Non-Equivalent Martingale Measures: An Example

Transcription

1 Non-Equivalent Martingale Measures: An Example Stephen F. LeRoy University of California, Santa Barbara April 27, 2005 The Fundamental Theorem of Finance (Dybvig and Ross [2]) states that the absence of arbitrage implies that there exists a linear functional that values all contingent claims. This functional may be represented in various forms: it may be a set of state prices, a stochastic process (the stochastic discount factor) or a probability measure (the risk-neutral measure). The Fundamental Theorem of Finance always holds without qualiþcation in Þnite settings, but not necessarily in inþnite settings. In inþnite settings the requisite linear functional may not exist even in the absence of arbitrage (Kreps [5]). If it does exist, it may not be representable by a measure. For example, Back and Pliska [1], Gilles and LeRoy [3], [4], Werner [7] and others have analyzed examples in which the pricing measure is not countably additive, contrary to the deþnition of a measure. In this paper we analyze an example in which there exists a valuation functional, and this functional is representable by a measure. However, that measure is not equivalent to the natural probability measure (in the sense that the two measures do not have the same sets of zero measure). The example is a general equilibrium representative-agent model, which has the convenient property that markets are effectively complete under very weak assumptions. In such settings the zero portfolio is always optimal. Therefore one can analyze valuation without even specifying the traded securities. To establish notation we will begin with the Þnite case, in which there is no problem deriving the equivalent martingale measure. 1 The Finite Case The representative agent maximizes TX 2 t E [ln(c t )]. (1) t=0 1

2 The agent s endowment equals 2 t,t=0, 1, 2,..., T if the state is high (H) att or at any date prior to t, and1 otherwise. Successive draws of the states (H and L) are independent and equally likely. The state at date 0, the initial date, is L. It is assumed that a Þnite number of securities is traded, that each security has apayoff that is nonzero at only a Þnite number of events, and that all portfolio strategies are Þnitely nonzero. This assumption rules out Ponzi schemes, implying (from the representative agent assumption) that the zero portfolio strategy is always optimal. Security prices can be calculated using the event prices that support the representative agent s endowment, so we now calculate these. The event prices at date t each of these is the price of one unit of consumption contingent upon a particular sequence of H s and L s up to date t is just the representative agent s marginal utility implied by adding the payoff of the corresponding Arrow security to consumption. This equals 2 3t if the state is high at t or at any date prior to t, and2 2t otherwise. Accordingly, the state price deßator ρ the ratio of event prices to probabilities is given by 2 2t if τ t, ρ t = (2) 2 t otherwise, since each partial sample path up to date t occurs with probability 2 t. Here τ is the date of the Þrst high endowment realization. Since the endowment realization after τ equals 2 t regardless of the state, it is clear that all uncertainty is resolved at date τ. The event that τ t has probability 1 2 t for any t. Note here that, because of the presence of the discount factor in the utility function (1), the state price deßator declines with t. However, it declines an order of magnitude faster when t>τthan otherwise, and this is what gives rise to the distinctive features of the example. Let r t be the gross one-period interest rate from t 1 to t. It is given by 4 if τ<t, r t = (3) 4(1 + 2 t ) 1 otherwise. The money-market account B is equal to the cumulated value of one unit of consumption invested at the one-period interest rate at date 0 and rolled over at each date. Its date-t value b t is ty 2 Q 2t τ i=1 (1 + 2 i ) 1 if τ t, b t = r i = i=1 2 Q (4) 2t t i=1 (1 + 2 i ) 1 otherwise. DeÞne M = {m t } as the product of ρ and B. From (2) and (4), M is given by 2

3 1 1/3 2/3 1/6 1/6 2/15 8/15 1/15 1/15 8/135 64/135 Figure 1: m t ρ t b t = Q τ i=1 (1 + 2 i ) 1 if τ t, 2 t Q t i=1 (1 + 2 i ) 1 otherwise. The process M is a martingale under the natural probabilities. 1 Its value at T is the Radon-Nikodym derivative of the change of measure from natural probabilities to risk-neutral probabilities. Therefore the risk-neutral probabilities of complete sample paths are given by 2 T m T, and the risk-neutral probabilities of partial sample paths from 0 to t are given by 2 t m t. The accompanying diagram shows the risk-neutral probabilities of the events up to date 3. The high realization of the endowment is indicated by the upward-sloping branch at each node. Observe that after the Þrst up move, risk-neutral probabilities maintain the same proportions with natural probabilities that is, they decrease at rate 2 t, like the natural probabilities. Risk neutral probabilities along the L, L, L,... path, on the other hand, decrease more slowly than natural probabilities. This is as would be expected in light of agents risk aversion. 2 The InÞnite Case As indicated in the introduction, difficulties arise when T =. The problem is that M is not a closed martingale (deþned as a martingale which has the property that 1 For example, the date-0 value of W is 1, its date-1 values are w H =2/3, w L =4/3, its date-2 values are w HH = w HL =2/3, w LH =8/15 and w LL =32/15. (5) 3

4 the pointwise limit of m t has the same expectation as the other terms). 2 To see this, note that the occurrence of t consecutive realizations of L from dates 0 to t, which has probability 2 t under the natural measure, has probability Q t i=1 (1+2 i ) 1 under the risk-neutral measure. This term converges to Therefore the expectation of the pointwise limit of m t is = , implying that it cannot deþne the Radon-Nikodym derivative of a change of measure. Therefore there does not exist an equivalent martingale measure. There does exist a measure such that the date-0 value of any payoff equals the integral of that payoff discounted by the money-market account with respect to that measure (summed over time in the case of multidate payoffs), but this measure is not equivalent to the natural probability measure since it gives measure to the event L, L, L..., which occurs with zero probability under the natural measure. 3 Interpretation For any security or portfolio strategy there exists a date n such that the payoff beyond n is zero with certainty. Then the value of that security or portfolio strategy equals the summed expectation of its discounted payoff, where the expectation is taken with respect to the risk-neutral probability measure for date n or any date beyond n. Correspondingly, the risk-neutral probability measure that applies at date n can be used to value payoffs terminating at or before n, but only these. If we wish to use a single probability measure to value all payoffs, this measure must be the limiting measure deþned in the preceding section. This measure assigns strictly positive weight to the sample path consisting of an inþnite sequence of L s, even though this sample path occurs with probability zero (and therefore does not affect the representative agent s expected utility) under the natural measure. It follows that when the equivalent martingale measure is in fact not equivalent to the natural probabilities, as here, one must be very cautious in analyzing the role of zero-probability events in determining equilibrium. In the present case, even though the sample path L, L, L,... occurs with probability zero, and therefore contributes nothing to the expected utility of the representative agent s consumption path, this eventdoescontributetothevalueofanysecuritythepayoff of which is nonzero in the event of a realization of L, L, L,... up to date n. References [1] Kerry Back and Stanley R. Pliska. On the fundamental theorem of asset pricing with an inþnite state space. Journal of Mathematical Economics, 20:1 18, In continuous time analyses, such as Loewenstein and Willard [6], the analogue of the martingale is the local martingale, and the analogue of the closed martingale is the martingale. 4

5 [2] Philip H. Dybvig and Stephen A. Ross. Arbitrage. In M. Milgate J. Eatwell and P. Newman, editors, The New Palgrave: A Dictionary of Economics. McMillan, [3] Christian Gilles and Stephen F. LeRoy. Bubbles and charges. International Economic Review, 33: , [4] Christian Gilles and Stephen F. LeRoy. Arbitrage, martingales and bubbles. Economics Letters, 60: , [5] David M. Kreps. Arbitrage and equilibrium in economies with inþnitely many commodities. Journal of Mathematical Economics, pages 15 35, [6] Mark Loewenstein and Gregory A. Willard. Rational equilibrium asset-pricing bubbles in continuous-trading models. Journal of Economic Theory, 91:17 58, [7] Jan Werner. Arbitrage, bubbles and valuation. International Economic Review, 38: ,