ࡱ>   |~#` abjbjmm 8&@@@@@@@T66686:T12d;S"*S*S*S `BM`i`<>>>>>>$chb@\a^ `\a\ab@@*S*S kkk\a@*S@*S<k\a<kkb$@@*SX; `Z6 elV<01^]i]]@,y`"`k``y`y`y`bbkXy`y`y`1\a\a\a\aTTTd26TTT6TTT@@@@@@ Lecture 2: Pricing by Arbitrage Readings: Ingersoll Chapter 2 Dybvig & Ross Arbitrage, New Palgrave entry Ross A Simple Approach to the Valuation of Risky Streams, Journal of Business, 1978 Here we will take a first look at a financial market using a simple state space model. We first develop some structure then examine the implications of the absence of arbitrage. Often in finance problems, uncertainty is characterized by the use of a set of random variables with a particular joint distribution, perhaps something like  EMBED Equation.3 ~ N((, (). Here, we characterize uncertainty by considering a state space tableau of payoffs on the primitive assets. We assume that there are a finite number of states of nature and that each security has its payoffs written explicitly as a function of the realized state of nature. We index states by s = 1, 2, , S (not a problem for S = ( but intuition can be lost as we look at this for the first time) and assets by i = 1, 2, , N. The 2-date investment problem can be characterized by the tableau of per share dollar payoffs on the N assets in each of the S states at date 2 (Y) and a set of current prices (v). Y a"  EMBED Equation.3  S states and N assets ( S N matrix We want to impose some structure on Y right off. In the investment decision, the agent can make choices only over outcomes (states) which can be distinguished by different patterns of payoffs on the marketed assets. Thus, for the investment decision, if there are states with identical payoffs on all of the assets, then we cannot distinguish between the two so we can collapse them into a single state (that is, the payoff matrix should not have 2 identical rows). Example:  EMBED Equation.3  (  EMBED Equation.3  but,  EMBED Equation.3  and  EMBED Equation.3  are both fine from this perspective. To complete the description of the technology or opportunities of the model, we use the vector v =  EMBED Equation.3  to represent the current price per share of each asset. The decision makers choice variable is a portfolio (an N 1 vector) n where ni is the number of shares of asset i held in the portfolio. The final payoff on the portfolio n is an (S 1) vector:  EMBED Equation.3 = Yn that gives the dollar payoff in each state for the chosen portfolio. The price of this portfolio is n2 v, so the budget constraint the decision maker is faced with can be written n2 v = W0 where W0 is the initial (after date 1 consumption) investable wealth. Thus, we might think of solving the following economic problem:  EMBED Equation.3  E[u( EMBED Equation.3 )] subject to  EMBED Equation.3  = Yn v2 n = W0 n  EMBED Equation.3  or,  EMBED Equation.3   EMBED Equation.3  subject to  EMBED Equation.3   EMBED Equation.3  = W0 and n  EMBED Equation.3 . where, for now, we will assume only that u() is increasing. Y is the technological restriction in this model and we are interested in the space of payoffs spanned by the columns of Y, i.e. what different returns patterns can be generated by trading in the marketed securities (n is a vector in EMBED Equation.3 ) this, the budget constraint, and any other restrictions on n (short sales restrictions etc.) define the opportunity set for the agent. It will often be convenient to describe a market by a returns tableau rather than a payoff tableau plus initial prices. Z a"  EMBED Equation.3  S states and N assets ( again an S N matrix where [zsi] = [ysi][diag(vi)]-1, or zsi = ysi / vi. Thus, Z is a matrix of gross returns. Z can also be thought of as a payoff tableau where the number of shares of each asset is adjusted so that all current prices per share (vi) are $1. Note: The construction of Z (from Y) requires only that all initial prices are not zero (i.e. we are not modeling a market of futures contracts). Zero prices on some assets are not a problem as long as there is at least one asset whose current price is non-zero. If so we can construct a new set of assets by adding the payoff of this asset to that of all of the others. Since we are ultimately interested in the set of returns possibilities spanned by the columns of Y or Z and we do not change this by forming a new basis for this space, we will usually assume that it is not the case that all prices are zero and that any assets with a zero price have been transformed as just described. This is not a big stretch since in most applications the primitive securities have limited liability which implies a non-zero price as we will see. Negative prices are not a problem, but, conceptually, is this really an asset? Portfolios in Returns Space: We consider the vector ( where i = nivi is the dollar amount committed to each asset i. Thus, 12 ( = Wo is the budget constraint and Z( is the vector of dollar payoffs on the portfolio (, the equivalent of Yn. We often normalize ( by initial wealth (W0) and consider wi = (nivi)/W0 so the vector w is a vector of portfolio weights. The budget constraint is then written 12 w =  EMBED Equation.3  = 1 and Zw is the vector of gross returns (per dollar invested) on the portfolio w (often written as Zw). Simple Numerical Example: Suppose Y =  EMBED Equation.3  and v =  EMBED Equation.3 . Then we can write Z =  EMBED Equation.3  (this is a very special case). Consider a commitment of W0 = $5 divided as  EMBED Equation.3  =  EMBED Equation.3  (not meant to be in any way optimal). Then the payoff is Z EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  this is return per dollar times the number of dollars invested or simply the dollar payoff. This could also be written in terms of Y and (since (i = nivi) n =  EMBED Equation.3  (one share of asset 1 costs $2 and 3 shares of asset 2 costs $3) so that the dollar payoff is again given as Yn =  EMBED Equation.3 . As portfolio weights, this is w =  EMBED Equation.3  with gross returns per dollar invested of Zw =  EMBED Equation.3 , then you need to know the number of dollars invested ($5) to determine the dollar payoff. An arbitrage portfolio  EMBED Equation.3  ( EMBED Equation.3  in the text) is a nontrivial vector of dollar commitments that sum to zero. That is, an arbitrage or zero investment portfolio is a vector  EMBED Equation.3  where  EMBED Equation.3   EMBED Equation.3  0 with 12  EMBED Equation.3  = 0 =  EMBED Equation.3 . We distinguish this from a positive investment portfolio, . If all prices, all vi, are positive, as is usual, then it must be that  EMBED Equation.3 j < 0 for some (at least one) asset j as the long positions in  EMBED Equation.3  are financed with short positions in other assets. No normalization is done for an arbitrage portfolio, and so we talk of  EMBED Equation.3  = (5, -5) as being the same portfolio as  EMBED Equation.3  = (20, -20). Thus, an arbitrage portfolio is scale free. So, for any scalar  EMBED Equation.3 , if  EMBED Equation.3  is an arbitrage portfolio, then  EMBED Equation.3  EMBED Equation.3 is the same arbitrage portfolio. Dont confuse an arbitrage portfolio with an arbitrage opportunity (which we will discuss shortly). If an arbitrage opportunity exists, it is often possible to exploit it with an arbitrage portfolio and it can thus be run at an unlimited scale, its profits being limited only by the supply of assets or (more likely) price reaction to trading but they are different concepts. More on the Construction of Z: Note that the characteristics of the market will be very important for our analysis, so it pays to spend some time on its construction up front. (1) Redundant Assets Definition: Let w1 be a specific positive investment (12 w1 = 1) portfolio. We call this portfolio duplicable if there exists a distinct (w2  EMBED Equation.3  w1) portfolio w2 (12 w2 = 1) with Zw1 = Zw2. Equivalently, if there exists an arbitrage portfolio  (which must be nontrivial  `" 0, you will recall) with payoff exactly equal to zero in all states (Z EMBED Equation.3  = 0, 1' = 0), then there exists a duplicable portfolio. (Why is this equivalent?) Clearly, this is a property of the set of returns as a whole. If w1 and w2 duplicate each other, then for any portfolio w,  EMBED Equation.3  = w + w1 w2 = w +  EMBED Equation.3 ( EMBED Equation.3  w) will duplicate w. Duplicability is, therefore, usually expressed as a redundancy in the primitive assets, i.e. the columns of Z. A redundant asset is one such that if it were removed from the market there would be no change in the space of returns possibilities spanned by the marketed securities. Redundancy is formally stated as: If the  augmented returns tableau  EMBED Equation.3  = (Z2 , 1)( =  EMBED Equation.3  has rank less than N, then some assets are redundant. Loosely& we want to have N linearly independent assets (really the columns of  EMBED Equation.3 ) or else we have a redundancy. If we dont have this situation, then any asset or portfolio is duplicable, but only the assets contributing to the collinearity are redundant assets. The row of 1s in  EMBED Equation.3 is important to ensure that we consider only equivalent investment duplications of assets (otherwise we could label an arbitrage opportunity as a redundancy). Consider the following example: Z = EMBED Equation.3 . None of the assets is redundant since  EMBED Equation.3 = -1. Thus, the augmented returns tableau has full rank of N=3. Clearly, assets 1 and 2 are linearly dependent but the augmented assets 1 and 2 are not. As far as the opportunities for the investor go, both assets are very important in this economy and we would substantially alter an investors opportunity set by eliminating either asset. We will generally assume that redundant assets have been eliminated from Z meaning the augmented tableau has full column rank, N (though Z may only have column rank N-1). LeRoy and Werner consider an alternative definition: essentially imposing a restriction of the absence of arbitrage before they consider the issue of redundant assets (their definition is actually in terms of Y but that is not of consequence). Definition: The right inverse of Z is a matrix R such that  EMBED Equation.3  and the left inverse of Z is a matrix L such that  EMBED Equation.3 . If Rank(Z) = N = S, then Z is a square matrix and has an inverse. If N  EMBED Equation.3  S, then Z cant have an inverse. Z, however, may have a left (if Z has full column rank) or a right (if Z has full row rank) inverse in this case. Their definition: Lack of redundant assets  EMBED Equation.3   EMBED Equation.3  a left inverse for Z. Suppose there are no redundant assets, Rank(Z) = N, and N d" S, find L such that LZ = IN. (Z2 Z)-1(Z2 Z) = IN if (Z2 Z)-1 exists and if Rank(Z) = N then it does exist (in fact, it s iff) Then L is uniquely defined as L  EMBED Equation.3  (Z2 Z)-1Z2 . Pick any portfolio w of the marketed securities generating return Zw. Now, apply the left inverse: Zw = Zw so LZw = LZw or w = LZw (since LZ = IN) So, the portfolio that generates the returns Zw is uniquely defined by the left inverse of Z, L. Thus, there can be no redundant assets. Again this is a bit more restrictive than the definition used by Ingersoll but has a bit easier intuition. Now, further suppose S = N and Rank(Z) = S = N We know that (AB)-1 = B-1A-1 if A and B are square matrices. Now, L = (Z2 Z)-1Z2 = (Z-1Z2 -1)Z2 = Z-1(Z2 -1Z2 ) = Z-1 (the same is true of R) So, if N = S and Rank(Z) = N = S, then L = R = Z-1. Again, since they add nothing to the uncertainty spanned by the marketed returns matrix (i.e. the returns that can be generated by an investors portfolio) we usually assume that all redundant assets have been removed from Z (or Y). (2) Insurable States Definition: If you can construct a portfolio (from the assets in Z) that pays off only in one state, then that state is said to be insurable. More precisely, state s is insurable if there exists a solution s to Zs = is ( Arrow-Debreu security for state s (such as for s = 2: i2 =  EMBED Equation.3 ), and, 12 s is the cost of insurance, per dollar received, against the occurrence of state s. Theorem: A state is insurable iff the asset returns in that state are linearly independent from the asset returns in all other states (i.e. the sth row of Z is linearly independent from the other rows of Z). Said another way: zs. is not collinear with z1., z2., , zs-1., zs+1., , zS. Proof: If zs. is linearly dependent on the returns in other states, then zs. =  EMBED Equation.3  That is, zs. can be written as a linear combination of the asset returns in other states,  EMBED Equation.3  for some set of scalars . So, for every portfolio EMBED Equation.3 , we have z2 s. EMBED Equation.3  =  EMBED Equation.3  But, if state s is insurable, then for some  EMBED Equation.3 s the right hand side of the equality must be zero, term by term, and cannot sum to one as required (by the left hand side). So, if the sth row of Z is linearly dependent on the other rows of Z, that state is not insurable. If zs. is linearly independent from the other rows, then Rank(Z, is) = Rank(Z) and by the rule of rank, a solution  EMBED Equation.3 s to Z2  EMBED Equation.3 s = is exists. ( Rule of Rank: A non-homogenous system of equations  EMBED Equation.3  +  EMBED Equation.3  + & +  EMBED Equation.3  =  EMBED Equation.3  has a solution iff Rank( EMBED Equation.3 ,  EMBED Equation.3 , ,  EMBED Equation.3 ) = Rank( EMBED Equation.3 ,  EMBED Equation.3 , ,  EMBED Equation.3 ,  EMBED Equation.3 ). If all states are insurable, the market is said to be complete. This requires that all the rows of Z are mutually linearly independent. So, for our example above: Z =  EMBED Equation.3   EMBED Equation.3 1 =  EMBED Equation.3   EMBED Equation.3 2 =  EMBED Equation.3  so, the market is complete (which is one of the ways this was a very special case). What is the cost of insurance in each state? Whats going on here? What do these insurance costs mean? This illustrates basic dominance (another special aspect of this example). Connection: Suppose that all states are insurable: Rank(Z) = S. This requires N e" S (there may be redundant securities). Then Z has a right inverse = R ZR =  EMBED Equation.3  What is R? ZZ2 (ZZ2 )-1 =  EMBED Equation.3  if (ZZ2 )-1 exists, which it will iff Rank(Z) = S, or full row rank ZR =  EMBED Equation.3  where R = Z2 (ZZ2 )-1 A complete market is a special structure, not an innocuous transformation as is assuming a lack of redundant assets. We draw an explicit distinction between complete and incomplete markets. (3) Riskless Asset or Portfolio Definition: A positive investment portfolio w with the same return in every state is called a riskless asset. That is Zw = R1 with 12 w = 1 (this is a very special asset). The existence of a riskless asset (or lack thereof) has a large impact on many issues in finance. We call R the gross riskless or risk free return and R-1 = rf the risk free rate (net return). Often, no riskless asset or portfolio will exist. In this case there is a shadow riskless return for the economy that will depend upon other aspects of the economy we havent highlighted so far (preferences for example). The shadow risk free return can be bounded in this context, below by the largest return that can be guaranteed for some portfolio (i.e. the best lower bound for any portfolios return) and above by the largest return that can be achieved by every portfolio (i.e. the worst maximum return on any portfolio). Or,  EMBED Equation.3 = maxw [mins(Zw)]  EMBED Equation.3 = minw [maxs(Zw)], these bounds are determined by dominance. Example 2 assets and 3 states: Z =  EMBED Equation.3 With 2 assets, any positive investment portfolio w = EMBED Equation.3  can be written as w = EMBED Equation.3  (simply from 12 w = 1) so it fits easily in a picture. The returns on any portfolio are given by Zw  or as a function of w1 in this 2 asset case:  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  We can graph the returns in the states {1, 2, 3} as a function of w1. Clearly, there can be no risk free portfolio with 12 w = 1 as there is no point (w1) where the three lines meet.  Zws Zw1    EMBED Equation.3    EMBED Equation.3    EMBED Equation.3   EMBED Equation.3  w1 Zw2 Zw3 At w1 < the lowest return is in state 1. For w1 > the lowest return on any portfolio is in state 3. Z1 increases in w1 and Z3 decreases in w1 so the max of this min is at w1 = and this identifies  EMBED Equation.3  = 1.50. Similarly,  EMBED Equation.3  = 2.0 at w1 = 1.0 from dominance again. If we were to introduce an asset with Z1 = Z2 = Z3 = 1.20, what would be true? (4) Dominance and Arbitrage An arbitrage opportunity is an investment strategy that guarantees a positive payoff in some contingency (date 1 wealth or at some date 2 state or states) with no possibility of a negative payoff in any contingency (i.e. a money pump). The modern study of arbitrage is the study of the implications of the absence of arbitrage opportunities in the financial market for the pricing of assets. The assumption of the absence of arbitrage opportunities is an appealing place to start for a number of reasons: It is a more primitive concept than is equilibrium. We will show that AA is a necessary but not sufficient condition for equilibrium. Only relatively few rational agents are needed to bid away arbitrage opportunities as opposed to all agents optimizing as in standard equilibria. We need only assume increasing preferences. Definition: Dominance A (positive investment) portfolio (or asset) w1 dominates portfolio w2 if Zw1 e" Zw2 (strict in at least one element). Recall that any portfolio has 12 w =1 (spreads $1 around). So, the initial price of the two positions is the same. Such a circumstance would be an example of an arbitrage opportunity. It is a special example in that it requires dominance of one asset over another state by state. It is clear that no investor preferring more to less would ever hold a positive amount of a dominated asset (any investor with strictly increasing preferences would hold instead the dominating asset). It is further true that no non-satiated investor can find an internal optimal portfolio (a finite optimum) if w1 and w2 are both available, even if neither would be held by the investor. Proof: Suppose w1 dominates w2 and w is any portfolio. Define X EMBED Equation.3  Z(w1 - w2)  EMBED Equation.3  0 (by dominance). Any portfolio w is dominated by w +  EMBED Equation.3 (w1 - w2)  EMBED Equation.3   EMBED Equation.3  > 0. Note that 12 ( w +  EMBED Equation.3  (w1 - w2)) = 1, so this is a feasible portfolio, and that Z( w +  EMBED Equation.3  (w1 - w2)) = Zw +  EMBED Equation.3 X e" Zw. So, w +  EMBED Equation.3  (w1 - w2) is a portfolio that dominates w. Furthermore, the portfolio return is increasing in  EMBED Equation.3  so because it is costless to do so and it increases the portfolio return in at least one state, every investor with increasing preferences seeks to always increase the scale of his position in (w1 w2). Thus, no investor has an internal optimal portfolio. Alternatively, dominated assets cant exist price pressure will quickly eliminate the dominance. This illustrates the idea of an arbitrage opportunity.  EMBED Equation.3 = (w1 w2) is an arbitrage portfolio that exploits the opportunity. It also illustrates why AA is necessary for equilibrium with non-satiated agents; there can be no equilibrium if no agent has a finite optimum. From our example above, Z =  EMBED Equation.3  Asset one dominates asset two; this lies behind the negative insurance price. In this example, for any portfolio  =  EMBED Equation.3   EMBED Equation.3  represents an arbitrage opportunity (and a special kind). Dominance is a special form of arbitrage, others are easily defined. Definition: Riskless Arbitrage  not only arbitrage, but a certain payoff A riskless arbitrage opportunity is any  EMBED Equation.3  such that 12  EMBED Equation.3  d" 0 and Z  EMBED Equation.3  =  EMBED Equation.3 , then  EMBED Equation.3  guarantees a riskless payoff when no (or possibly negative) investment is required. The existence of such an  EMBED Equation.3  does not imply the existence of a riskless asset: e.g. let Z =  EMBED Equation.3  then  EMBED Equation.3  =  EMBED Equation.3  is a riskless opportunity: Z EMBED Equation.3  =  EMBED Equation.3  and 12  EMBED Equation.3  = 0. But, for any w with 12 w = 1& Zw =  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  for any w1 or R so no positive investment portfolio with a riskless payoff exists. If a riskless asset is available in an economy with a riskless arbitrage opportunity, it is possible to create a risk free asset with any level of return. Proof: If w is a riskless asset, the 12 w = 1 and Zw = R1 Let  EMBED Equation.3  be a risk free arbitrage 12  EMBED Equation.3  = 0 and Z EMBED Equation.3  = k1 Then w +  EMBED Equation.3  EMBED Equation.3  is a positive investment portfolio (12 ( w +  EMBED Equation.3  EMBED Equation.3 ) = 1) with return Z(w +  EMBED Equation.3  EMBED Equation.3 ) = (R+  EMBED Equation.3 k)1 and a judicious choice of  EMBED Equation.3  gets you any desired level of riskless return Conversely, if there exists a riskless asset a riskless arbitrage opportunity exists whenever there exists a solution to 12 w = 1 and Zw = k1 where k ( R More generally: Definition: 1st Type (arbitrage in LeRoy &Werner) An arbitrage opportunity of the 1st type is defined as any  EMBED Equation.3  such that 12  EMBED Equation.3   EMBED Equation.3  0 and Z EMBED Equation.3   EMBED Equation.3  0. ( This says that no (or possibly negative) investment buys a limited liability payoff that is strictly positive in at least one state at the future date. Definition: 2nd Type (strong arbitrage in LeRoy & Werner) An arbitrage opportunity of the 2nd type is defined as any  such that 12  EMBED Equation.3  < 0 and Z EMBED Equation.3   EMBED Equation.3  0. ( This says that you can generate money now and have at worst a limited liability payoff at the future date. Exercise  In equation (33) in Ingersoll, there is an arbitrage opportunity of the second type. Describe how to implement it. The existence of arbitrage opportunities (of the 1st and/or 2nd types) can be succinctly stated as: arbitrage opportunities exist if there exists some  (or some n), for which the following is true:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  or,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  When there is no such  EMBED Equation.3  (or n) then there are said to be no arbitrage opportunities (of the 1st or 2nd types) in the market represented by Z (or Y) ( an Absence of Arbitrage. __ Definition: A supporting state price vector (or linear pricing rule): An S1 vector p (more generally, a pricing function p( )) is said to support the market Z if Z2 p = 1 (or Y2 p = v). That is, the vector p  prices all assets correctly in that it relates future payoffs to current prices (Z and 1 or Y and v). Example: Z =  EMBED Equation.3  think of the elements of Z as a $ payoff per dollar invested Then Z2 p = 1  EMBED Equation.3  2p1 + 2p2 + p3 = 1 p1 + 3p2 + 2p3 = 1 (when you solve this system of equations you will find)  EMBED Equation.3  p1 = + p3 p2 = - p3 p3 = p3 is a set of supporting prices as a function of p3 By construction, if we multiply either of the primitive securities in Z, or any portfolio of these securities, by the vector p we will get the current price of the security/portfolio, 1. Linearity of the pricing rule: is the same as the lack of monopoly power in the financial market. (1) The cost of money in state r is independent of how much is purchased in state s so there are no economies of scope (although the possible payoffs in states r and s in portfolios of the traded assets may be tied by any incompleteness in Z). (2) The cost of $2 in state s is just exactly twice the cost of $1 in state s so there are no economies of scale that exist. Our goal is to derive implications for p from the absence of arbitrage opportunities. An important implication is going to be  EMBED Equation.3  p > 0 such that Z2 p = 1. (Why?) What does p look like? What would happen if we were to multiply an Arrow-Debreu security (in Y) by the vector p? The elements of p can be seen as state (or insurance) prices. Since ps is the current cost of purchasing a dollar in state s and zero elsewhere, ps d" 0 is a clear arbitrage. Example: To more completely show the link between a positive price vector and arbitrage: Let Z =  EMBED Equation.3  Each asset s current price is $1 since we are working with Z. Z2 p =  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  Suppose p1 < 0. Then from (1) p1 =  EMBED Equation.3  so p1 < 0  EMBED Equation.3  z11 < 0 In this case, just shorting asset 1 is an arbitrage (receive $1 now) since z11 < 0 implies that a short position in asset 1 has a payoff > 0. Similarly, from equation (2): p2 =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Consider the portfolio w: w1 =  EMBED Equation.3  w2 =  EMBED Equation.3  . w1 + w2 = 1 and, Zw =  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  So, if p2 < 0 shorting the portfolio w is an arbitrage opportunity. The secret behind this exampleThis is a complete market which implies that we can form any returns pattern on the assets we would like. This makes the relation easy to see since we can then form portfolios that pay off only in one state (i.e. an Arrow-Debreu security) where the payoff in a state and the state price are explicitly related. The Law of One Price This is a weak (less restrictive) version of the absence of arbitrage, i.e., it requires less of the market. It states that perfect substitutes must have the same price, i.e., two distinct assets with identical payoffs must have the same current price. Clearly, this requires a redundancy in the primitive assets. It is a consequence of the absence of arbitrage but does not imply it. Formally: If n  EMBED Equation.3  n* and Yn = Yn* then v2 n = v2 n* or, If  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 * and Z EMBED Equation.3  = Z EMBED Equation.3 * then 12  EMBED Equation.3  = 12  EMBED Equation.3 * The law of one price is equivalent to the existence of a supporting price vector but places no restrictions on it. Consider the following problem: define  EMBED Equation.3 1 =  EMBED Equation.3    EMBED Equation.3 *, Choose (1 to Min 12  EMBED Equation.3 1 subject to Z EMBED Equation.3 1 = 0 (find the smallest difference in cost between two portfolios with identical payoffs) If the law of one price holds, clearly the solution to this problem is a minimum of zero. A finite solution to the primal problem implies that the dual program is feasible. The dual to this minimization problem is written: Maxp 02 p subject to Z2 p = 1 (where p is the vector of Lagrange multipliers from the primal problem) Because the primal problem has a finite solution this dual problem is feasible, i.e. there is a finite p that solves the maximization problem. Thus, if the law of one price holds, some p exists with Z2 p = 1, i.e. there exists a supporting price vector. The constraint in the primal problem is an equality constraint so p is unconstrained in the dual. Now if we consider the existence of a supporting state price vector the maximization problem is feasible and its objective must be zero. The theorem of duality implies then that the minimization problem has an objective of zero so the law of one price holds. Alternatively, if  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 * and Z EMBED Equation.3  = Z EMBED Equation.3 * but 12  EMBED Equation.3  `" 12  EMBED Equation.3 * then no vector p can exist such that p'Z = 1' and p'Z* = 1'* (which are not equal). See also Theorem 2 in Ingersoll. Considering an absence of arbitrage we get The Fundamental Theorem of Asset Pricing. Theorem: The following are equivalent: The absence of arbitrage. The existence of a strictly positive supporting price vector. The existence of an internal optimum (portfolio) for some agent with strictly increasing preferences. Proof: This version of the proof uses Stiemkes Lemma or the Theorem of the alternative: Let A be a matrix in  EMBED Equation.3 . Then one and only one of the following is true There exists an x  EMBED Equation.3   EMBED Equation.3  s.t. 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hW5hWhWhWH*jhC`UjˤhC`hC`EHU/p.:<VXZ\bdrĬƬ*@HJLPȮܮݮ  !ǿǰǔxkjhC`hWGEHUjQ}xC hWGCJUVaJjhC`hWGEHUj?}xC hWGCJUVaJjhC`hWGEHUjI}xC hWGCJUVaJjhWGUhWGhReh: hhmi' hW5h%~hW5huhW hWhW hg&X5hg&X*!'()<=>?BCDWXYZ`*,.068:<bdfhjlİưȰʰ̰԰ְذܰް hWG5H* hWGhWGhPEjqhC`hWGEHUj|hC`hWGEHUhb(hWG5jhC`hWGEHUhRUhWG5jhC`hWGEHUjQ}xC hWGCJUVaJjhWGU hWG5hWG60^`bz~ Ͳ+RScdwxyzҳӳ㼴zljHC hOhmUVjfhOh=9EHUjE h=9CJUVaJjhOhmUhlhm5hl hOhmh1Dhm6h1Dh9 hm>* hm5>* hm56h9hm6h"khmh)}hWhPEhRe hWG5hWG(ӳԳճֳ׳ppppppp@pBpDpFpHpJppprptpvppppppෲzmiaiihlhLf5hLfjhOh=9EHUjE h=9CJUVaJjhOhmEHUjeHC hOhmUVhlh]5h]U hm5 hl5hlhm5j~hOh=9EHUjE h=9CJUVaJ hOhmjhOhmUjhOhmEHU$exists an n  EMBED Equation.3   EMBED Equation.3  with A2 n  EMBED Equation.3  0.  (This is based on a separating hyperplane argument if you are familiar with them from your economics classes.) Let  EMBED Equation.3  =  EMBED Equation.3  so M = S+1 and N = N Suppose (a) is true. Then using Stiemkes Lemma,  EMBED Equation.3  x  EMBED Equation.3  EMBED Equation.3  such that Ax = 0. Write this as:  EMBED Equation.3   EMBED Equation.3  = 0 Now explicitly write out the multiplication of  EMBED Equation.3 x 1st row col xs  EMBED Equation.3  = 0 2nd row  EMBED Equation.3  = 0  EMBED Equation.3  Nth row  EMBED Equation.3  = 0 Now we know x  EMBED Equation.3   EMBED Equation.3  so all the  EMBED Equation.3  are strictly positive, in particular  EMBED Equation.3  > 0. Divide both sides of each equation by  EMBED Equation.3  and rearrange  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  Now define  EMBED Equation.3  = ps > 0  EMBED Equation.3  s  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  or, Y2 p = v with p > 0 So, (a) is equivalent to  EMBED Equation.3  p > 0 s.t. Y2 p = v and the alternative (b) cannot hold. Now, suppose that (b) holds. Recall, we let A =  EMBED Equation.3 . (b) says  EMBED Equation.3  n  EMBED Equation.3  s.t. A2 n e" 0. A2 =  EMBED Equation.3  So, there exists an n such that A2 n  EMBED Equation.3  0 or  EMBED Equation.3 ( 0 This implies that Yn ( 0 and v2 n ( 0, where at least one of these inequalities is strict. This is just the general definition of the existence of arbitrage opportunities of the first and/or second types. Therefore, we have shown that either (a) holds which is the existence of a strictly positive supporting price vector and (b), the existence of arbitrage opportunities, does not or (b) holds which is the presence of arbitrage and (a) does not proving the equivalence of the absence of arbitrage and the existence of a strictly positive price vector. Now, part (3) of the Theorem ( part (1) (or not 1 ( not 3): An investors maximization problem can be written:  EMBED Equation.3 . So, maximize expected utility of current consumption and date 2 wealth. This allows for state dependent utility using us(,) for generality; requiring only increasing preferences so  EMBED Equation.3 s us(,) increases in all arguments (current spending and future wealth in all states). If there is an arbitrage opportunity n*, then the investors problem cannot have a finite optimum if preferences are increasing since for any n:  EMBED Equation.3  is strictly increasing in k. With an arbitrage opportunity the investor can increase current consumption or future wealth, in at least one future state, without sacrificing current consumption or wealth in other contingencies (other states) and will thus desire to do so without bound. Now, (1) ( (3): If there is no arbitrage then there exists a positive supporting price vector p. Let W0 = 0. Consider us(co, c1s) = -exp[-(W0  v(n)] -  EMBED Equation.3 exp[-(Yn)s]. Each us( ) is strictly increasing and strictly concave, infinitely differentiable and additively separable. Using the fact that p is strictly positive and the relation v2 = p2 Y you can show (and you should) that with this utility function the FOCs for a maximum (which are necessary and sufficient by concavity) are satisfied at n = 0, therefore this investor would find an internal optimum. More intuitivelythe absence of arbitrage means that consumption at date 1 or in any state at date 2 has a positive price, i.e. it can only be increased at the expense of consumption elsewhere, either at t=1 or in some t=2 state. So, pick any strictly concave strictly increasing utility function with u2 (-") = " and u2 (") = 0 and standard convex programming arguments show that this function will have an interior optimum due to the tradeoffs implied by the positive state prices. Example: Let Z =  EMBED Equation.3 . Thus, our set of assets is a simple gamble. Clearly no arbitrage opportunities exist. What is the supporting price vector? Z2 p = 1 ( p1 + 2p2 = 1 or p1 = 1  2p2 For any p2 with 0 < p2 < these define strictly positive supporting state prices. p1 =  EMBED Equation.3  and p2 =  EMBED Equation.3  both support this market. It is also clear that p3 =  EMBED Equation.3  also supports this market. What does this mean? The result doesnt say that there are no non-positive supporting price vectors, only that the existence of one strictly positive price vector is equivalent to the absence of arbitrage. In general, the supporting state prices are not unique. Z is an SN matrix, Z2 p = 1 places only (if N < S) N restrictions on S prices leaving S  N degrees of freedom and we will commonly have at least one non-positive vector p that satisfies this equation, even if there is an Absence of Arbitrage. If state s is insurable, then the element ps of p is unique across all supporting vectors p. Recall that a state being insurable means:  EMBED Equation.3  Thus, for all supporting vectors p (not necessarily positive)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  indicating that ps is the price of insurance against state s occurring. This is true for all supporting vectors p and if the law of one price holds (and it does since there is a supporting price vector) then  EMBED Equation.3  is the same for all p such that  EMBED Equation.3  and all  EMBED Equation.3  such that  EMBED Equation.3 so that ps is unique. If Z has full row rank, i.e. all rows of Z are linearly independent so all states are insurable (i.e. the market is complete) then p itself is unique (requires N e" S). Full row rank implies there exists a unique right inverse, R, for Z so p is unique: p2 Z = 12  EMBED Equation.3  p2 Z(Z2 (ZZ2 )-1) = 12 (Z2 (ZZ2 )-1) p2 I = p2 = 12 (Z2 (ZZ2 )-1) = 12 R which is uniquely determined by R. Riskless Asset: If there exists a riskless asset, the sum of all state prices must equal:  EMBED Equation.3  for all supporting price vectors p. Assume there exists a wR such that 12 wR = 1 and ZwR = R1. Then, p2 (ZwR) = (p2 Z)wR = 12 wR = 1 and, p2 (ZwR) = p2 (R1) = R(p2 1) = R ( EMBED Equation.3 ) so, 1 = R ( EMBED Equation.3 )  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  p s.t. Z2 p = 1 Example: re-examined Let Z =  EMBED Equation.3  From Z2 p = 1 we found  EMBED Equation.3  and  EMBED Equation.3  and  EMBED Equation.3  For p > 0 we require 0 < p3 < 1/3. Look at the  edges of this range for p3& For p3  near zero p3  EMBED Equation.3  0 p1  EMBED Equation.3  1/4 p2  EMBED Equation.3  . And,  EMBED Equation.3  =  EMBED Equation.3  R = 2 For p3 near 1/3 p3  EMBED Equation.3  1/3 p1  EMBED Equation.3  1/3 p2  EMBED Equation.3  0. And,  EMBED Equation.3  = 2/3  EMBED Equation.3  R = 1.5 Recall that earlier using this example we found no riskless asset but bounds on the shadow riskless return given by:  EMBED Equation.3  = 2 and  EMBED Equation.3  = 1.5. Thus, these bounds are  tight : in the sense that we can find a vector p with Z2 p = 1 such that  EMBED Equation.3  = R for any R in the interval  EMBED Equation.3  < R <  EMBED Equation.3 . This is generally true, here it occurs because  EMBED Equation.3  = + p3 is continuous and monotonic in p3. Representation Theorem: The following are equivalent: The existence of a positive linear pricing rule. The existence of positive risk neutral probabilities and an associated riskless rate. The existence of a positive state price density. (1) Linear Pricing Rule, our basic representation Any asset or portfolio is correctly priced by p: Y2 p = v or Z2 p = 1 So, for any portfolio n or w& p2 Yn = v2 n p2 Zw = 12 w = 1 or,  EMBED Equation.3   EMBED Equation.3  (2) Risk Neutral Probabilities (indicated via the *) The current price of any asset or portfolio is given by the expected payoff under the risk neutral probabilities discounted by the associated risk free rate. So, v2 n =  EMBED Equation.3   EMBED Equation.3  or,  EMBED Equation.3   EMBED Equation.3  To show the equivalence between (1) and (2) simply set  EMBED Equation.3  and R* =  EMBED Equation.3  or  EMBED Equation.3 . Clearly, if the economy has a riskless asset then R* = R for all valid p. Here, as in the proof of existence of the state prices, we simply require that all traders believe that the same set of states are possible. For equivalence we require that the same states have positive probability under both measures. Every investor agrees on the set of valid ps (if all believe the same set of states are possible) so all will necessarily agree on the set of valid risk neutral probabilities. Thus, all investors price assets the same under both approaches. The use of risk neutral probabilities can also be thought of as developing a market based certainty equivalent measure for any risky asset. Since it is a certainty equivalent, the proper discount rate is the associated risk free rate. 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There, v =  EMBED Equation.3 . Here, we explicitly recognize the state dependence of the cash flows and discounting each at its appropriate state dependent rate eliminates the need to risk adjust our discount rate. The size of the cash flows state-by-state does this for us as opposed to considering only expected cash flow and a risk adjusted discount rate that applies to the expectation. This suggests that the relation between state contingent payoffs for an asset, state probabilities, and state prices will be important will be important in the risk adjusting of a discount rate. (3) State Price Density The current price of any asset or portfolio is given by the expectation of the product of the state price density (() and the assets payoff. v2 n = E((Yn) 1 = E((Zw) (Note no * s)  EMBED Equation.3   EMBED Equation.3  Equivalence follows from defining  EMBED Equation.3  or  EMBED Equation.3  Note: E(() = 1/R if there exists a riskless asset or 1/R* if not. Clearly, ( is positive for all states s iff p is positive. This representation is most valuable when we move to a continuum of states since p(s) and  EMBED Equation.3 (s) may be zero for all states s yet (s) may be well defined. Note: we can write 1 = E( EMBED Equation.3 Zw) as: 1 = E( EMBED Equation.3 )E(Zw) + cov( EMBED Equation.3 , Zw) or, E(Zw) =  EMBED Equation.3  = R Rcov( EMBED Equation.3 , Zw) (assume there exists a riskless asset) So, if cov( EMBED Equation.3 , Zw) = 0, the asset has no risk premium. In other words, the same message we see in other asset pricing frameworks is illustrated here: (1) some risk is not priced, (2) the expected return on risky assets is the risk free return plus a premium, and (3) marginal risk is determined by covariances. Note also that the correlation between the state price density (state prices and probabilities) and payoffs appears as was suggested above. Idiosyncratic Risk Illustration: Suppose payoffs can be written as: a + f( EMBED Equation.3  where E[f()] =  EMBED Equation.3  = 0 (all expectations are reflected in a). Then, the price of any asset is given by:  EMBED Equation.3  If, under the risk neutral probabilities, the expectation of  EMBED Equation.3  is also zero (E*[ EMBED Equation.3 ] = 0) then the risk or variability of this component of returns does not affect this assets price, i.e. the risk of  EMBED Equation.3  does not imply a differential expected return. Looked at another way (using the state price density):  EMBED Equation.3  (Assuming there exists a riskless asset.) Consider E[] =  EMBED Equation.3  =  EMBED Equation.3  = R-1 EMBED Equation.3  = R-1 EMBED Equation.3  = R-1 E*[ EMBED Equation.3 ] = 0 iff E*[ EMBED Equation.3 ] = 0 So, if E*[ EMBED Equation.3 ] = 0 then E[ EMBns)0=Sv?@YZ\ *,.0NPTV|~ƾƶƲjؐhGhGEHUjЖC hGCJUVaJjhGUhGhG5 hpShGhGh:hG5 jlh:5hL3(hRUhthl7hh0h.hpShA4jhA4Uj[hA4hA4EHU2PRh01vwxgdX & F&gdGgdg&Xgd~gdA4&(*,8:`bdfhz| @BDJLN µؕ|xph`hhG5h0hG6h:hG6h: jlh:5 hg&XH* jlhg&X5hg&XhI@j~hGhGEHUjC hGCJUVaJjhGhGEHUjC hGCJUVaJ hA4hA4jhGUjhGhGEHUj,C hGCJUVaJhG" "HJLN!'(;<=>ACKL_`abdefsuyzӚӅxӦӅpjhGUjhLfhLfEHUhpShG5jhLfhLfEHUh0h:hG5 hG5jhLfhLfEHUj`xC hLfCJUVaJhReh:hG6hGjɚhLfhLfEHUjxC hLfCJUVaJhLfjhLfU) /0p~ǸӣӟӟӛӗǸ˗}y}uӗqh)}hl7hu h0h hX5jhLfhLfEHUhXh.hg&Xh:hG5jhLfhLfEHUj`xC hLfCJUVaJhLfjhLfUhGhpSjhGUjhGhXEHUjC hXCJUVaJ, !"#$56IJKLabfp$%8uqiehLfjhLfUh+$jhFShFSEHUjC hFSCJUVaJjƭhFShFSEHUjtC hFSCJUVaJhZDjhFShFSEHUj?C hFSCJUVaJjhFShFSEHUjՙC hFSCJUVaJjhFSUhFShX hX5>*%89:;<=>?|-.23FGHIMUstuv8״ףןחwӓӓדsksjhFUhFj¶hT hFEHUj>C hFCJUVaJhT jhT Uh:hc:{jhFShFSEHUjC hFSCJUVaJjhFSUhl7h0hZDhFSjhLfUjhLfh{EHUjxC h{CJUVaJ)8:<>HJprtv6Ȼ׳׳wha׳YUhLfjhLfU hFhFjhFh0EHH*Uj,5E h0CJUVaJjZhFhFEHH*UjC hFCJUVaJ hFH*jhFH*UhFhFH*jۻhFhFEHUjC hFCJUVaJhFjhFUjhFhFEHUjΛC hFCJUVaJ68:<JPXZ&'()/0CDEFIJ]^˯ˠwjjhLfhLfEHUjxC hLfCJUVaJj&hLfhLfEHUj2hLfh{xEHUjxC h{xCJUVaJUj?hLfhLfEHUjLhLfhLfEHUhLfh0hF6hFjhLfUjYhLfhLfEHUjxC hLfCJUVaJ'ED Equation.3  EMBED Equation.3 ] = E[ EMBED Equation.3 ]E[ EMBED Equation.3 ] + cov( EMBED Equation.3 ,  EMBED Equation.3 ) = 0 and so it must be that cov( EMBED Equation.3 ,  EMBED Equation.3 ) = 0 since we assumed that E[ EMBED Equation.3 ] = 0. Risk is not priced it carries no risk premium if it is uncorrelated with the state price density, (. Note Almost none of what we have said so far has had to do with actual or subjective probabilities of the states a seemingly strange omission when talking about asset pricing. The message is thisDominance and arbitrage are dependent upon possibilities not probabilities state by state comparisons (with no regard for the likelihood of each state). Though this is, in some sense, blunt, it carries us a long way.     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