《金融投资学》教学资源（英文文献）Arbitrage, State prices and Portfolio Theory

Arbitrage,State Prices and Portfolio Theory Handbook of the Economics of Finance Philip Dybvig Stephen A.Ross Washington University in Saint Louis MIT First draft:September,2001 This draft:September 19,2002

Arbitrage, State Prices and Portfolio Theory Handbook of the Economics of Finance Philip Dybvig Washington University in Saint Louis Stephen A. Ross MIT First draft: September, 2001 This draft: September 19, 2002

Abstract Neoclassical financial models provide the foundation for our understanding of finance.This chapter introduces the main ideas of neoclassical finance in a single- period context that avoids the technical difficulties of continuous-time models,but preserves the principal intuitions of the subject.The starting point of the analysis is the formulation of standard portfolio choice problems. A central conceptual result is the Fundamental Theorem of Asset Pricing,which asserts the equivalence of absence of arbitrage,the existence of a positive linear pricing rule,and the existence of an optimum for some agent who prefers more to less.A related conceptual result is the Pricing Rule Representation Theorem, which asserts that a positive linear pricing rule can be represented as using state prices,risk-neutral expectations,or a state-price density.Different equivalent rep- resentations are useful in different contexts. Many applied results can be derived from the first-order conditions of the portfolio choice problem.The first-order conditions say that marginal utility in each state is proportional to a consistent state-price density,where the constant of proportion- ality is determined by the budget constaint.If markets are complete,the implicit state-price density is uniquely determined by investment opportunities and must be the same as viewed by all agents,thus simplifying the choice problem.Solv- ing first-order conditions for quantities gives us optimal portfolio choice,solving them for prices gives us asset pricing models,solving them for utilities gives us preferences,and solving them for for probabilities gives us beliefs. We look at two popular asset pricing models,the CAPM and the APT,as well as complete-markets pricing.In the case of the CAPM,the first-order conditions link nicely to the traditional measures of portfolio performance. Further conceptual results include aggregation and mutual fund separation theory, both of which are useful for understanding equilibrium and asset pricing

Abstract Neoclassical financial models provide the foundation for our understanding of finance. This chapter introduces the main ideas of neoclassical finance in a single￾period context that avoids the technical difficulties of continuous-time models, but preserves the principal intuitions of the subject. The starting point of the analysis is the formulation of standard portfolio choice problems. A central conceptual result is the Fundamental Theorem of Asset Pricing, which asserts the equivalence of absence of arbitrage, the existence of a positive linear pricing rule, and the existence of an optimum for some agent who prefers more to less. A related conceptual result is the Pricing Rule Representation Theorem, which asserts that a positive linear pricing rule can be represented as using state prices, risk-neutral expectations, or a state-price density. Different equivalent rep￾resentations are useful in different contexts. Many applied results can be derived from the first-order conditions of the portfolio choice problem. The first-order conditions say that marginal utility in each state is proportional to a consistent state-price density, where the constant of proportion￾ality is determined by the budget constaint. If markets are complete, the implicit state-price density is uniquely determined by investment opportunities and must be the same as viewed by all agents, thus simplifying the choice problem. Solv￾ing first-order conditions for quantities gives us optimal portfolio choice, solving them for prices gives us asset pricing models, solving them for utilities gives us preferences, and solving them for for probabilities gives us beliefs. We look at two popular asset pricing models, the CAPM and the APT, as well as complete-markets pricing. In the case of the CAPM, the first-order conditions link nicely to the traditional measures of portfolio performance. Further conceptual results include aggregation and mutual fund separation theory, both of which are useful for understanding equilibrium and asset pricing

The modern quantitative approach to finance has its original roots in neoclassical economics.Neoclassical economics studies an idealized world in which markets work smoothly without impediments such as transaction costs,taxes,asymme- try of information,or indivisibilities.This chapter considers what we learn from single-period neoclassical models in finance.While dynamic models are becom- ing more and more common,single-period models contain a surprisingly large amount of the intuition and intellectual content of modern finance,and are also commonly used by investment practitioners for the construction of optimal port- folios and communication of investment results.Focusing on a single period is also consistent with an important theme.While general equilibrium theory seeks great generality and abstraction,finance has work to be done and seeks specific models with strong assumptions and definite implications that can be tested and implemented in practice. 1 Portfolio Problems In our analysis,there are two points of time,0 and 1,with an interval of time in between during which nothing happens.At time zero,our champion (the agent) is making decisions that will affect the allocation of consumption between non- random consumption,co,at time 0,and random consumption fco}across states o=1,2,...,revealed at time 1.At time 0 and in each state at time 1,there is a single consumption good,and therefore consumption at time 0 or in a state at time 1 is a real number.This abstraction of a single good is obviously not"true" in any literal sense,but this is not a problem,and indeed any useful theoretical model is much simpler than reality.The abstraction does,however face us with the question of how to interpret our simple model(in this case with a single good) in a practical context that is more complex (has multiple goods).In using a single- good model,there are two usual practices:either use nominal values and measure consumption in dollars,or use real values and measure consumption in inflation- adjusted dollars.Depending on the context,one or the other can make the most sense.In this article,we will normally think of the consumption units as being the numeraire,so that“cash flows'”or“claims to consumption”have the same meaning. Following the usual practice from general equilibrium theory of thinking of units 2

The modern quantitative approach to finance has its original roots in neoclassical economics. Neoclassical economics studies an idealized world in which markets work smoothly without impediments such as transaction costs, taxes, asymme￾try of information, or indivisibilities. This chapter considers what we learn from single-period neoclassical models in finance. While dynamic models are becom￾ing more and more common, single-period models contain a surprisingly large amount of the intuition and intellectual content of modern finance, and are also commonly used by investment practitioners for the construction of optimal port￾folios and communication of investment results. Focusing on a single period is also consistent with an important theme. While general equilibrium theory seeks great generality and abstraction, finance has work to be done and seeks specific models with strong assumptions and definite implications that can be tested and implemented in practice. 1 Portfolio Problems In our analysis, there are two points of time, 0 and 1, with an interval of time in between during which nothing happens. At time zero, our champion (the agent) is making decisions that will affect the allocation of consumption between non￾random consumption, c0, at time 0, and random consumption ￾ cω ✁ across states ω ✂ 1 ✄ 2 ✄✆☎✝☎✝☎✞✄ Ω revealed at time 1. At time 0 and in each state at time 1, there is a single consumption good, and therefore consumption at time 0 or in a state at time 1 is a real number. This abstraction of a single good is obviously not “true” in any literal sense, but this is not a problem, and indeed any useful theoretical model is much simpler than reality. The abstraction does, however face us with the question of how to interpret our simple model (in this case with a single good) in a practical context that is more complex (has multiple goods). In using a single￾good model, there are two usual practices: either use nominal values and measure consumption in dollars, or use real values and measure consumption in inflation￾adjusted dollars. Depending on the context, one or the other can make the most sense. In this article, we will normally think of the consumption units as being the numeraire, so that “cash flows” or “claims to consumption” have the same meaning. Following the usual practice from general equilibrium theory of thinking of units 2

of consumption at various times and in different states of nature as different goods, a typical consumption vector is C fco,c1,..,co,where the real number co de- notes consumption of the single gdod at time zero,and the vector c tc1,...,co of real numbers c1,...,co denotes random consumption of the single good in each state 1,...,at time 1. If this were a typical exercise in general equilibrium theory,we would have a price vector for consumption across goods.For example,we might have the following choice problem,which is named after two great pioneers of general equilibrium theory,Kenneth Arrow and Gerard Debreu: Problem 1 Arrow-Debreu Problem Choose consumptions C co,c1,.,co to maximize utility of consumption U'C-subject to the budget constraint 2 (1) Poco=W. 0.1 Here,U'.-is the utility function that represents preferences,p is the price vector, and W is wealth,which might be replaced by the market value of an endowment. We are taking consumption at time 0 to be the numeraire,and p is the price of the Arrow-Debreu security which is a claim to one unit of consumption at time 1 in state o The first-order condition for Problem 1 is the existence of a positive Lagrangian multiplierA(the marginal utility of wealth)such that UcoA,and for all= 1,,2 Uoco-hpo: This is the usual result from neoclassical economics that the gradient of the util- ity function is proportional to prices.Specializing to the leading case in finance of time-separable von Neumann-Morgenstern preferences,named after John von Neumann and Oscar Morgenstern,two great pioneers of utility theory,we have that U'c-cWe will take v and u to be differentiable, strictly increasing (more is preferred to less),and strictly concave(risk averse). 3

of consumption at varioustimes and in different states of nature as different goods, a typical consumption vector is C ￾ ￾ c0 ✄ c1 ✄ ☎✞☎✝☎✞✄ cΩ ✁ , where the real number c0 de￾notes consumption of the single good at time zero, and the vector c ￾ ￾ c1 ✄ ☎✝☎✞☎✝✄ cΩ ✁ of real numbers c1 ✄ ☎✝☎✞☎✝✄ cΩ denotes random consumption of the single good in each state 1 ✄ ☎✞☎✝☎✞✄ Ω at time 1. If this were a typical exercise in general equilibrium theory, we would have a price vector for consumption across goods. For example, we might have the following choice problem, which is named after two great pioneers of general equilibrium theory, Kenneth Arrow and Gerard Debreu: Problem 1 Arrow-Debreu Problem Choose consumptions C ￾ ￾ c0 ✄ c1 ✄ ☎✝☎✞☎✝✄ cΩ ✁ to maximize utility of consumption U ✁ C ✂ subject to the budget constraint c0 ✄ Ω ∑ ω☎1 (1) pωcω ✂ W ☎ Here, U ✁✝✆ ✂ is the utility function that represents preferences, p is the price vector, and W is wealth, which might be replaced by the market value of an endowment. We are taking consumption at time 0 to be the numeraire, and pω is the price of the Arrow-Debreu security which is a claim to one unit of consumption at time 1 in state ω. The first-order condition for Problem 1 is the existence of a positive Lagrangian multiplier λ (the marginal utility of wealth) such that U ✞ 0 ✁ c0 ✂ ✂ λ, and for all ω ✂ 1 ✄✆☎✝☎✝☎✞✄ Ω, U ✞ ω ✁ cω ✂ ✂ λpω ☎ This is the usual result from neoclassical economics that the gradient of the util￾ity function is proportional to prices. Specializing to the leading case in finance of time-separable von Neumann-Morgenstern preferences, named after John von Neumann and Oscar Morgenstern, two great pioneers of utility theory, we have that U ✁ C ✂ ✂ v ✁ c0 ✂ ✄ ∑ Ω ω☎1 πωu ✁ cω ✂ . We will take v and u to be differentiable, strictly increasing (more is preferred to less), and strictly concave (risk averse). 3

Here,nt is the probability of state o.In this case,the first-order condition is the existence of A such that (2)v'(co)=入， and for all o=1,2,…,n, (3)元od(co)=po or equivalently (4)u(co)=APo; where po=po/nt is the state-price density (also called the stochastic discount factor or pricing kernel),which is a measure of priced relative scarcity in state of nature o.Therefore,the marginal utility of consumption in a state is pro- portional to the relative scarcity.There is a solution if the problem is feasible. prices and probabilities are positive,the von Neumann-Morgenstern utility func- tion is increasing and strictly concave,and there is satisfied the Inada condition limet(c)-0.There are different motivations of von Neumann-Morgenstern preferences in the literature and the probabilities may be objective or subjective. What is important for us that the von Neumann-Morgenstern utility function rep- resents preferences in the sense that expected utility is higher for more preferred consumption patterns.2 Using von Neumann-Morganstern preferences has been popular in part because of axiomatic derivations of the theory (see,for example,Herstein and Milnor [1953]or Luce and Raiffa [1957],chapter 2).There is also a large literature on alternatives and extensions to von Neumann-Morgenstern preferences.For Proving the existence of an equilibrium requires more assumptions in continuous-state mod- els. 2Later,when we look at multiple-agent results,we will also make the neoclassical assumption of identical beliefs,which is probably most naturally motivated by symmetric objective informa- tion. 4

Here, πω is the probability of state ω. In this case, the first-order condition is the existence of λ such that v✞ ✁ (2) c0 ✂ ✂ λ ✄ and for all ω ✂ 1 ✄ 2 ✄✆☎✝☎✝☎✞✄ n, πωu✞ ✁ (3) cω ✂ ✂ λpω or equivalently u✞ ✁ (4) cω ✂ ✂ λρω ✄ where ρω ￾ pω ￾πω is the state-price density (also called the stochastic discount factor or pricing kernel), which is a measure of priced relative scarcity in state of nature ω. Therefore, the marginal utility of consumption in a state is pro￾portional to the relative scarcity. There is a solution if the problem is feasible, prices and probabilities are positive, the von Neumann-Morgenstern utility func￾tion is increasing and strictly concave, and there is satisfied the Inada condition limc✁∞ u✞ ✁ c ✂ ✂ 0.1 There are different motivations of von Neumann-Morgenstern preferences in the literature and the probabilities may be objective or subjective. What is important for us that the von Neumann-Morgenstern utility function rep￾resents preferences in the sense that expected utility is higher for more preferred consumption patterns.2 Using von Neumann-Morganstern preferences has been popular in part because of axiomatic derivations of the theory (see, for example, Herstein and Milnor [1953] or Luce and Raiffa [1957], chapter 2). There is also a large literature on alternatives and extensions to von Neumann-Morgenstern preferences. For 1Proving the existence of an equilibrium requires more assumptions in continuous-state mod￾els. 2Later, when we look at multiple-agent results, we will also make the neoclassical assumption of identical beliefs, which is probably most naturally motivated by symmetric objective informa￾tion. 4