《金融投资学》教学资源（英文文献）Intertemporal Asset Pricing Theory

2.7 Breeden's Consumption-Based CAPM The consumption-based capital asset pricing model (CAPM)of Breeden [1979 extends the results of Rubinstein [1976]by showing that,if agents have additive utility that is,locally,quadratic,then expected asset returns are linear with respect to their covariances with aggregate consumption,as will be stated more carefully shortly.Notably,the result does not depend on complete markets.Locally quadratic additive utility is an extremely strong assumption.(It does not violate monotonicity,as utility need not be quadratic at all levels.)Breeden actually worked in a continuous-time setting of Brownian information,reviewed shortly,within which smooth ad- ditive utility functions are automatically locally quadratic,in a sense that is sufficient to recover a continuous-time analogue of the following consumption- based CAPM.8 In a one-period setting,the consumption-based CAPM cor- responds to the classical CAPM of Sharpe [1964. First,we need some preliminary definitions.The return at time t+1 on a trading strategy 0 whose market value 0.St is non-zero is R1=B~(S+1+d+ 0:·St There is short-term riskless borrowing if,for each given time t<T,there is a trading strategy 0 with F-conditionally deterministic return,denoted r.We refer to the sequence {ro,r1,...,rT-1}of such short-term risk-free returns as the associated "short-rate process,"even though rr is not defined. Conditional on F,we let vart()and covi()denote variance and covariance, respectively. Proposition.(Consumption-Based CAPM)Suppose,for each agenti,that the utility U:()is of the additive form U:(e)=Eua(c),and more- over that,for equilibrium consumption processes,..cm),we have(c) ai+bac,where ait and ba0 are constants.Let S be the associated equi- librium price process of the securities.Then,for any time t, S:=AtE(d+1+S+1）-B:E[(S+1+d+1)et+1], for adapted strictly positive scalar processes A and B.For a given time t, suppose that there is riskless borrowing at the short rate rt.Then there is 8For a theorem and proof,see Duffie and Zame [1989]. 16

2.7 Breeden’s Consumption-Based CAPM The consumption-based capital asset pricing model (CAPM) of Breeden [1979] extends the results of Rubinstein [1976] by showing that, if agents have additive utility that is, locally, quadratic, then expected asset returns are linear with respect to their covariances with aggregate consumption, as will be stated more carefully shortly. Notably, the result does not depend on complete markets. Locally quadratic additive utility is an extremely strong assumption. (It does not violate monotonicity, as utility need not be quadratic at all levels.) Breeden actually worked in a continuous-time setting of Brownian information, reviewed shortly, within which smooth ad￾ditive utility functions are automatically locally quadratic, in a sense that is sufficient to recover a continuous-time analogue of the following consumption￾based CAPM.8 In a one-period setting, the consumption-based CAPM cor￾responds to the classical CAPM of Sharpe [1964]. First, we need some preliminary definitions. The return at time t + 1 on a trading strategy θ whose market value θt · St is non-zero is Rθ t+1 = θt · (St+1 + δt+1) θt · St . There is short-term riskless borrowing if, for each given time t<T, there is a trading strategy θ with Ft-conditionally deterministic return, denoted rt. We refer to the sequence {r0, r1,...,rT −1} of such short-term risk-free returns as the associated “short-rate process,” even though rT is not defined. Conditional on Ft, we let vart(·) and covt(·) denote variance and covariance, respectively. Proposition. (Consumption-Based CAPM) Suppose, for each agent i, that the utility Ui(·) is of the additive form Ui(c) = E hPT t=0 uit(ct) i , and more￾over that, for equilibrium consumption processes c(1),...,c(m) , we have u0 it(c (i) t ) = ait + bitc (i) t , where ait and bit > 0 are constants. Let S be the associated equi￾librium price process of the securities. Then, for any time t, St = AtEt(δt+1 + St+1) − Bt Et[(St+1 + δt+1)et+1], for adapted strictly positive scalar processes A and B. For a given time t, suppose that there is riskless borrowing at the short rate rt. Then there is 8For a theorem and proof, see Duffie and Zame [1989]. 16

a trading strategy with the property that its return Ri has maximal F- conditional correlation with the aggregate consumption e (among all trad- ing strategies).Suppose,moreover,that there is riskless borrowing at the short rate r and that var(R)is strictly positive.Then,for any trading strategy 0 with return E(R+1-T)=E(Rt+1-r), where cov:(R+1,Ri+1) vart(R1) The essence of the result is that expected returns of any security,in excess of risk-free rates,are increasing in the degree to which the security's return de- pends(in the sense of regression)on aggregate consumption.This is natural; there is an average preference in favor of securities that are hedges against aggregate economic performance.While the consumption-based CAPM does not depend on complete markets,its reliance on locally-quadratic expected utility,and otherwise perfect markets,is limiting,and its empirical perfor- mance is mixed,at best.For some evidence,see for example Hansen and Jaganathan 1990]. 2.8 Arbitrage and Martingale Measures This section shows the equivalence between the absence of arbitrage and the existence of a "risk-neutral"probabilities,under which,roughly speaking, the price of a security is the sum of its expected discounted dividends.This idea,stemming from Cox and Ross [1976],was developed into the notion of equivalent martingale measures by Harrison and Kreps [1979. We suppose throughout this subsection that there is short-term riskless borrowing at some uniquely defined short-rate process r.We can define,for any times t and T≤T, R,x=(1+r)(1+rt+1)…(1+Tr-1), the payback at time r of one unit of account borrowed risklessly at time t and "rolled over"in short-term borrowing repeatedly until date T. It would be a simple situation,both computationally and conceptually, if any security's price were merely the expected discounted dividends of the 17

a trading strategy with the property that its return R∗ t+1 has maximal Ft￾conditional correlation with the aggregate consumption et+1 (among all trad￾ing strategies). Suppose, moreover, that there is riskless borrowing at the short rate rt and that vart(R∗ t+1) is strictly positive. Then, for any trading strategy θ with return Rθ t+1, Et(Rθ t+1 − rt) = βθ t Et(R∗ t+1 − rt), where βθ t = covt(Rθ t+1, R∗ t+1) vart(R∗ t+1) . The essence of the result is that expected returns of any security, in excess of risk-free rates, are increasing in the degree to which the security’s return de￾pends (in the sense of regression) on aggregate consumption. This is natural; there is an average preference in favor of securities that are hedges against aggregate economic performance. While the consumption-based CAPM does not depend on complete markets, its reliance on locally-quadratic expected utility, and otherwise perfect markets, is limiting, and its empirical perfor￾mance is mixed, at best. For some evidence, see for example Hansen and Jaganathan [1990]. 2.8 Arbitrage and Martingale Measures This section shows the equivalence between the absence of arbitrage and the existence of a “risk-neutral” probabilities, under which, roughly speaking, the price of a security is the sum of its expected discounted dividends. This idea, stemming from Cox and Ross [1976], was developed into the notion of equivalent martingale measures by Harrison and Kreps [1979]. We suppose throughout this subsection that there is short-term riskless borrowing at some uniquely defined short-rate process r. We can define, for any times t and τ ≤ T, Rt,τ = (1 + rt)(1 + rt+1)···(1 + rτ−1), the payback at time τ of one unit of account borrowed risklessly at time t and “rolled over” in short-term borrowing repeatedly until date τ . It would be a simple situation, both computationally and conceptually, if any security’s price were merely the expected discounted dividends of the 17

security.Of course,this is unlikely to be the case in a market with risk-averse investors.We can nevertheless come close to this sort of characterization of security prices by adjusting the original probability measure P.For this,we define a new probability measure Q to be equivalent to p if Q and P assign zero probabilities to the same events.An equivalent probability measure Q is an equivalent martingale measure if S t<T, where EQ denotes expectation under Q,and E(X)=ER(XF)for any random variable X. It is easy to show that a is an equivalent martingale measure if and only if,for any trading strategy 0, 0·S= t<T. (10) =t+1 We will show that the absence of arbitrage is equivalent to the existence of an equivalent martingale measure. The deflator y defined by y=Ro.defines the discounted gain process G, byGC欧=eS:+∑=17,.The word"martingale"in the term“equivalent martingale measure"comes from the following equivalence. Lemma.A probability measure Q equivalent to P is an equivalent martin- gale measure for (8,S)if and only if Sr=0 and the discounted gain process Gy is a martingale with respect to Q. If,for example,a security pays no dividends before T,then the property described by the lemma is that the discounted price process is a Q-martingale. We already know that the absence of arbitrage is equivalent to the exis- tence of a state-price density T.A probability measure Q equivalent to P can be defined in terms of a Radon-Nikodym derivative,a strictly positive random variable with E()-1,via the definition of expectation with respect to Q given by E(Z)=E(),for any random variable Z.We will consider the measure Q defined by=Er,where ET=TTRo.T TO 18

security. Of course, this is unlikely to be the case in a market with risk-averse investors. We can nevertheless come close to this sort of characterization of security prices by adjusting the original probability measure P. For this, we define a new probability measure Q to be equivalent to P if Q and P assign zero probabilities to the same events. An equivalent probability measure Q is an equivalent martingale measure if St = EQ t X T j=t+1 δj Rt,j ! , t < T, where EQ denotes expectation under Q, and EQ t (X) = EQ(X | Ft) for any random variable X. It is easy to show that Q is an equivalent martingale measure if and only if, for any trading strategy θ, θt · St = EQ t X T j=t+1 δθ j Rt,j ! , t < T. (10) We will show that the absence of arbitrage is equivalent to the existence of an equivalent martingale measure. The deflator γ defined by γt = R−1 0,t defines the discounted gain process Gγ, by Gγ t = γtSt + Pt j=1 γjδj . The word “martingale” in the term “equivalent martingale measure” comes from the following equivalence. Lemma. A probability measure Q equivalent to P is an equivalent martin￾gale measure for (δ, S) if and only if ST = 0 and the discounted gain process Gγ is a martingale with respect to Q. If, for example, a security pays no dividends before T, then the property described by the lemma is that the discounted price process is a Q-martingale. We already know that the absence of arbitrage is equivalent to the exis￾tence of a state-price density π. A probability measure Q equivalent to P can be defined in terms of a Radon-Nikodym derivative, a strictly positive random variable dQ dP with E( dQ dP ) = 1, via the definition of expectation with respect to Q given by EQ(Z) = E( dQ dP Z), for any random variable Z. We will consider the measure Q defined by dQ dP = ξT , where ξT = πT R0,T π0 . 18

(Indeed,one can check by applying the definition of a state-price density to the payoff Ro.r that er is strictly positive and of expectation 1.)The density process for Q is defined by =E(Er).Bayes Rule implies that for any times t and j>t,and any Fi-measurable random variable Zj, 1 (Z,)=EE,2 (11) Fixing some time t<T,consider a trading strategy 0 that invests one unit of account at time t and repeatedly rolls the value over in short-term riskless borrowing until time T,with final value Rr.That is,6S=1 and 4=RT.Relation(3)then implies that T=E(TTRLT)= E(πrRo,D=E:(5ro_o Rot Ro.t Rot (12) From(11),(12),and the definition of a state-price density,(10)is satisfied,so Q is indeed an equivalent martingale measure.We have shown the following result. Theorem.There is no arbitrage if and only if there erists an equivalent martingale measure.Moreover,n is a state-price density if and only if an equivalent martingale measure Q has the density process g defined by Et= Ro.tTt/TO. This martingale approach simplifies many asset-pricing problems that might otherwise appear to be quite complex,and applies much more generally than indicated here.For example,the assumption of short-term borrowing is merely a convenience,and one can typically obtain an equivalent martingale measure after normalizing prices and dividends by the price of some partic- ular security (or trading strategy).Girotto and Ortu 1996 present general results of this type for this finite-dimensional setting.Dalang,Morton,and Willinger [1990]gave a general discrete-time result on the equivalence of no arbitrage and the existence of an equivalent martingale measure,covering even the case with infinitely many states. 2.9 Valuation of Redundant Securities Suppose that the dividend-price pair (S)for the N given securities is arbitrage-free,with an associated state-price density m.Now consider the 19

(Indeed, one can check by applying the definition of a state-price density to the payoff R0,T that ξT is strictly positive and of expectation 1.) The density process ξ for Q is defined by ξt = Et(ξT ). Bayes Rule implies that for any times t and j>t, and any Fj -measurable random variable Zj , EQ t (Zj ) = 1 ξt Et(ξjZj ). (11) Fixing some time t<T, consider a trading strategy θ that invests one unit of account at time t and repeatedly rolls the value over in short-term riskless borrowing until time T, with final value Rt,T . That is, θt · St = 1 and δθ T = Rt,T . Relation (3) then implies that πt = Et(πT Rt,T ) = Et(πTR0,T ) R0,t = Et(ξT π0) R0,t = ξtπ0 R0,t . (12) From (11), (12), and the definition of a state-price density, (10) is satisfied, so Q is indeed an equivalent martingale measure. We have shown the following result. Theorem. There is no arbitrage if and only if there exists an equivalent martingale measure. Moreover, π is a state-price density if and only if an equivalent martingale measure Q has the density process ξ defined by ξt = R0,tπt/π0. This martingale approach simplifies many asset-pricing problems that might otherwise appear to be quite complex, and applies much more generally than indicated here. For example, the assumption of short-term borrowing is merely a convenience, and one can typically obtain an equivalent martingale measure after normalizing prices and dividends by the price of some partic￾ular security (or trading strategy). Girotto and Ortu [1996] present general results of this type for this finite-dimensional setting. Dalang, Morton, and Willinger [1990] gave a general discrete-time result on the equivalence of no arbitrage and the existence of an equivalent martingale measure, covering even the case with infinitely many states. 2.9 Valuation of Redundant Securities Suppose that the dividend-price pair (δ, S) for the N given securities is arbitrage-free, with an associated state-price density π. Now consider the 19

introduction of a new security with dividend process 6 and price process S. We say that 6 is redundant given(6,S)if there exists a trading strategy 0, with respect to only the original security dividend-price process(,S),that replicates 6,in the sense that 8o=6t,t>1. If 6 is redundant given(,S),then the absence of arbitrage for the "aug- mented"dividend-price process [(6,6),(S,S)]implies that S=Yi,where t<T. If this were not the case,there would be an arbitrage,as follows.For example, suppose that for some stopping time T,we have S,>Y,and that r T with strictly positive probability.We can then define the strategy: (a)Sell the redundant security 6 at time r for S,and hold this position until T. (b)Invest 0,S,at time r in the replicating strategy 0,and follow this strategy until T. Since the dividends generated by this combined strategy (a)-(b)after r are zero,the only dividend is at r,for the amount S,-Y>0,which means that this is an arbitrage.Likewise,if S<for some non-trivial stopping time r,the opposite strategy is an arbitrage.We have shown the following. Proposition.Suppose (6,S)is arbitrage-free with state-price density Let 6 be a redundant dividend process with price process S.Then the augmented dividend-price pair [(6,6),(S,S)is arbitrage-free if and only if it has n as a state-price density. In applications,it is often assumed that (6,S)generates complete mar- kets,in which case any additional security is redundant,as in the classical "binomial"model of Cox,Ross,and Rubinstein [1979],and its continuous- time analogue,the Black-Scholes option pricing model,coming up in the next section. Complete markets means that every new security is redundant. Theorem.Suppose that Fr =F and there is no arbitrage.Then markets are complete if and only if there is a unique equivalent martingale measure. Banz and Miller 1978 and Breeden and Litzenberger [1978 explore the ability to deduce state prices from the valuation of derivative securities. 20

introduction of a new security with dividend process ˆδ and price process Sˆ. We say that ˆδ is redundant given (δ, S) if there exists a trading strategy θ, with respect to only the original security dividend-price process (δ, S), that replicates ˆδ, in the sense that δθ t = ˆδt, t ≥ 1. If ˆδ is redundant given (δ, S), then the absence of arbitrage for the “aug￾mented” dividend-price process [(δ, ˆδ),(S, Sˆ)] implies that Sˆt = Yt, where Yt = 1 πt Et X T j=t+1 πj ˆδj ! , t < T. If this were not the case, there would be an arbitrage, as follows. For example, suppose that for some stopping time τ , we have Sˆτ > Yτ , and that τ ≤ T with strictly positive probability. We can then define the strategy: (a) Sell the redundant security ˆδ at time τ for Sˆτ , and hold this position until T. (b) Invest θτ · Sτ at time τ in the replicating strategy θ, and follow this strategy until T. Since the dividends generated by this combined strategy (a)-(b) after τ are zero, the only dividend is at τ , for the amount Sˆτ − Yτ > 0, which means that this is an arbitrage. Likewise, if Sˆτ < Yτ for some non-trivial stopping time τ , the opposite strategy is an arbitrage. We have shown the following. Proposition. Suppose (δ, S) is arbitrage-free with state-price density π. Let ˆδ be a redundant dividend process with price process Sˆ. Then the augmented dividend-price pair [(δ, ˆδ),(S, Sˆ)] is arbitrage-free if and only if it has π as a state-price density. In applications, it is often assumed that (δ, S) generates complete mar￾kets, in which case any additional security is redundant, as in the classical “binomial” model of Cox, Ross, and Rubinstein [1979], and its continuous￾time analogue, the Black-Scholes option pricing model, coming up in the next section. Complete markets means that every new security is redundant. Theorem. Suppose that FT = F and there is no arbitrage. Then markets are complete if and only if there is a unique equivalent martingale measure. Banz and Miller [1978] and Breeden and Litzenberger [1978] explore the ability to deduce state prices from the valuation of derivative securities. 20