《金融投资学》教学资源（英文文献）Intertemporal Asset Pricing Theory.pdf_大学文库

1 Introduction This is a survey of "classical"intertemporal asset pricing theory.A central objective of this theory is to reduce asset-pricing problems to the identifica- tion of "state prices,"a notion of Arrow [1953 from which any security has an implied value as the weighted sum of its future cash flows,state by state, time by time,with weights given by the associated state prices.Such state prices may be viewed as the marginal rates of substitution among state-time consumption opportunities,for any unconstrained investor,with respect to a numeraire good.Under many types of market imperfections,state prices may not exist,or may be of relatively less use or meaning.While market im- perfections constitute an important thrust of recent advances in asset pricing theory,they will play a limited role in this survey,given the limitations of space and the priority that should be accorded to first principles based on perfect markets. Section 2 of this survey provides the conceptual foundations of the broader theory in a simple discrete-time setting.After extending the basic modeling approach to a continuous-time setting in Section 3,we turn in Section 4 to term-structure modeling,in Section 5 to derivative pricing,and in Section 6 to corporate securities. The theory of optimal portfolio and consumption choice is closely linked to the theory of asset pricing,for example through the relationship between state prices and marginal rates of substitution at optimality.While this connection is emphasized,for example in Sections 2.3-2.4 and 3.12-3.13,the theory of optimal portfolio and consumption choice,particularly in dynamic incomplete-markets settings,has become so extensive as to defy a proper summary in the context of a reasonably sized survey of asset-pricing theory. The interested reader is especially directed to the treatments of Karatzas and Shreve [1998,Schroder and Skiadas [1999],and Schroder and Skiadas [20001. For ease of reference,as there is at most one theorem per sub-section,we refer to a theorem by its subsection number,and likewise for lemmas and propositions.For example,the unique proposition of Section 2.9 is called “Proposition2.9.” 3

1 Introduction This is a survey of “classical” intertemporal asset pricing theory. A central objective of this theory is to reduce asset-pricing problems to the identification of “state prices,” a notion of Arrow [1953] from which any security has an implied value as the weighted sum of its future cash flows, state by state, time by time, with weights given by the associated state prices. Such state prices may be viewed as the marginal rates of substitution among state-time consumption opportunities, for any unconstrained investor, with respect to a numeraire good. Under many types of market imperfections, state prices may not exist, or may be of relatively less use or meaning. While market imperfections constitute an important thrust of recent advances in asset pricing theory, they will play a limited role in this survey, given the limitations of space and the priority that should be accorded to first principles based on perfect markets. Section 2 of this survey provides the conceptual foundations of the broader theory in a simple discrete-time setting. After extending the basic modeling approach to a continuous-time setting in Section 3, we turn in Section 4 to term-structure modeling, in Section 5 to derivative pricing, and in Section 6 to corporate securities. The theory of optimal portfolio and consumption choice is closely linked to the theory of asset pricing, for example through the relationship between state prices and marginal rates of substitution at optimality. While this connection is emphasized, for example in Sections 2.3-2.4 and 3.12-3.13, the theory of optimal portfolio and consumption choice, particularly in dynamic incomplete-markets settings, has become so extensive as to defy a proper summary in the context of a reasonably sized survey of asset-pricing theory. The interested reader is especially directed to the treatments of Karatzas and Shreve [1998], Schroder and Skiadas [1999], and Schroder and Skiadas [2000]. For ease of reference, as there is at most one theorem per sub-section, we refer to a theorem by its subsection number, and likewise for lemmas and propositions. For example, the unique proposition of Section 2.9 is called “Proposition 2.9.” 3

2 Basic Theory Radner [1967 and Radner [1972 originated our standard approach to a dy- namic equilibrium of "plans,prices,and expectations,"extending the static approach of Arrow [1953]and Debreu ([1953].2 After formulating this stan- dard model,this section provides the equivalence of no arbitrage and state prices,and shows how state prices may be derived from investors'marginal rates of substitution among state-time consumption opportunities.Given state prices,we examine pricing derivative securities,such as European and American options,whose payoffs can be replicated by trading the underlying primitive securities. 2.1 Setup We begin for simplicity with a setting in which uncertainty is modeled as some finite set of states,with associated probabilities.We fix a set F of events,called a tribe,also known as a o-algebra,which is the collection of subsets of n that can be assigned a probability.The usual rules of probability apply.3 We let P(A)denote the probability of an event A. There are T+1 dates:0,1,...,T.At each of these,a tribeFF is the set of events corresponding to the information available at time t. Any event in F is known at time t to be true or false.We adopt the usual convention that Fc Fs whenever t s,meaning that events are never "forgotten."For simplicity,we also take it that events in Fo have probability 0 or 1,meaning roughly that there is no information at time t =0.Taken altogether,the filtration F={Fo,...,Fr,sometimes called an information structure,represents how information is revealed through time.For any random variable Y,we let E(Y)=E(YF)denote the conditional expectation of Y given Ft.In order to simplify things,for any two random variables Y and Z,we always write "Y=2"if the probability that Y≠Z is zero. An adapted process is a sequence X={Xo,...,Xr}such that,for each t,X:is a random variable with respect to (F).Informally,this means 2The model of Debreu [1953]appears in Chapter 7 of Debreu [1959].For more details in a finance setting,see Dothan [1990].The monograph by Magill and Quinzii [1996]is a comprehensive survey of the theory of general equilibrium in a setting such as this. 3The triple (F,P)is a probability space,as defined for example by Jacod and Protter [2000]

2 Basic Theory Radner [1967] and Radner [1972] originated our standard approach to a dynamic equilibrium of “plans, prices, and expectations,” extending the static approach of Arrow [1953] and Debreu [1953].2 After formulating this standard model, this section provides the equivalence of no arbitrage and state prices, and shows how state prices may be derived from investors’ marginal rates of substitution among state-time consumption opportunities. Given state prices, we examine pricing derivative securities, such as European and American options, whose payoffs can be replicated by trading the underlying primitive securities. 2.1 Setup We begin for simplicity with a setting in which uncertainty is modeled as some finite set Ω of states, with associated probabilities. We fix a set F of events, called a tribe, also known as a σ-algebra, which is the collection of subsets of Ω that can be assigned a probability. The usual rules of probability apply.3 We let P(A) denote the probability of an event A. There are T + 1 dates: 0, 1,...,T. At each of these, a tribe Ft ⊂ F is the set of events corresponding to the information available at time t. Any event in Ft is known at time t to be true or false. We adopt the usual convention that Ft ⊂ Fs whenever t ≤ s, meaning that events are never “forgotten.” For simplicity, we also take it that events in F0 have probability 0 or 1, meaning roughly that there is no information at time t = 0. Taken altogether, the filtration F = {F0,..., FT }, sometimes called an information structure, represents how information is revealed through time. For any random variable Y , we let Et(Y ) = E(Y | Ft) denote the conditional expectation of Y given Ft. In order to simplify things, for any two random variables Y and Z, we always write “Y = Z” if the probability that Y 6= Z is zero. An adapted process is a sequence X = {X0,...,XT } such that, for each t, Xt is a random variable with respect to (Ω, Ft). Informally, this means 2The model of Debreu [1953] appears in Chapter 7 of Debreu [1959]. For more details in a finance setting, see Dothan [1990]. The monograph by Magill and Quinzii [1996] is a comprehensive survey of the theory of general equilibrium in a setting such as this. 3The triple (Ω, F, P) is a probability space, as defined for example by Jacod and Protter [2000]. 4

that Xt is observable at time t.An adapted process X is a martingale if,for any times t and s >t,we have Et(Xs)=Xt. A security is a claim to an adapted dividend process,say 6,with 6t denot- ing the dividend paid by the security at time t.Each security has an adapted security-price process S,so that St is the price of the security,ex dividend,at time t.That is,at each time t,the security pays its dividend ot and is then available for trade at the price St.This convention implies that do plays no role in determining ex-dividend prices.The cum-dividend security price at time t is S:+6t. We suppose that there are N securities defined by an RN-valued adapted dividend process 6=(6(1),...,(N)).These securities have some adapted price process S=(S(),...,S(N)).A trading strategy is an adapted process 0 in RN.Here,represents the portfolio held after trading at time t.The dividend process 6 generated by a trading strategy 0 is defined by 69=0-1…(S:+d)-0·S, (1) with“o_l”taken to be zero by convention. 2.2 Arbitrage,State Prices,and Martingales Given a dividend-price pair (S)for N securities,a trading strategy 0 is an arbitrage if 60>0(that is,if 60>0 and 60).An arbitrage is thus a trading strategy that costs nothing to form,never generates losses,and, with positive probability,will produce strictly positive gains at some time. One of the precepts of modern asset pricing theory is a notion of efficient markets under which there is no arbitrage.This is reasonable axiom,for in the presence of an arbitrage,any rational investor who prefers to increase his dividends would undertake such arbitrages without limit,so markets could not be in equilibrium,in a sense that we shall see more formally later in this section.We will first explore the implications of no arbitrage for the representation of security prices in terms of "state prices,"the first step toward which is made with the following result. Proposition.There is no arbitrage if and only if there is a strictly positive adapted process n such that,for any trading strategy 0, 5

that Xt is observable at time t. An adapted process X is a martingale if, for any times t and s>t, we have Et(Xs) = Xt. A security is a claim to an adapted dividend process, say δ, with δt denoting the dividend paid by the security at time t. Each security has an adapted security-price process S, so that St is the price of the security, ex dividend, at time t. That is, at each time t, the security pays its dividend δt and is then available for trade at the price St. This convention implies that δ0 plays no role in determining ex-dividend prices. The cum-dividend security price at time t is St + δt. We suppose that there are N securities defined by an RN -valued adapted dividend process δ = (δ(1),...,δ(N) ). These securities have some adapted price process S = (S(1),...,S(N) ). A trading strategy is an adapted process θ in RN . Here, θt represents the portfolio held after trading at time t. The dividend process δθ generated by a trading strategy θ is defined by δθ t = θt−1 · (St + δt) − θt · St, (1) with “θ−1” taken to be zero by convention. 2.2 Arbitrage, State Prices, and Martingales Given a dividend-price pair (δ, S) for N securities, a trading strategy θ is an arbitrage if δθ > 0 (that is, if δθ ≥ 0 and δθ 6= 0). An arbitrage is thus a trading strategy that costs nothing to form, never generates losses, and, with positive probability, will produce strictly positive gains at some time. One of the precepts of modern asset pricing theory is a notion of efficient markets under which there is no arbitrage. This is reasonable axiom, for in the presence of an arbitrage, any rational investor who prefers to increase his dividends would undertake such arbitrages without limit, so markets could not be in equilibrium, in a sense that we shall see more formally later in this section. We will first explore the implications of no arbitrage for the representation of security prices in terms of “state prices,” the first step toward which is made with the following result. Proposition. There is no arbitrage if and only if there is a strictly positive adapted process π such that, for any trading strategy θ, E X T t=0 πtδθ t ! = 0. 5

meaning roughly that the market value of a trading strategy is, at any time, the state-price discounted expected future dividends generated by the strategy. The gain process G for (δ, S) is defined by Gt = St + Pt j=1 δj , the price plus accumulated dividend. Given a deflator γ, the deflated gain process Gγ is defined by Gγ t = γtSt + Pt j=1 γjδj . We can think of deflation as a change of numeraire. Theorem. The dividend-price pair (δ, S) admits no arbitrage if and only if there is a state-price density. A deflator π is a state-price density if and only if ST = 0 and the state-price-deflated gain process Gπ is a martingale. Proof: It can be shown as an easy exercise that a deflator π is a state-price density if and only if ST = 0 and the state-price-deflated gain process Gπ is a martingale. Suppose there is no arbitrage. Then ST = 0, for otherwise the strategy θ is an arbitrage when defined by θt = 0, t<T, θT = −ST . By the previous proposition, there is some deflator π such that E( PT t=0 δθ t πt) = 0 for any strategy θ. We must prove (2), or equivalently, that Gπ is a martingale. Doob’s Optional Sampling Theorem states that an adapted process X is a martingale if and only if E(Xτ ) = X0 for any stopping time τ ≤ T. Consider, for an arbitrary security n and an arbitrary stopping time τ ≤ T, the trading strategy θ defined by θ(k) = 0 for k 6= n and θ (n) t = 1,t<τ , with θ (n) t = 0, t ≥ τ . Since E( PT t=0 πtδθ t ) = 0, we have E −S(n) 0 π0 +Xτ t=1 πtδ (n) t + πτS(n) τ ! = 0, implying that the π-deflated gain process Gn,π of security n satisfies Gn,π 0 = E (Gn,π τ ). Since τ is arbitrary, Gn,π is a martingale, and since n is arbitrary, Gπ is a martingale. This shows that absence of arbitrage implies the existence of a state-price density. The converse is easy. The proof is motivated by those of Harrison and Kreps [1979] and Harrison and Pliska [1981] for a similar result to follow in this section regarding the notion of an “equivalent martingale measure.” Ross [1987], Prisman [1985], Kabanov and Stricker [2001], and Schachermayer [2001] show the impact of taxes or transactions costs on the state-pricing model. 7

2.3 Individual Agent Optimality We introduce an agent, defined by a strictly increasing4 utility function U on the set L+ of nonnegative adapted “consumption” processes, and by an endowment process e in L+. Given a dividend-price process (δ, S), a trading strategy θ leaves the agent with the total consumption process e + δθ. Thus the agent has the budget-feasible consumption set C = {e + δθ ∈ L+ : θ ∈ Θ}, and the problem sup c ∈ C U(c). (4) The existence of a solution to (4) implies the absence of arbitrage. Conversely, if U is continuous,5 then the absence of arbitrage implies that there exists a solution to (4). (This follows from the fact that the feasible consumption set C is compact if and only if there there is no arbitrage.) Assuming that (4) has a strictly positive solution c∗ and that U is continuously differentiable at c∗, we can use the first-order conditions for optimality to characterize security prices in terms of the derivatives of the utility function U at c∗. Specifically, for any c in L, the derivative of U at c∗ in the direction c is g0 (0), where g(α) = U(c∗ + αc) for any scalar α sufficiently small in absolute value. That is, g0 (0) is the marginal rate of improvement of utility as one moves in the direction c away from c∗. This directional derivative is denoted ∇U(c∗; c). Because U is continuously differentiable at c∗, the function that maps c to ∇U(c∗; c) is linear. Since δθ is a budget-feasible direction of change for any trading strategy θ, the first-order conditions for optimality of c∗ imply that ∇U(c∗ ; δθ )=0, θ ∈ Θ. We now have a characterization of a state-price density. Proposition. Suppose that (4) has a strictly positive solution c∗ and that U has a strictly positive continuous derivative at c∗. Then there is no arbitrage 4A function f : L → R is strictly increasing if f(c) > f(b) whenever c>b. 5For purposes of checking continuity or the closedness of sets in L, we will say that cn converges to c if E[ PT t=0 |cn(t) − c(t)|] → 0. Then U is continuous if U(cn) → U(c) whenever cn → c. 8

An example of a habit-formation utility is some U : L+ → R with U(c) = E " X T t=0 u(ct, ht) # , where u : R+ × R → R is continuously differentiable and, for any t, the “habit” level of consumption is defined by ht = Pt j=1 αj ct−j for some α ∈ RT +. For example, we could take αj = γj for γ ∈ (0, 1), which gives geometrically declining weights on past consumption. A natural motivation is that the relative desire to consume may be increased if one has become accustomed to high levels of consumption. By applying the chain rule, we can calculate the Riesz representation π of the gradient of U at a strictly positive consumption process c as πt = uc(ct, ht) + Et X s>t uh(cs, hs)αs−t ! , where uc and uh denote the partial derivatives of u with respect to its first and second arguments, respectively. The habit-formation utility model was developed by Dunn and Singleton [1986] and in continuous time by Ryder and Heal [1973], and has been applied to asset pricing problems by Constantinides [1990], Sundaresan [1989], and Chapman [1998]. Recursive utility, inspired by Koopmans [1960], Kreps and Porteus [1978], and Selden [1978], was developed for general discrete-time multi-period assetpricing applications by Epstein and Zin [1989], who take a utility of the form U(c) = V0, where the “utility process” V is defined recursively, backward in time from T, by Vt = F(ct, ∼ Vt+1 | Ft), where ∼ Vt+1 | Ft denotes the probability distribution of Vt+1 given Ft, where F is a measurable real-valued function whose first argument is a non-negative real number and whose second argument is a probability distribution, and fi- nally where we take VT +1 to be a fixed exogenously specified random variable. One may view Vt as the utility at time t for present and future consumption, noting the dependence on the future consumption stream through the conditional distribution of the following period’s utility. As a special case, for example, consider F(x, m) = f (x, E[h(Ym)]), (7) where f is a function in two real variables, h(·) is a “felicity” function in one variable, and Ym is any random variable whose probability distribution is m. 10