1 No-arbitrage pricing

Transcription

1 BURNABY SIMON FRASER UNIVERSITY BRITISH COLUMBIA Paul Klein Office: WMC 3635 Phone: TBA paul klein URL: Economics 809 Advanced macroeconomic theory Spring 2012 Lecture 8: Finance 1 No-arbitrage pricing How far can we get in determining the correct price of assets by just assuming that there are no arbitrage opportunities? Notice that this is a weaker assumption than (competitive) equilibrium. Surprisingly far, as it turns out. In particular, we can work out how to price derivatives claims that are defined in terms of other securities, e.g. options. Arbitrage free pricing is a lot like expressing a vector as the linear combination of basis vectors. 1

2 1.1 A two period model Let t = 0, 1 (today and tomorrow). There are N securities. The price of security n at time t is denoted by St n. We write S t = S 1 t S 2 t. S N t. S 0 is deterministic, but S 1 is stochastic. We write S1(ω) 1 S 1 (ω) = S 2 1(ω). S1 N (ω) where ω Ω = {ω 1, ω 2,..., ω M }. Now define the matrix D via D = N M S1(ω 1 1 ) S1(ω 1 2 )... S1(ω 1 M ) S1(ω 2 1 ) S1 N (ω 1 ) S1 N (ω M ) [ = d 1 d 2... d M ]. Definition. A portfolio is a vector h R N. Interpretation: h n is the number of securities of type n purchased at t = 0. Remark: Fractional holdings as well as short positions (h n < 0) are allowed. 2

3 The value of a portfolio h at time t is given by V t (h) = N h n St n = h T S t. n=1 Definition. A vector h R N is called an arbitrage portfolio if V 0 (h) < 0 and V 1 (h) 0 for all ω Ω. Remark. We can weaken the condition for all ω Ω to with probability one if we want but in this context that wouldn t add much. Theorem. Let securities prices S be as above. Then there exists no arbitrage portfolio iff there exists a z R M + such that S 0 = Dz. Remark. This means that today s (period 0 s) price vector has to lie in the convex cone spanned by tomorrow s (period 1 s) possible prices vectors. (A convex cone is a subset C of a vector space X such that for any x, y C and α 0 we have (αx) C and (x + y) C. ) Proof. Absence of arbitrage opportunities means that the following system of inequalities has no solution for h. { h T S 0 < 0 h T d j 0 for each j = 1, 2,..., M. Geometric interpretation: there is no hyperplane that separates S 0 from the columns of D. Such a hyperplane would have an arbitrage portfolio as a normal 3

4 vector. Now according to Farkas lemma, the non existence of such a normal vector is equivalent to the existence of non negative numbers z 1, z 2,..., z M such that or, equivalently, where z R M +. S 0 = M z j d j j=1 S 0 = Dz We will prove Farkas lemma as a corollary of the Separating Hyperplane Theorem. Proposition (Farkas lemma). If A is a real matrix and if b R m, then m n exactly one of the following statements is true. 1. Ax = b for some x R n + 2. (y T A) R n + and y T b < 0 for some y R m. Proof. In what follows we will sometimes say that x 0 when x is a vector. This means that all the components are non negative. To prove that (1) implies that not (2), suppose that Ax = b for some x 0. Then y T Ax = y T b. But then (2) is not true. If it were, then y T A 0 and hence y T Ax 0. But then y T b 0. Hence (1) implies not (2). Next we show that not (1) implies (2). Let X be the convex cone spanned by the columns a 1, a 2,..., a n of A, i.e. { } n X = a R n : a = λ i a i ; λ i 0, i = 1, 2,..., n. i=1 Suppose there is no x 0 such that Ax = b. Then b / X. Since X is closed and convex, the Separating Hyperplane Theorem says that there is a y R m 4

5 such that y T a > y T b for all a X. Since 0 X, y T 0 > y T b and consequently y T b < 0. Now suppose (to yield a contradiction) that not y T A 0. Then there is a column in A, say a k, such that y T a k < 0. Since X is a convex cone and a k X, we have (αa k ) X for all α 0. But by the (absurd) supposition, for sufficiently large α, we have y T (αa k ) = α(y T a k ) < y T b, and this contradicts separation so the supposition cannot be true. Thus y T a i 0 for all columns a i of A, i.e. y T A 0. Separating Hyperplane Theorem. Let X R n be closed and convex, and suppose y / X. Then there is an a R and an h R n such that h T x > a > h T y for all x X. Proof. Omitted. A popular interpretation of the result S 0 = Dz is the following (but beware of over interpretation). Define where q i = z i β β = M z i. i=1 Then q can be thought of as a probability distribution on Ω and we may conclude the following. Theorem. The market S is arbitrage free iff there is a scalar β > 0 and a probability measure Q such that S 0 = βe Q [S 1 ] = β[q 1 S 1 (ω 1 ) + q 2 S 1 (ω 2 ) q M S 1 (ω M )] and we call this a martingale measure (for reasons that are not immediately obvious in this context). Economically speaking, the q i s are state prices. Relative prices of $1 in state i. Arrow Debreu prices. 5

6 Yet another approach is to define a so called pricing kernel. For an arbitrage free market S, there exists a (scalar) random variable m : Ω R such that such that S 0 = E[m S 1 ] where the expectation is now taken under the objective measure P. Provided the outcomes ω i all obtain with strictly positive probabilities, we have m(ω i ) = βq({ω i })/P ({ω i }) = βq i /p i. where the p i s are the objective probabilities. So the pricing kernel takes care both of discounting and the change of measure Pricing contingent claims Definition. A contingent claim is a mapping X : Ω R. We represent it as a vector x R M. Interpretation: the contract x entitles the owner to $x i in state ω i. Theorem. Let S be an arbitrage free market. Then there exist β, q such that if each contingent claim x is priced according to π 0 [X] = βq T x = βe Q [X] then the market consisting of all contingent claims is arbitrage free. Example. Let Ω = {ω 1, ω 2 }. Let S0 1 = b S0 2 = s S1(ω 1 1 ) = (1 + r)b S1(ω 1 2 ) = (1 + r)b S1(ω 2 1 ) = x S1(ω 2 2 ) = y where r > 0 and, without loss of generality, y x. Then [ ] (1 + r)b (1 + r)b D =. x y 6

7 Abusing the notation somewhat, define q = q 1. No arbitrage implies b = β[q(1 + r)b + (1 q)(1 + r)b] s = β[qx + (1 q)y] 0 q 1. We may conclude that β = r and s(1 + r) y q =, 1 q = x y The condition 0 q 1 then becomes That is to say, x y and 0 s(1 + r) y x y x s(1 + r). x y 1. y s(1 + r) x. This is not a surprising no arbitrage condition. In this case, it turns out that there is a unique martingale measure, so all contingent claims can be uniquely priced. Below we give a more general treatment of this phenomenon, and it turns out that this uniqueness is intimately related to the notion of market completeness Completeness and uniqueness of martingale measure Can contingent claims be prices in a unique way? One class of contingent claims certainly can: the hedgeable ones. Definition. A contingent claim X is said to be hedgeable if there is an h R N such that V 1 (h) X 7

8 i.e. for all ω Ω or for all i = 1, 2,..., M or h T S 1 (ω) = X(ω) h T d i = X(ω i ) h T D = x T i.e. hedgeability of X boils down to the corresponding x being in the row space of D. Proposition. Suppose the contingent claim X is hedgeable via h. Then the only price of X at 0 that is consistent with no arbitrage is π 0 [X] = h T S 0. Proof. Exercise. Proposition. Suppose X is a contingent claim that is hedgeable via h and that it is also hedgeable via g. Suppose also that there are no arbitrage opportunities. Then h T S 0 = g T S 0. Proof. Exercise. Thus any hedgeable contingent claim can be uniquely priced. circumstances can all contingent claims be uniquely priced? Under what Definition. hedgeable. A market S is said to be complete if all contingent claims are Proposition. A market S is complete iff rank(d) M. A necessary condition is N M. Proof. Exercise. 8

9 Meta theorem. A market S is generically arbitrage free and complete if N = M. If N > M it is generically not arbitrage free. If N < M it is incomplete. Proposition. Suppose a market is arbitrage free and complete. Then there is a unique probability measure Q such that every contingent claim is priced according to π 0 [X] = βe Q [X] where β = π[1] where 1 is the unit contingent claim that delivers $1 no matter what. Proof. π 0 [X] = h T S 0 = h T βe Q [S 1 ] = = βe Q [h T S 1 ] = βe Q [X]. Meanwhile, π 0 [1] = βe Q [1] = β. 2 Equilibrium pricing In general equilibrium, relative prices are given by marginal rates of substitution. This gives a way of determining prices even if they are not given exogenously. Lucas (1978) pioneered the pricing of assets in dynamic equilibrium. We will assume here that preferences are represented by [ ] E β t c1 α t. 1 α t=0 9

10 Now let s suppose that a single asset with stochastic return r t is available, so that the consumer s period-by-period budget constraint is a t+1 = w t + r t a t c t Suppose that the information flow is given by the filtration {F t } t=0 generated by the processes {r t } and {w t }. Applying the principles of stochastic dynamic optimization, we form the Hamiltonian: and the optimality conditions are H = β t c1 α t 1 α + λ t+1[w t + r t a t c t ] E[λ t+1 F t ]r t = λ t and β t c α t = E[λ t+1 F t ]. Apparently though it is not immediately obvious why this is relevant E[λ t+2 F t+1 ]r t+1 = λ t+1 and Thus and you may recall that β t+1 c α t+1 = E[λ t+2 F t+1 ] β t+1 c α t+1 r t+1 = λ t+1 (1) β t c α t = E[λ t+1 F t ]. Thus, taking expectations with respect to F t on each side of Equation (1), we get β t+1 E [ c α t+1 r ] t+1 F t = β t c α t 10

11 or, simplifying and rearranging, establishing that is a pricing kernel for this economy. 1 = β E[c α t+1 r t+1 F t ] c α t m t+1 = β c α t+1 r t+1 c α t 2.1 What kind of assets yield high average returns? It is tempting to conclude that, in general, if people are risk averse, then risky assets have to yield a high return to compensate for risk. That would be wrong, however. What matters is how good an asset is in providing insurance against consumption variance. An risky asset that pays more when consumption is low and less when it is high is actually better than a safe asset and will still be held even if the average return is lower than that of a safe asset. To establish this result, we will assume that consumption growth ln c t+1 ln c t is i.i.d. normally distributed with mean µ and variance σ 2. The normal distribution is convenient because the linear combination of two random variables is also normally distributed. We also need the following result. If X is normally distributed with mean µ X and variance σ 2 X, then { E[e X ] = exp µ X + 1 } 2 σ2 X. It follows that if X and Y are joint normal with means µ X and µ X, variances σ 2 X and σ2 Y and covariance σ XY then E[e X+Y ] = exp {µ X + µ Y + 12 σ2x + 12 } σ2y + σ XY. 11

12 With the assumptions we have made so far we are able to derive an explicit expression for the price of a bond a riskless asset that yields precisely 1 unit of consumption in period 1. Denote the period 0 price of a bond by q. Then [ c α ] q = E[m 1] = βe = t+1 c α t = βe[exp{ α(ln c t+1 ln c t )}] = β exp { αµ + 12 } α2 σ 2. On the other hand, the risk-free rate r is r = 1 q = β 1 exp {αµ 12 } α2 σ 2. (The risk free rate might have depended on time. But it doesn t in this case, so we omit the time subscript.) Now let s introduce an arbitrary risky asset with stochastic rate of return r t. Let s assume ln r t+1 and ln c t+1 ln c t are joint normal. Then Aiyagari (1993) claims (rightly of course) that E[r t+1 ] r = exp {αcov(ln r t+1, ln c t+1 ln c t )}. Let s try to verify this! First of all, we have the following beautiful and very general result which holds for all rates of return processes r t : E[m t+1 r t+1 ] = 1. for some non-negative stochastic process {m t } (adapted to {F t } t=0 ) With our specification of preferences, as we have seen, m t+1 = β exp{ α(ln c t+1 ln c t )} 12

13 and the equation E[m t+1 r t+1 ] = 1 becomes βe[exp{ln r t+1 α[ln c t+1 ln c t ]}] = 1 which implies { β exp µ ln rt σ2 ln r t+1 αµ + 1 } 2 α2 σ 2 αcov(ln r t+1, ln c t+1 ln c t ) = 1. This can be factorized to become β exp {µ ln rt } σ2ln exp { αµ + 12 } rt+1 σ2 exp { αcov(ln r t+1, ln c t+1 ln c t )} = 1 and the result follows. We take away from this that an asset earns a premium on average when its return is positively correlated with consumption growth or negatively correlated with future marginal utility. 2.2 The risky asset is a claim to consumption Here we will assume, in the spirit of Mehra and Prescott (1985), that the risky asset return is proportional to per-capita consumption growth, i.e. r t+1 = A ct+1 c t. for some A > 0. But then the covariance between ln r and ln c t+1 ln c t is just the variance of ln c t+1 ln c t! Plugging this into Aiyagari s formula, we get E[r t+1 ] r = exp{ασ 2 }. This is a very satisfying result, because it means that the ratio of returns depends on the variance of consumption growth and α only. 13

14 According to my calculations, based on data from the U.S. Bureau of Economic Analysis, the growth rate of annual per capita consumption in the United States has averaged about 2.0 percent between 1929 and 2005, with a standard deviation of about If we believe that r E = 1.06 and r B = 1.01 then we get α = Now if α = 125 and µ = 0.02 we can compute β from the formula r = β 1 exp {αµ 12 } α2 σ 2. The result is β So we can make the model fit the data. puzzle? Is there really an equity premium 2.3 The more general case Consider the approximate version of Aiyagari s formula: E[r] r αcov(ln r, ln c t+1 ln c t ). Now apply Cauchy-Schwarz inequality which says that, for any random variables X and Y COV(X, Y ) V(X) V(Y ) = STD(X) STD(Y ). We get (approximately at least) E[r] r αstd(ln r) STD(ln c t+1 ln c t ). 14

15 According to Aiyagari (1993), the standard deviation of the return on equity is about Thus the predicted equity premium is about α. If the equity premium is 0.05, we must have α 36. So do we really have an equity premium puzzle? Most economists believe there is. Indeed, the conventional wisdom is that none of the proposed solutions have been successful. Kocherlakota (1996) concludes that it is still a puzzle. This in spite of, for example, Constantinides (1990). Habit formation. Huggett (1993). Market incompleteness. 2.4 An infinitely-lived asset Like Mehra and Prescott (1985), I now assume that the risky asset is a claim to per-capita consumption in every period from the next one onwards. (Unlike them, I assume log-normal and i.i.d. consumption growth.) The purpose of this is to give an example of an asset whose rate of return is proportional to consumption growth. Denote the price of the risky asset by p t. Might there be a stationary equilibrium where p t = Ac t for some A > 0? If so, Ac t = βe t { c α t+1 cα t [c t+1 + Ac t+1 ] }. It follows that (since we assume c t to be known when expectations are evaluated) A = β(1 + A)E t {exp [(1 α)(ln c t+1 ln c t ]}. Hence A = β(1 + A) exp {(1 α)µ + 12 } (1 α)2 σ 2. 15

16 Rearrangement (for no obvious purpose right now) yields { 1 + A A = β 1 exp (1 α)µ 1 } 2 (1 α)2 σ 2. Given what we know already, E t [r t+1 ] = 1 + A {µ A exp + 12 } σ2. Substituting in our expression for (1 + A)/A we get { E[r t+1 ] = E t [r t+1 ] = β 1 exp αµ + α (1 12 ) } α σ 2. Summarizing our results so far, we have { E[r t+1 ] = β 1 exp { αµ + α ( α) σ 2} It follows from this that r = β 1 exp { αµ 1 2 α2 σ 2}. E[r t+1 ] r = exp{ασ 2 }. 2.5 The original Mehra-Prescott paper Mehra and Prescott (1985) begin their paper by noting the following fact. Between 1889 and 1978, the average return on equity was 7 percent per year and the average return on bonds was less than one percent. The difference (by definition) is called the equity premium. Jagannathan et al. (2000) claim that this premium has declined significantly, and that it was approximately zero But this new fact (if indeed it is a fact) was not known by Mehra and Prescott in Mehra and Prescott analyze an exchange (endowment) economy where a representative consumer maximizes [ ] E β t u(c t ) t=0 16

17 where u(c) = c1 σ 1 1 σ. Output follows an exogenous stochastic process satisfying y t+1 = x t+1 y t and since all agents are alike and there is no trade, we have c t = y t. Meanwhile, {x t } is a finite state Markov chain in discrete time. Specifically, x t {λ 1, λ 2,..., λ n } and P [x t+1 = λ j x t = λ i ] = φ ij. It follows that the distribution of x t satisfies µ t+1 = Φ T µ t. Suppose there is a unique stationary distribution π such that { π = Φ T π and for all µ 0 R n such that µ T 0 1 = 1. π T 1 = 1. lim µ t = π t Definition. A share of equity at t is a claim to {y s } s=t+1. Denote the price of this bundle of claims by p t. The price of equity satisfies the following difference equation. u (y t )p t = βe t [u (y t+1 )(p t+1 + y t+1 )]. 17

18 The presence of the conditional expectation is justified in my handout on stochastic dynamic optimization. Might there exist a pricing function p t = p(y, i) where y is current output and i is the current state of x? We can define it implicitly via p(y, i) = β n φ ij (λ j y) σ [p(λ j y, j) + λ j y]y σ. j=1 It turns out that this function not only exists but is linear in y! So there exist constants {ω 1, ω 2,..., ω n } such that p(y, i) = ω i y. A system of equations for these constants is given by n ω i = β φ ij λ 1 σ j (ω j + 1); i = 1,..., n. j=1 We now want to derive the average return on equity. We begin by defining the period return on equity when going from state i to state j via R e ij = p(λ jy, j) + λ j y p(y, i) p(y, i) The conditionally expected return on equity is n Ri e = φ ij Rij e j=1 = λ j(ω j + 1) ω i 1. and the unconditionally expected return (equal to the long run average by the LOLN) is R e = n π i Ri e. i=1 18

19 Now consider a riskless one period bond. How are we to price it, in terms of current consumption? Since it is a claim to 1 unit of consumption in each state, the formula is q(y, i) = β n j=1 With our specification of preferences, we get φ ij u (λ j y) u (y) 1. q(y, i) = β n j=1 φ ij λ σ j. Incidentally, βu (c t+1 ) u (c t ) constitutes a pricing kernel for this economy. We can now define the period return on a bond in state i via R b i = 1 q(y, i) 1 and the unconditionally expected return is R b = n π i Ri. b i=1 The equity premium is then defined as R e R b. But how are we to judge whether theory is consistent with R e = 0.07 and R b = 0.01? We calibrate! Here are the parameters we need to determine. (1) The states {λ 1, λ 2,..., λ n } and the transition probabilities φ ij ; i, j = 1, 2,..., n. (2) The preference parameters β and σ. We begin with (1). There is data on consumption growth rates and they have an average µ. 19

20 Impose symmetry so that Φ = Φ T. Set n = 2. λ 1 = 1 + µ + δ λ 2 = 1 + µ δ. [ φ 1 φ Φ = 1 φ φ ] Apparently π 1 = π 2 = 1 2. Finally, use δ and φ to match variance and autocorrelation! The (unconditional) variance of consumption growth is Meanwhile, the autocovariance is 1 2 [δ2 ] [δ2 ] = δ [φ δ2 (1 φ)δ 2 ] [φδ2 (1 φ)δ 2 ] and the autocorrelation is the autocovariance divided by the variance, i.e. 1 2 [φ 1 + φ]1 2 [φ 1 + φ] = 1 [4φ 2] = 2φ 1. 2 Mehra and Prescott (1985) find that, with annual data, and µ = δ = φ = 0.43 resulting from an autocorrelation of consumption growth equal to (2) Determining β and σ. It would, in principle, be feasible to choose β and σ so as to match R e and R b. Strangely, Mehra and Prescott (1985) do not 20

21 consider this alternative. If they did, they would find that σ must be be very large. Apparently Mehra and Prescott (1985) take as an axiom that σ < 10. So they conclude that the model cannot account for the facts. Apparently most economists are still uncomfortable with σ 1. It is not altogether clear why. But if one were to estimate β and σ in the way that I suggested, the parameter estimates would vary wildly with the sampling period. This suggests that there might be something fishy with the model. Many attempts have been made since Mehra and Prescott (1985) to alter preferences in such a way as to solve the equity puzzle. Have any of these attempts been successful and persuasive? Kocherlakota (1996) doesn t think so. Exercise 1 Suppose consumption c t satisfies ln c t+1 = ρ ln c t + ε t+1 where ε t is i.i.d. normal with mean 0 and variance σ 2. Assume the same preferences as in these lecture notes. Suppose 0 < ρ < 1. Find an expression for the period t riskless rate of return. 21

22 References Aiyagari, S. R. (1993). Explaining financial market facts: The importance of incomplete markets and transactions. Federal Reserve Bank of Minneapolis Quarterly Review 17 (1), Constantinides, G. (1990). Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98 (3). Huggett, M. (1993). The risk free rate in heterogeneous-agents, incomplete insurance economies. Journal of Economic Dynamics and Control 17 (5/6), Jagannathan, R., E. R. McGrattan, and A. Scherbina (2000). The declining U.S. equity premium. Federal Reserve Bank of Minneapolis Quarterly Review 24 (4), Kocherlakota, N. R. (1996, March). The equity premium: It s still a puzzle. Journal of Economic Literature 34 (1), Lucas, R. E. (1978, November). Asset prices in an exchange economy. Econometrica 46 (6), Mehra, R. and E. Prescott (1985). The equity premium: A puzzle. Journal of Monetary Economics 15,