Asset Pricing John H. Cochrane June 12, 2000

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CHAPTER 3 CONTINGENT CLAIMS MARKETS price planes “perpendicular” to pc.) When vectors are finite-dimensional, the prime notation is commonly used for inner products, pc0x. This notation does not extend well to infinitedimensional spaces. The notation hpc|xi is also often used for inner products. Planes of constant price move out linearly, and the origin x =0must have a price of zero. If payoff y =2x, then its price is twice the price of x, p(y) = X pc(s)y(s) = X pc(s)2x(s) =2p(x). s Similarly, a payoff of zero must have a price of zero. We can think of p(x) as a pricing function, a map from the state space or payoff space in which x lies (RS in this case) to the real line. We have just deduced from the definition (3.50) that p(x) is a linear function, i.e. that p(ax + by) =ap(x)+bp(y). The constant price lines in Figure 7 are of course exactly what one expects from a linear function from RS to R. (One might draw the price on the z axis coming out of the page. Then the price function would be a plane going through the origin and sloping up with isoprice lines as given in Figure 7.) Figure 7 also includes the payoffs to a contingent claim to the first state. This payoff is one in the first state and zero in other states and thus located on the axis. The plane of price = 1 payoffs is the plane of asset returns; the plane of price = 0 payoffs is the plane of excess returns. A riskfree unit payoff (the payoff to a risk-free pure discount bond) lies on the (1, 1) point in Figure 7; the riskfree return lies on the intersection of the 45o line (same payoff in both states) and the price = 1 plane (the set of all returns). Geometry with m in place of pc. The geometric interpretation of Figure 7 goes through with the discount factor m in the place of pc. We can define an inner product between the random variables x and y by s x · y ≡ E(xy), and retain all the mathematical properties of an inner product. For this reason, random variables for which E(xy) =0are often called “orthogonal.” This language may be familiar from linear regressions. When we run a regression of y on x, y = b 0 x + ε we find the linear combination of x that is “closest” to y, by minimizing the variance or “size” of the residual ε. We do this by forcing the residual to be “orthogonal” to the right hand variable E(xε) = 0. The projection of y on x is defined as the fitted value, proj(y|x) = 62
SECTION 3.5 STATE DIAGRAM AND PRICE FUNCTION b 0 x =E(xx 0 ) −1 E(yx 0 )x. This ideal is often illustrated by a residual vector ε that is perpendicular to a plane definedbytherighthandvariablesx. Thus, when the inner product is defined by a second moment, the operation “project y onto x” isaregression. (If x does not include a constant, you don’t add one.) The geometric interpretation of Figure 7 also is valid if we generalize the setup to an infinite-dimensional state space, i.e. if we think of continuously-valued random variables. Instead of vectors, which are functions from R S to R, random variables are (measurable) functions from Ω to R. Nonetheless, we can still think of them as vectors. The equivalent of R s is now a Hilbert space L 2 , which denotes spaces generated by linear combinations of square integrable functions from Ω to the real line, or the space of random variables with finite second moments. We can still define an “inner product” between two such elements by x · y = E(xy), andp(x) =E(mx) can still be interpreted as “m is perpendicular to (hyper)planes of constant price.” Proving theorems in this context is a bit harder. You can’t just say things like “we can take a line perpendicular to any plane,” such things have to be proved. Sometimes, finite-dimensional thinking can lead you to errors, so it’s important to prove things the right way, keeping the finite dimensional pictures in mind for interpretation. Hansen and Richard (1987) is a very good reference for the Hilbert space machinery. 63
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Asset Pricing John H. Cochrane June
Page 3 and 4:
Contents Acknowledgments 2 Preface
Page 5 and 6:
9.2 Intertemporal Capital Asset Pri
Page 7 and 8:
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
Page 13 and 14: Chapter 1. Consumption-based model
Page 15 and 16: SECTION 1.1 BASIC PRICING EQUATION
Page 17 and 18: 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: State 2 Payoff SECTION 3.5 STATE DI
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 and 70: 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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