERAL EQUILIBRIUM mig
Page 45 and 46: C t+1 SECTION 2.2 GENERAL EQUILIBRI
Page 47 and 48: SECTION 2.3 CONSUMPTION-BASED MODEL
Page 49 and 50: SECTION 2.4 ALTERNATIVE ASSET PRICI
Page 51 and 52: SECTION 2.5 PROBLEMS Since the opti
Page 53 and 54: that function. From SECTION 2.5 PRO
Page 55 and 56: SECTION 3.2 RISK NEUTRAL PROBABILIT
Page 57 and 58: The discount factor prices the asse
Page 59 and 60: SECTION 3.4 RISK SHARING Asset pric
Page 61 and 62: State 2 Payoff SECTION 3.5 STATE DI
Page 63 and 64: SECTION 3.5 STATE DIAGRAM AND PRICE
Page 65 and 66: SECTION 4.1 LAW OF ONE PRICE AND EX
Page 67 and 68: SECTION 4.1 LAW OF ONE PRICE AND EX
Page 69: SECTION 4.2 NO-ARBITRAGE AND POSITI
Page 73 and 74: SECTION 4.2 NO-ARBITRAGE AND POSITI
Page 75 and 76: so SECTION 4.3 AN ALTERNATIVE FORMU
Page 77 and 78: Chapter 5. Mean-variance frontier a
Page 79 and 80: SECTION 5.1 EXPECTED RETURN -BETA R
Page 81 and 82: SECTION 5.2 MEAN-VARIANCE FRONTIER:
Page 83 and 84: SECTION 5.3 AN ORTHOGONAL CHARACTER
Page 85 and 86: SECTION 5.3 AN ORTHOGONAL CHARACTER
Page 87 and 88: SECTION 5.4 SPANNING THE MEAN-VARIA
Page 89 and 90: SECTION 5.5 A COMPILATION OF PROPER
Page 91 and 92: SECTION 5.5 A COMPILATION OF PROPER
Page 93 and 94: SECTION 5.6 MEAN-VARIANCE FRONTIERS
Page 95 and 96: SECTION 5.6 MEAN-VARIANCE FRONTIERS
Page 97 and 98: SECTION 5.7 PROBLEMS Hansen-Jaganna
Page 99 and 100: SECTION 6.1 FROM DISCOUNT FACTORS T
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Asset Pricing John H. Cochrane June 12, 2000

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