preference-free in the sense that they do not depend on any specific assumptions about preferences but only depend on an assumption that agents prefer more to less.Central to this section is the notion of an arbitrage,which is a"money pump" or a"free lunch".If there is arbitrage,linearity of the neoclassical problem im- plies that any candidate optimum can be dominated by adding the arbitrage.As a result,no agent who prefers more to less would have an optimum if there ex- ists arbitrage.Furthermore,this seemingly weak assumption is enough to obtain two useful theorems.The Fundamental Theorem of Asset Pricing says that the following are equivalent:absence of arbitrage,existence of a consistent positive linear pricing rule,and existence of an optimum for some hypothetical agent who prefers more to less.The Pricing Rule Representation Theorem gives different equivalent forms for the consistent positive linear pricing rule,using state prices, risk-neutral probabilites(martingale valuation),state-price density (or stochastic discount factor or pricing kernel),or an abstract positive linear operator.The re- sults in this section are from Cox and Ross [1975],Ross [1976c,1978b],and Dybvig and Ross [1987].The results have been formalized in continuous time by Harrison and Kreps [1979]and Harrison and Pliska [1981]. Occasionally,the theorems in this section can be applied directly to obtain an in- teresting result.For example,linearity of the pricing rule is enough to derive put- call parity without constructing the arbitrage.More often,the results in this sec- tion help to answer conceptual questions.For example,an option pricing formula that is derived using absence of arbitrage is always consistent with equilibrium, as can be seen from the Fundamental Theorem.By the Fundamental Theorem. absence of arbitrage implies there is an optimum for some hypothetical agent who prefers more to less;we can therefore construct an equilibrium in the single-agent pure exchange economy in which this agent is endowed with the optimal holding By construction the equilibrium in this economy will have the desired pricing,and therefore any no-arbitrage pricing result is consistent with some equilibrium. In this section,we will work in the context of Problem 2.An arbitrage is a change in the portfolio that makes all agents who prefer more to less better off.We make all such agents better off if we increase consumption sometime,and in some state of nature,and we never decrease consumption.By combining the two constraints in Problem 2,we can write the consumption C associated with any portfolio choice 10
preference-free in the sense that they do not depend on any specific assumptions about preferences but only depend on an assumption that agents prefer more to less. Central to this section is the notion of an arbitrage, which is a “money pump” or a “free lunch”. If there is arbitrage, linearity of the neoclassical problem implies that any candidate optimum can be dominated by adding the arbitrage. As a result, no agent who prefers more to less would have an optimum if there exists arbitrage. Furthermore, this seemingly weak assumption is enough to obtain two useful theorems. The Fundamental Theorem of Asset Pricing says that the following are equivalent: absence of arbitrage, existence of a consistent positive linear pricing rule, and existence of an optimum for some hypothetical agent who prefers more to less. The Pricing Rule Representation Theorem gives different equivalent forms for the consistent positive linear pricing rule, using state prices, risk-neutral probabilites (martingale valuation), state-price density (or stochastic discount factor or pricing kernel), or an abstract positive linear operator. The results in this section are from Cox and Ross [1975], Ross [1976c, 1978b], and Dybvig and Ross [1987]. The results have been formalized in continuous time by Harrison and Kreps [1979] and Harrison and Pliska [1981]. Occasionally, the theorems in this section can be applied directly to obtain an interesting result. For example, linearity of the pricing rule is enough to derive putcall parity without constructing the arbitrage. More often, the results in this section help to answer conceptual questions. For example, an option pricing formula that is derived using absence of arbitrage is always consistent with equilibrium, as can be seen from the Fundamental Theorem. By the Fundamental Theorem, absence of arbitrage implies there is an optimum for some hypothetical agent who prefers more to less; we can therefore construct an equilibrium in the single-agent pure exchange economy in which this agent is endowed with the optimal holding. By construction the equilibrium in this economy will have the desired pricing, and therefore any no-arbitrage pricing result is consistent with some equilibrium. In this section, we will work in the context of Problem 2. An arbitrage is a change in the portfolio that makes all agents who prefer more to less better off. We make all such agents better off if we increase consumption sometime, and in some state of nature, and we never decrease consumption. By combining the two constraints in Problem 2, we can write the consumptionC associated with any portfolio choice 10
using the stacked matrix equation c-[8]+[] The first row,W-P',is consumption at time 0,which is wealth W less the cost of our portfolio.The remaining rows,X,give the random consumption across states at time 1. Now,when we move from the portfolio choice e to the portfolio choice +n, the initial wealth term cancels and the change in consumption can now be written as sc- This will be an arbitrage it AC is never negative and is positive in at least one component,which we will write as4 AC>0 or [ n>0. Some authors describe taxonomies of different types of arbitrage,having perhaps a negative price today and zero payoff tomorrow,a zero price today and a non- negative but not identically zero payoff tomorrow,or a negative price today and a positive payoff tomorrow.These are all examples of arbitrages that are sub- sumed by our general formula.The important thing is that there is an increase in consumption in some state of nature at some point of time and there is never any decrease in consumption. Fundamental Theorem of Asset Pricing Theorem 1 Fundamental Theorem of Asset Pricing The following conditions on prices P and payoffs X are equivalent: ()Absence ofarbre:(A向）( p! 4 We use the following terminology for vector inequalities:(x≥y)台(i)(ex,≥z),(r>y)台 (r≥y)&(3i)(>y),and(x>y)台(i)(x>). 11
Θ using the stacked matrix equation C ✂ W 0 ✁ ✄ P✞ X ✁ Θ ☎ The first row, W P✞ Θ, is consumption at time 0, which is wealth W less the cost of our portfolio. The remaining rows, XΘ, give the random consumption across states at time 1. Now, when we move from the portfolio choice Θ to the portfolio choice Θ ✄ η, the initial wealth term cancels and the change in consumption can now be written as ∆C ✂ P✞ X ✁ η ☎ This will be an arbitrage it ∆C is never negative and is positive in at least one component, which we will write as4 ∆C ✂ 0 or P✞ X ✁ η ✂ 0 ☎ Some authors describe taxonomies of different types of arbitrage, having perhaps a negative price today and zero payoff tomorrow, a zero price today and a nonnegative but not identically zero payoff tomorrow, or a negative price today and a positive payoff tomorrow. These are all examples of arbitrages that are subsumed by our general formula. The important thing is that there is an increase in consumption in some state of nature at some point of time and there is never any decrease in consumption. Fundamental Theorem of Asset Pricing Theorem 1 Fundamental Theorem of Asset Pricing The following conditions on prices P and payoffs X are equivalent: ✁ i✂ Absence of arbitrage: ✁☎✄✆ η ✂✞✝ P✞ X ✁ η ✂ 0✟ . 4We use the following terminology for vector inequalities: ✠ x ✡ y☛✌☞✍✠✏✎i☛✑✠ xi ✡ yi ☛ , ✠ x ✒ y☛✓☞ ✠✔✠ x ✡ y ☛ & ✠✖✕i☛✗✠ xi ✒ yi ☛✔☛ , and ✠ x ✒✘✒ y☛✓☞✙✠✏✎i☛✗✠ xi ✒ yi ☛ . 11
(ii)Existence of a consistent positive linear pricing rule (positive state prices): (3p>>0)(P=p'X) (iii)Some agent with strictly increasing preferences U has an optimum in Prob- lem 2. PROOF We prove the equivalence by showing (i)=(ii),(ii)=(iii),and (iii) () (i)=(ii):This is the most subtle part,and it follows from a separation theorem or the duality theorem from linear programming.From the definition of absence of arbitrage,we have that the sets ={‖]nn∈ and S2={x∈r2+1x>0} must be disjoint.Therefore,there is a separating hyperplane=such thatx=0 for all x E S andx >0 for all xS2.(See...,theorem ...Normalizing so that the first component(the shadow price of time zero consumption)is 1,we will see that p defined by (1p)=z/zo is the consistent linear pricing rule we seek.Constancy of zx for xES implies that (1p) ] =0,which is to say that p=p'X,i.e. p is a consistent linear pricing rule.Furthermore,x positive forxS2 implies >>0 and consequently p >>0,and p is indeed the desired consistent positive linear pricing rule. (ii)=(iii):This part is proven by construction.Let U(C)=(1p)C,then =0 solves Problem 2.To see this,note that the objective function U(C)is constant and equal to W for all⊙： U(C)=(1p)C p(8]+[x]o) = W+(-P'+pX)Θ =W. 12
✁ ii ✂ Existence of a consistent positive linear pricing rule (positive state prices): ✁ ✆ p ✂✂ 0✂ ✁ P✞ ✂ p✞ X ✂ . ✁ iii✂ Some agent with strictly increasing preferences U has an optimum in Problem 2. PROOF We prove the equivalence by showing ✁ i ✂ ✁ ✁ ii ✂ , ✁ ii ✂ ✁ ✁ iii ✂ , and ✁ iii ✂ ✁ ✁ i✂ . ✁ i✂ ✁ ✁ ii✂ : This is the most subtle part, and it follows from a separation theorem or the duality theorem from linear programming. From the definition of absence of arbitrage, we have that the sets S1 ✂ P✞ X ✁ η ✄ η ☎ ℜ n ✆ and S2 ✞✝ x ☎ ℜ Ω✟1 ✄ x ✂ 0 ✠ must be disjoint. Therefore, there is a separating hyperplane z such that z ✞ x ✂ 0 for all x ☎ S1 and z✞ x ✂ 0 for all x ☎ S2. (See ..., theorem ...) Normalizing so that the first component (the shadow price of time zero consumption) is 1, we will see that p defined by ✁ 1p✞ ✂ ✂ zz0 is the consistent linear pricing rule we seek. Constancy of zx for x ☎ S1 implies that ✁ 1p✞ ✂ P✞ X ✁ ✂ 0, which is to say that P✞ ✂ p✞ X, i.e. p is a consistent linear pricing rule. Furthermore, z ✞ x positive for x ☎ S2 implies z ✂✂ 0 and consequently p ✂✂ 0, and p is indeed the desired consistent positive linear pricing rule. ✁ ii ✂ ✁ ✁ iii✂ : This part is proven by construction. Let U ✁ C ✂ ✂ ✁ 1p ✞ ✂ C, then Θ ✂ 0 solves Problem 2. To see this, note that the objective function U ✁ C ✂ is constant and equal to W for all Θ: U ✁ C ✂ ✂ ✁ 1p✞ ✂ C ✂ ✁ 1p✞ ✂ ✝ W 0 ✁ ✄ P✞ X ✁ Θ✟ ✂ W ✄ ✁ P✞ ✄ p✞ X ✂ Θ ✂ W ☎ 12
(The motivation this construction is observation that the existence of the consistent linear pricing rule with state prices p implies that all feasible consumptions satisfy (1p)C=W) (iii)(i):This part is obvious,since any candidate optimum is dominated by adding the arbitrage,and therefore there can be no arbitrage if there is an optimum More formally,adding an arbitrage implies the change of consumption AC>0, which implies an increase in U(C). One feature of the proof that may seem strange is the degeneracy (linearity)of the utility function whose existence is constructed.This was all that was needed for this proof,but it could also be constructed to be strictly concave,additively separable over time,and of the von Neumann-Morgenstern class for given proba- bilities.Assuming any of these restrictions on the class would make some parts of the theorem weaker ((iii)implies (i)and(ii))at the same time that it makes other parts stronger ((i)or (ii)implies (iii)).The point is that the theorem is still true if(iii)is replaced by a much more restrictive class that imposes on U any or all of strict concavity,some order of differentiability,additive separability over time, and a von Neumann-Morgenstern form with or without specifying the probabil- ities in advance.All of these classes are restrictive enough to rule out arbitrage, and general enough to contain a utility function that admits an optimum when there is no arbitrage. The statement and proof of the theorem is a little more subtle if the state space is infinite-dimensional.The separation theorem is topological in nature,so we must restrict our attention to a topologically relevant subset of the nonnegative random variables.Also,we may lose the separating hyperplane theorem because the in- terior of the positive orthant is empty in most of these spaces(unless we use the sup-norm topology,in which case the dual is very large and includes dual vectors that do not support state prices).However,with some definition of arbitrage in limits,the economic content of the Fundamental Theorem can be maintained. 13
(The motivation this construction is observation that the existence of the consistent linear pricing rule with state prices p implies that all feasible consumptionssatisfy ✁ 1p✞ ✂ C ✂ W.) ✁ iii✂ ✁ ✁ i ✂ : This part is obvious, since any candidate optimum is dominated by adding the arbitrage, and therefore there can be no arbitrage if there is an optimum. More formally, adding an arbitrage implies the change of consumption ∆C ✂ 0, which implies an increase in U ✁ C ✂ . One feature of the proof that may seem strange is the degeneracy (linearity) of the utility function whose existence is constructed. This was all that was needed for this proof, but it could also be constructed to be strictly concave, additively separable over time, and of the von Neumann-Morgenstern class for given probabilities. Assuming any of these restrictions on the class would make some parts of the theorem weaker ( ✁ iii✂ implies ✁ i ✂ and ✁ ii ✂ ) at the same time that it makes other parts stronger ( ✁ i ✂ or ✁ ii ✂ implies ✁ iii ✂ ). The point is that the theorem is still true if (iii) is replaced by a much more restrictive class that imposes on U any or all of strict concavity, some order of differentiability, additive separability over time, and a von Neumann-Morgenstern form with or without specifying the probabilities in advance. All of these classes are restrictive enough to rule out arbitrage, and general enough to contain a utility function that admits an optimum when there is no arbitrage. The statement and proof of the theorem is a little more subtle if the state space is infinite-dimensional. The separation theorem is topological in nature, so we must restrict our attention to a topologically relevant subset of the nonnegative random variables. Also, we may lose the separating hyperplane theorem because the interior of the positive orthant is empty in most of these spaces (unless we use the sup-norm topology, in which case the dual is very large and includes dual vectors that do not support state prices). However, with some definition of arbitrage in limits, the economic content of the Fundamental Theorem can be maintained. 13
Pricing Rule Representation Theorem Depending on the context,there are different useful ways of representing the pric- ing rule.For some abstract applications(like proving put-call parity),it is easiest to use a general abstract representation as a linear operator L(c)such thatc)0 L(c))0.For asset pricing applications,it is often useful to use either the the state-price representation we used in the Fundamental Theorem,L(c)=Pc. or risk-neutral probabilities,L(c)=(1+r*)→E*'cal=(l+r*)→∑oπ"co.The intuition behind the risk-neutral representation(or martingale representation)is that the price is the expected discounted value computed using a shadow risk-free rate (equal to the actual risk-free rate if there is one)and artificial risk-neutral probabilities n*that assign positive probability to the same states as do the true probabilities.Risk-neutral pricing says that all investments are fair gambles once we have adjusted for time preference by discounting and for risk preference by adjusting the probabilities.The final representation using the state-price density (or stochastic discount factor)p to write L(c)=E poco=o tPoCo.The state price density simplifies first-order conditions of portfolio choice problems because the state-price density measures priced scarcity of consumption.The state-price density is also handy for continuous-state models in which individual states have zero state probabilities and state prices but there exists a well-defined positive ratio of the two. Theorem 2 Pricing Rule Representation Theorem The consistent positive lin- ear pricing rule can be represented equivalently using (i)an abstract linear functional L(c)that is positive:(c)0)(L(c))0) (ii)positive state prices p))0:L(c)Poco (ii)）positive risk-neutral probabilitiesπ*））/0 summing to1 with associated shadow risk-free rater*:L(c)=(1+r*)→E*'co]月(1+r*)→∑orca (iw)positive state-price densities p)）O:L(c）=Epd≡∑oToPoCo- PROOF (ii):This is the known form of a linear operator in 9 5The reason for calling the term"martingale representation"is that using the risk-neutral prob- abilities makes the discounted price process a martingale,which is a stochastic process that does not increase or decrease on average. 14
Pricing Rule Representation Theorem Depending on the context, there are different useful ways of representing the pricing rule. For some abstract applications (like proving put-call parity), it is easiest to use a general abstract representation as a linear operator L ✁ c✂ such that c ✂ 0 ✁ L ✁ c✂ ✂ 0. For asset pricing applications, it is often useful to use either the the state-price representation we used in the Fundamental Theorem, L ✁ c ✂ ✂ ∑ω pωcω, or risk-neutral probabilities, L ✁ c ✂ ✂ ✁ 1 ✄ r ✂ ✁1E cω ✁ ✂ ✁ 1 ✄ r ✂ ✁1 ∑ω πω cω. The intuition behind the risk-neutral representation (or martingale representation5 ) is that the price is the expected discounted value computed using a shadow risk-free rate (equal to the actual risk-free rate if there is one) and artificial risk-neutral probabilities π that assign positive probability to the same states as do the true probabilities. Risk-neutral pricing says that all investments are fair gambles once we have adjusted for time preference by discounting and for risk preference by adjusting the probabilities. The final representation using the state-price density (or stochastic discount factor) ρ to write L ✁ c ✂ ✂ E ρωcω ✁ ✂ ∑ω πωρωcω. The state price density simplifies first-order conditions of portfolio choice problems because the state-price density measures priced scarcity of consumption. The state-price density is also handy for continuous-state models in which individual states have zero state probabilities and state prices but there exists a well-defined positive ratio of the two. Theorem 2 Pricing Rule Representation Theorem The consistent positive linear pricing rule can be represented equivalently using ✁ i✂ an abstract linear functional L ✁ c ✂ that is positive: ✁ c ✂ 0 ✂ ✁ ✁ L ✁ c✂ ✂ 0✂ ✁ ii ✂ positive state prices p ✂✂ 0: L ✁ c✂ ✂ ∑ Ω ω☎1 pωcω ✁ iii✂ positive risk-neutral probabilities π ✂✂ 0 summing to 1 with associated shadow risk-free rate r : L ✁ c ✂ ✂ ✁ 1 ✄ r ✂ ✁1E cω ✁ ✁ 1 ✄ r ✂ ✁1 ∑ω πω cω ✁ iv ✂ positive state-price densities ρ ✂✂ 0: L ✁ c✂ ✂ E ρc✁ ∑ω πωρωcω. PROOF ✁ i ✂ ✁ ✁ ii ✂ : This is the known form of a linear operator in ℜΩ. 5The reason for calling the term “martingale representation” is that using the risk-neutral probabilities makes the discounted price process a martingale, which is a stochastic process that does not increase or decrease on average. 14