Proof:Let e denote the space of trading strategies.For any 0 and in and scalars a and b,we have a+b=8+.Thus,the marketed subspace M=f8o:0e}of dividend processes generated by trading strategies is a linear subspace of the space L of adapted processes. Let L+=fcL:c>0.There is no arbitrage if and only if the cone L+and the marketed subspace M intersect precisely at zero.Suppose there is no arbitrage.The Separating Hyperplane Theorem,in a version for closed convex cones that is sometimes called Stiemke's Lemma (see Appendix B of Duffie [2001)implies the existence of a nonzero linear functional F such that F(x)<F(y)for each x in M and each nonzero y in L+.Since M is a linear subspace,this implies that F(x)=0 for each x in M,and thus that F(y)>0 for each nonzero y in L+.This implies that F is strictly increasing. By the Riesz representation theorem,for any such linear function F there is a unique adapted process m,called the Riesz representation of F,such that F(z) x∈L. As F is strictly increasing,m is strictly positive,that is,P(m>0)=1 for all t. The converse follows from the fact that if >0 and m is a strictly positive process,then0. For convenience,we call any strictly positive adapted process a deflator. A deflator n is a state-price density if,for all t, St= (2) i= A state-price density is sometimes called a state-price deflator,a pricing kernel,or a marginal-rate-of-substitution process. For t =T,the right-hand side of(2)is zero,so ST=0 whenever there is a state-price density.It can be shown as an exercise that a deflator m is a state-price density if and only if,for any trading strategy 0, 0·S= t<T, (3) 6
Proof: Let Θ denote the space of trading strategies. For any θ and ϕ in Θ and scalars a and b, we have aδθ +bδϕ = δaθ+bϕ. Thus, the marketed subspace M = {δθ : θ ∈ Θ} of dividend processes generated by trading strategies is a linear subspace of the space L of adapted processes. Let L+ = {c ∈ L : c ≥ 0}. There is no arbitrage if and only if the cone L+ and the marketed subspace M intersect precisely at zero. Suppose there is no arbitrage. The Separating Hyperplane Theorem, in a version for closed convex cones that is sometimes called Stiemke’s Lemma (see Appendix B of Duffie [2001]) implies the existence of a nonzero linear functional F such that F(x) < F(y) for each x in M and each nonzero y in L+. Since M is a linear subspace, this implies that F(x) = 0 for each x in M, and thus that F(y) > 0 for each nonzero y in L+. This implies that F is strictly increasing. By the Riesz representation theorem, for any such linear function F there is a unique adapted process π, called the Riesz representation of F, such that F(x) = E X T t=0 πtxt ! , x ∈ L. As F is strictly increasing, π is strictly positive, that is, P(πt > 0) = 1 for all t. The converse follows from the fact that if δθ > 0 and π is a strictly positive process, then E PT t=0 πtδθ t > 0. For convenience, we call any strictly positive adapted process a deflator. A deflator π is a state-price density if, for all t, St = 1 πt Et X T j=t+1 πjδj ! . (2) A state-price density is sometimes called a state-price deflator, a pricing kernel, or a marginal-rate-of-substitution process. For t = T, the right-hand side of (2) is zero, so ST = 0 whenever there is a state-price density. It can be shown as an exercise that a deflator π is a state-price density if and only if, for any trading strategy θ, θt · St = 1 πt Et X T j=t+1 πjδθ j ! , t < T, (3) 6
meaning roughly that the market value of a trading strategy is,at any time, the state-price discounted expected future dividends generated by the strat- egy The gain process G for (8,S)is defined by G=+,the price plus accumulated dividend.Given a deflator y,the deflated gain process G is defined by G?We can think of deflation as a change of numeraire. Theorem.The dividend-price pair (S)admits no arbitrage if and only if there is a state-price density.A deflator n is a state-price density if and only if Sr=0 and the state-price-deflated gain process G"is a martingale. Proof:It can be shown as an easy exercise that a deflator m is a state-price density if and only if Sr=0 and the state-price-deflated gain process G"is a martingale. Suppose there is no arbitrage.Then ST =0,for otherwise the strategy a is an arbitrage when defined by 0=0,t<T,er=-ST.By the previous proposition,there is some deflator such that E0for any strategy 0. We must prove (2),or equivalently,that G"is a martingale.Doob's Optional Sampling Theorem states that an adapted process X is a martingale if and only if E(X,)=Xo for any stopping time TT.Consider,for an arbitrary security n and an arbitrary stopping time r<T,the trading strategy 0 defined by ()=0 for ktn and of)=1,t<T,with om)= 0,t≥T. Since E(∑t-ortd)=0,we have =0 implying that the m-deflated gain process G of security n satisfies Go= E(G).Since T is arbitrary,Gm.is a martingale,and since n is arbitrary, G 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 Harri- son 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
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 increasing 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(,S),a trading strategy 6 leaves the agent with the total consumption process e+80.Thus the agent has the budget-feasible consumption set C={e+e∈L+:0eΘ}， and the problem sup U(c). (4) c∈C The existence of a solution to (4)implies the absence of arbitrage.Con- versely,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 con- sumption 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 contin- uously differentiable at c*,we can use the first-order conditions for optimality to characterize security prices in terms of the derivatives of the utility func- tion U at c*.Specifically,for any c in L,the derivative of U at c*in the direction c is g(0),where g(a)=U(c*+ac)for any scalar a sufficiently small in absolute value.That is,g'(0)is the marginal rate of improvement of utility as one moves in the direction c away from c*.This directional deriva- tive is denoted VU(c*;c).Because U is continuously differentiable at c*,the function that maps c to VU(c";c)is linear.Since 8 is a budget-feasible direction of change for any trading strategy 0,the first-order conditions for optimality of c*imply that 7U(c*:6)=0,0∈日 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:LR 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 c.converges to cif E∑f-olcn()-c(t训→0.Then U is continuous if U(c,)→U(c whenever cn→c. 8
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
and a state-price density is given by the Riesz representation n of VU(c*), defined by VU(c';x) x∈L. The Riesz Rrepresentation of the utility gradient is also sometimes called the marginal-rates-of-substitution process.Despite our standing assumption that U is strictly increasing,VU(c*;.)need not in general be strictly increasing, but is so if U is concave. As an example,suppose U has the additive form U(c) c∈L+ (5) for some ut:R>R,t >0.It is an exercise to show that if VU(c)exists, then VU(c;z)= (6) If,for all t,ut is concave with an unbounded derivative and e is strictly positive,then any solution c*to (4)is strictly positive. Corollary.Suppose U is defined by(5).Under the conditions of the Propo- sition,for any time t<T, St= n() This result is often called the stochastic Euler equation,made famous in a time-homogeneous Markov setting by Lucas [1978.A precursur is due to LeRoy [1973]. 2.4 Habit and Recursive Utilities The additive utility model is extremely restrictive,and routinely found to be inconsistent with experimental evidence on choice under uncertainty,as for example in Plott [1986].We will illustrate the state pricing associated with some simple extensions of the additive utility model,such as "habit- formation'”utility and“recursive utility.” 9
and a state-price density is given by the Riesz representation π of ∇U(c∗), defined by ∇U(c∗ ; x) = E X T t=0 πtxt ! , x ∈ L. The Riesz Rrepresentation of the utility gradient is also sometimes called the marginal-rates-of-substitution process. Despite our standing assumption that U is strictly increasing, ∇U(c∗; ·) need not in general be strictly increasing, but is so if U is concave. As an example, suppose U has the additive form U(c) = E " X T t=0 ut(ct) # , c ∈ L+, (5) for some ut : R+ → R, t ≥ 0. It is an exercise to show that if ∇U(c) exists, then ∇U(c; x) = E " X T t=0 u0 t(ct)xt # . (6) If, for all t, ut is concave with an unbounded derivative and e is strictly positive, then any solution c∗ to (4) is strictly positive. Corollary. Suppose U is defined by (5). Under the conditions of the Proposition, for any time t<T, St = 1 u0 t(c∗ t ) Et u0 t+1(c∗ t+1)(St+1 + δt+1 . This result is often called the stochastic Euler equation, made famous in a time-homogeneous Markov setting by Lucas [1978]. A precursur is due to LeRoy [1973]. 2.4 Habit and Recursive Utilities The additive utility model is extremely restrictive, and routinely found to be inconsistent with experimental evidence on choice under uncertainty, as for example in Plott [1986]. We will illustrate the state pricing associated with some simple extensions of the additive utility model, such as “habitformation” utility and “recursive utility.” 9
An example of a habit-formation utility is some U:L+-R with U(c) uc.h) where u:R+x R->R is continuously differentiable and,for any t,the habit"level of consumption is defined by for some For example,we could take aj=y for y(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 m of the gradient of U at a strictly positive consumption process c as Tt ue(ct;ht)+E un(Cs;hs)as-t s>t where uc and un 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 asset- pricing applications by Epstein and Zin [1989],who take a utility of the form U(c)=Vo,where the "utility process"V is defined recursively,backward in time from T,by Vi=F(c~V+F), whereV+F denotes the probability distribution of V+i given F,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 Vr-to be a fixed exogenously specified random variable. One may view Vi as the utility at time t for present and future consumption, noting the dependence on the future consumption stream through the con- ditional distribution of the following period's utility.As a special case,for example,consider F(x,m)=f(,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
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