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Equilibrium asset price ranges
Equilibrium
Asset Price Ranges
YAACOV Z. BERGMAN
The purpose of this paper is to ascertain what statements can be made about price ranges in equilibrium. This paper shows that equilibrium conditions impose restrictions on ranges of price processes. Specifically, it shows that the upper and the lower barriers on the price of a risky asset at any given time are themselves bounded by the discounted upper and lower barriers, respectively, on the price at any later time. It is also shown that, as a result, in the presence of a positive riskless yield, a constant finite barrier on the price of a risky asset is inconsistent with equilibrium, with an analogous result for the lower barrier. It is shown, on the other hand, that if there is a finite upper barrier on prices, then the barrier must grow at least as fast as the interest rate. The paper stresses that these restrictions must be obeyed when modeling asset prices.
I. INTRODUCTION With regard to the laws that govern the dynamics of asset prices, several possibilities exist with respect to the statements that differ in their scope. One possibility is to completely specify the stochastic process of equilibrium asset prices in terms of exogenous stochastic processes of other observables; e.g., Lucas (1978) and Cox, Ingersoll and Ross (1985). Another possibility is to derive relationships between moments of asset prices distributions that must hold in equilibrium; e.g., Sharpe-Lintner CAPM. Yet another possibility is to consider the ranges in which asset prices may move. An example for the latter is Cootner (1964):
Prices will behave as a random walk with reflecting tend to move like a random walk.
barriers. Prices within those upper and lower limits will
When discussing Cootner’s suggestion, Samuelson cations of some diffusion price processes.
(1981) expresses discontent
with impli-
From now on I can stick to Cootner’s belief that blind chance is held in by walls of economic law. The Brownian dance of Bechelier, Einstein, and Wiener implies that the price of a Cadillac can become anything relative to that of a pea or a share of IBM. Yaacov Z. Bergman - School of Business and the Center for Rationality and Interactive Decision Theory, The Hebrew University, Mount Scopus, .Ierusalem*91905, Israel; E-Mail: [email protected]; Tel: +972-2-X38-31 16; Fax: +972-2-588-1341. International Review of Financial Analysis, Vol. 5, No. 3, 1996, pp. 161-169 Coptyright 0 1996 by JAI PRESS Inc., All Rights of reproduction in any form reserved.
ISSN: 10574219
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According to Samuelson, the fact that the Brownian diffusion implies an infinite price range is somewhat troublesome.1 The purpose of this paper is to see what statements can be made about price ranges in equilibrium. In particular, we will investigate whether it is feasible to have, as Samuelson puts it, blind chance being held in by walls of economic law. Our main result is stated and proved in Section II. This proposition asserts that the upper and the lower barriers-in a sense to be made precise-on the price of a risky asset at a given time are themselves bounded by the discounted upper and lower barriers, respectively, on the price at any later time. The rationale for this statement relies on the following general observation. In an arbitragefree, frictionless price system there should be no positive probability event in which the returns on one asset dominate those of the other. But this is exactly what happens if the stated relationship between the barriers at different times is violated. In Section III, we show that this result implies that, if the (real) riskless yield to some finite maturity is positive (negative), then the price of a dividendless risky asset cannot have a constant upper (lower) barrier during any time interval that spans that maturity. On the other hand, nonexistence of a finite upper barrier on the price is consistent with equilibrium. Further applications are discussed in Section III. Conclusions are summarized in Section IV.
II. RELATIONSHIPS
BETWEEN
PRICE BOUNDS AT DIFFERENT
TIMES
In order to cast the results in general terms, we will use a general model of information revelation. Let R be the set of all possible states of the world, which agents commonly believe could materialize in a given economy. A “state of the world’ w E Q is a full description of one possible sequence of circumstances from time 0 onward. The o-algebra % describes the collection of events, subsets of Q to which agents can commonly assign probabilities, which, in turn, are described by the measure f’r(.). Assume that it is known at time 0 that a frictionless asset market will be open at least at two subsequent times, tl and t2 (0 < tl < t2). Two goods, which do not pay dividends2 will then be traded; a numeraire, and a risky asset (e.g., gold and silver). Let {P(o, t): t IO} denote a stochastic process3 on the probability space, so that if the market is open at time 7, let the price of the risky asset-in terms of the numeraire-be equal to P(o,T).~ This formulation of the price process is quite general, and it includes diffusion, jump, diffusion-jump, and binomial processes as special cases. Using the support5 of the random variable P(o,r), define P(z) := sup [support P(0, r)] fi(z) := inf [support P(w, r)]. Intuitively, the support is the set of probable realizations of a random variable, and P(Z) and P(z) are, respectively, the upper and the lower barriers on the price of the risky asset at time z. These barriers need not be finite. For example, if the price process is a geometric Brownian motion, then for all z > 0: f(z) = 0, and P(T) = 00. What is important for our purpose is to note that the price at time z falls inside [e(z), P(T)] with probability 1. We also assume that at time 0 it is known that later on, when the asset market is open at time t1, a default-free zero-coupon bond, which guarantees a payoff of one unit of the
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Equilibrium Asset Price Ranges
numeraire at maturity factor of D(tl, t2). Pl:
time t2, will be traded, implying
In a frictionless
a deterministi&
positive8 discount
asset market, the following is a necessary condition for equilibrium
D(t1, t2) e(n)
5 e s P(t1) I D(t1, t2) P(t2).
Proposition 1 asserts that the upper and the lower barriers on the price of a risky asset at a given time are themselves bounded by the discounted upper and lower barriers, respectively, on the price at any later time. For the proof, we need the following two lemmas. LEMMA 1. rium: Given the t1,9 such that for on usset i (i = 1,
In a frictionless asset market, the following is inconsistent with equilibinformation at time 0, let there exist a non-zero probability event E at time all w CITE:R,(o) > R2(0), where R,(o) is the (tl, t2) holding-period return 2).
Proofi Compare the following two investment strategies. Strategy I: Do nothing from time 0 until time tl. If event E does not materialize at time tl, then remain idle until time t2. But if it does materialize at time tl, then buy asset 1 and hold it until time 12.Strategy 2: Same as 1, but substitute asset 2 for 1. Clearly, strategy 1 dominates strategy 2 because, for every state w in Q, the (to, t2) holding-period-return on strategy 1 is either larger than (for o E E ) or equal to (for o E Q/E) that of strategy 2. In a frictionless market, this is inconsistent with equi1ibrium.B LEMMA 2.
In a frictionless
market, equilibrium
implies:
Pr{o: D(t,, t2) e(t2) 5 P(0, tl) I D(t,, t2) P(t2)) = 1. Proofi Denote: A := {w : D(t,, t2) P(t2) < P(o, t,)}, B := {o : D(t,, t2) F’(t2) 5 P(w, tl) C D(t,, t2) P(t2)}, c := {w : P(0, t,) < D(t,, tz) _P(t2). We want to prove that in equilibrium Pr(B) = 1. 10 Suppose that this is not the case, so Pr(B) < 1. Since A, B, and C partition R, it must be either that Pr(A) > 0 or Pr(C) > 0 (or both). Suppose that Pr(A) > 0. We will show that there exists a non-zero probability event (at time t,) in which the safe bond dominates the risky asset. Denote: G := {o : P(o, t2) 2 P(Q)_ Obviously, Pr(G) = 1. It followst t that Pr(AnG) > 0. By definition of A, if o E A then l/D(t,, t2) > p(t,)lP(w, t,). By definition of G, if o E G then P(w, t2) I P(t2). Hence, it follows from both last inequalities that forallwc
AnG:
1 ~>------. Wt,, t2)
P(m tr) P(W t, 1
But the left side of the last inequality is just the (t,, t2) holding-period return on the safe asset, and the right side is the (tl, t2) holding-period return on the risky asset. We now have the antecedent of Lemma 1, where the role of the set E of that lemma is played by AnG. Consequently, the assumption Pr(A) > 0 is inconsistent with equilibrium. The proof, that Pr(C) > 0 is also inconsistent with equilibrium, is analogous. Hence, only Pr(B) = 1 remains consistent with equilibrium.=
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ProafofProposition I: We want to prove first that in equilibrium (tt) 5 D(tt, t2) (t*). Suppose the opposite, that &tt) > D(tt, t2) P(Q. Then there exists an E > 0 such that D(tt, t2) P(t2) < &tt) - E. Denote H := ((0 : p(tl> - E < P(o, tl) I @tl) + E}. By definition of the support and its supremum, Pr(H) > 0. On the other hand, H L A (A is the set from Lemma 2). Hence, 0 < Pr(H) 2 Pr(A), or 0 < Pr(A), which by Lemma 2 is inconsistent with equilibrium. The left inequality of Proposition 1 is proved along the same lines. The cases, where any of the bounds is infinite, can be treated separately using similar ideas, and are, therefore, 0mitted.m
III. APPLICATIONS
Constant Barriers Proposition 1 is now brought to bear on the question raised in the introduction, namely, whether there can exist barriers on a price of a risky asset. We find that, in general, if such barriers exist, they cannot be constant. COROLLARY 1. Suppose that the riskless yield to somefinite maturity is positive (negative), then the price of a risky asset that does not pay dividends cannot have a constant upper (lower) barrier during any time interval that contains that maturity. The intuition behind ruling out a constant barrier on the risky asset’s price is that there is a non-zero probability for the latter to reach close enough to the barrier, where it becomes dominated by the risk-free asset. Proof Suppose that riskless yield at time tl for maturity at time t-2is positive. This implies that the corresponding discount factor is less than unity, i.e. D 1.a As an application of Corollary 1, consider the often observed behavior of a central bank that announces that it is going to keep the foreign exchange rate within a fixed, constant band. By Corollary 1, such an announcement cannot be credible, since it is equivalent to the introduction of an arbitrage opportunity into the capital market. When the price of the domestic currency reaches close enough to the announced upper boundary, arbitrageurs will sell short vast amounts of the foreign currency for the domestic currency to invest in domestic govemment bonds thus making arbitrage profits. As an example for the corollary, consider an economy with a riskless bond returning at a constant positive interest rate r and maturing at some time Tin the future, paying a face value of one unit of the numeraire. In addition, there also exists a dividendless risky asset, whose price S(2) is governed by the following stochastic differential equation:‘2 dS(t)=
[$(b-S)-F,S]dt+J/S(b-S)ldW(t) 0 I t,
0 < S(0) < b,
0 2 6,, 6,
(1)
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Equilibrium Asset Price Ranges
where W(t) is a standard Wiener process. This specification of the price system in (1) seems innocuous at first sight. However, a closer examination reveals that the risky asset’s price is confined within the interval [0, b] for any time after 0. 13 Consequently, this specification is ruled out by Corollary 1 as inconsistent with equilibrium. To see the arbitrage opportunity that is admitted by (1) consider the strategy of waiting until the first time t that the risky asset’s price hits the value b.exp[-r(TT)], the discounted barrier. According to the specification of the price process in (l), this event happens with positive probability. At such time the strategy calls for selling the risky asset short and using the proceeds to buy b units of the bond, then holding them until maturity T. At that time, the portfolio is worth b - S(T) units of the numeraire-a quantity that is non-negative with probability one, and that is positive with positive probability. If the risky asset’s price is already above the discounted barrier, then the trade should be executed right away. If it never hits the discounted barrier before maturity, then the strategy winds up with a zero value at time T. The rationale behind the arbitrage strategy is straightforward. An upper bound on the risky price process serves to limit the potential loss on a short position in that asset. Harrison and Kreps’s (1979) characterization of viable price systems can also be used to detect the inconsistency introduced by the prices in our example. They have shown that a necessary condition for the viability of a price system is that the original probability measure and the risk-neutral transformed measure are equivalent: i.e., that the two measures assign zero probability to the same events. But in our example the measures are not equivalent. This is because the probability-under the original measure-for the price to reach values above the barrier is zero, whereas it is positive under the risk-neutral measure. To see that, note that the risk-neutral transformed price process of (1) is. dS*(t)
= [6,(b-S*)-&S*]dt-dmdW(t),for
t2O.But
in this
transformed
process the drift at the upper barrier (S* = b) is no longer zero, as is the case with the original process. Rather, it is positive. Consequently, the transformed process pierces through that barrier and travels beyond with non-zero probability. In a recent paper on valuation of American options, Kim and Yu (1996, p. 72) discuss an example in which they postulate a price process that is in direct conflict with Corollary 1 in the present paper. They write: We consider the case in which the [risky] security price follows a lognormal diffusion but with an absorbing barrier at a constant positive number A. This price behavior of this process is that it follows [a geometric Brownian motion] as long as the price is greater than A. Once it hits the barrier A from above, the process is absorbed at state A permanently.
According to this specification, conditioned on the risky price process hitting from above the absorbing barrier at A-an event of positive probability-it cannot rise again. Thereafter, A becomes a constant upper barrier for that price. But this is ruled out by Corollary 1. The arbitrage opportunity that is introduced by this specification is as follows. At the first time that the risky security price hits the absorption barrier at A, the risky security is sold short for A dollars and the proceeds are invested in the risk-free asset. These two positions are held for a finite time interval, and then both positions are closed. Since the price of the risky asset was absorbed at A dollars, this amount is needed to buy back the risky security. But the investment in the riskless asset generates the principal of A dollars plus the interest. The latter part is the arbitrage profit.
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To stress our point, let us summon Samuelson’s metaphor from the Introduction. Suppose that, in an economy with a positive-yield riskless bond that matures in one day from now, we try to put a cap on the price of a pea at a constant upper barrier of 101ut) Cadillacs for the duration of that day only. We find that this is impossible; an arbitrage opportunity creeps in. On the other hand ‘the Brownian dance of Bechelier, Einstein, and Wiener [that] implies that the price of a pea can become anything relative to that of a Cadillac’--e.g., a geometric Brownian motion with no upper bound-would not suffer from a similar affliction.
Non-Constant Price Supports Corollary 1 does not rule out non-constant supports for risky asset’s prices. As a matter of fact, it clearly follows from Proposition 1 that, in the case of constant positive interest rate, it is possible for a risky asset’s price to be bounded from above as long as the bound increases at least as fast as the interest rate (with an analogous statement holding for negative interest rates.) As an example, consider a European put option on a dividendless risky underlying asset in the framework of the Black-Scholes option pricing model. It is straightforward to verify that, in that case, the rate of increase of the upper barrier on a put option price is exactly equal to the interest rate. An observations which is not captured by Corollary 1, but which is true for similar reasons, is the following. Even if the interest rate is zero (D(tt , t2) = I), a constant upper (lower) barrier on the risky asset’s price, which is a reflecting barrier rather than absorbing, is inconsistent with equilibrium for similar reasons. When the risky asset’s price hits the reflecting upper (lower) barrier, it becomes dominated by (it dominates) the riskless asset.
Detection of Inconsistencies in Bond Pricing by Arbitrage Cox, Ingersoll and Ross (CIR) (1985) demonstrate the advantage of bond pricing in a fullequilibrium framework rather than by arbitrage methods, where certain parameters have to be specified exogenously. They show that, when the latter method is followed, arbitrage opportunities may be inadvertently embedded in the economy, whereas this peril is avoided when using the former. We refer specifically to CIR (1985, p. 398) and use their notation. The instantaneous interest rate in their example follows a diffusion described by dr =
K(e
-r)dt+
o&dz,,
(2)
where dz,, is a one-dimensional Wiener process. Also, if o2 > 20, then zero is an accessible reflecting barrier for the interest rate; it can reach zero but not below. CIR show that, in their economy, a bond maturing at time T paying then one unit of the numeraire has at time t a price of
n(r, t, T) = [A(t,
T)](K@mW”)‘K’ exp [-rB(t,
T)],
whereA(r, 7) < 1 for r< T. CIR stress, that a choice of Yu # 0 in (3), which would seem innocuous from an arbitragepricing point of view, is rejected as inviable in a full-equilibrium analysis. This is because at a time when the instantaneous riskless interest rate hits zero, the rate of return on the bond, which instantaneously also becomes riskless, is then equal to -Y&t, 7), which is not zero.
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It is interesting to observe how Proposition 1 detects the arbitrage inducing specifications of the CIR example. To that end, we will assign the role of the risky asset in our framework to a share in a money-market type mutual fund in CIR’s economy, which accumulates gains at that stochastic riskless interest rate. If a share in that fund has a unit value at time 0, then the lower barrier on its price at any later time is also unity.14 Formally, P(T) = P(T) = 1 for all t > 0.
(4)
In this framework, the discount factor D(t, 7) is given by a the concurrent bond price II(0, t, 7). Proposition 1 implies, then, that a necessary condition for equilibrium is D(t, r)E(7’) SE(t), which using (4) can be rewritten in the current notation as: II(O,t,r)<
1.
(5)
In view of the fact that A(t, n < 1 for t < T, inequality (5) can be true only if Yo < K@. In other words, if Yu > KCO,then the price system specification contains an arbitrage opportunity, which presents itself whenever the instantaneous interest rate becomes zero. In that situation, the price of a bond with a promised payment of one unit of the numeraire is more than one. Hence buying and holding the bond until maturity is dominated by investment in the money market fund in which a share’s price never declines, because of the sure non-negativity of the interest rate in (2). Notice, though, that, whereas CIR observe an arbitrage opportunity for any specification of Yo that is different from zero, our Proposition 1 detects such opportunities only for Y() > K@. The following is an explanation for this dissimilarity. CIR’s interest rate process (2) has the following intriguing property. Consider any finite time interval-as short as desired-commencing at a time when the interest rate hits zero level. Then there is zero probability that it will cctntinuousZ~ sojourn at that level for the duration of that time interval. Nevertheless, the set of times when the interest rate again hits zero level in that interval has a positive measure. As mentioned earlier, if Yo f ~0 and ‘l’u I K@ the CIR prescription for an arbitrage strategy is to wait until the instantaneous interest rate hits zero, then to open opposite offsetting positions in the interest rate instrument and in the bond. Now, if this strategy is carried out at one point in time only, then the size of the arbitrage profits that are produced is infinitisimal-not finite. But infinitisimal profits are not inconsistent with equilibrium; therefore, strategies that produce them should not be recognized as arbitrage. Indeed, Proposition 1 in the paper recognizes as arbitrage only such strategies that produce finite-sized profits; it does not recognize infinitesimal profits as arbitrage profits. It may still be argued, though, that a finite-sized profit can be produced by the following strategy. During a finite time interval, whenever the instantaneous interest rate is zero,$&e opposite offsetting positions in the interest rate instrument and in the bond are opened; whenever the interest rate is positive-the positions are closed. But, as mentioned above, since the interest rate does not continuously stay at zero level for any finite duration, it follows that the suggested strategy is discontinuous at every point in the interval with probability one. Consequently, such strategy is not right-continuous either; thus, it is ruled out in the Duffie and Huang (1985) type economy, which we postulate in the current paper. (A strategy that calls for infinite offsetting positions at a single happenstance of a zero interest rate is ruled out for similar reasons. It is not even well defined in our framework.) It should be stressed, however, that the CIR example does demonstrate an arbitrage opportunity in our framework for any non-zero Yu that is larger than K@.
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IV. SUMMARY We have show that equilibrium conditions impose restrictions on ranges of price processes. Specifically, we have shown that the upper and the lower barriers on the price of a risky asset at any given time are themselves bounded by the discounted upper and lower barriers, respectively, on the price at any later time. We also showed that, as a result, in the presence of a positive riskless yield, a constant finite barrier on the price of a risky asset is inconsistent with equilibrium (with an analogous result for the lower barrier). We showed, on the other hand, that, if there is a finite upper barrier on prices, then the barrier must grow at least as fast as the interest rate. Our findings stress that these restrictions must be obeyed when modeling asset prices.
ACKNOWLEDGMENTS
For their helpful comments, I would like to thank the Hebrew University, New York University, and Chi-Fu Huang, John B. Long, and Steve Ross. All been supported by a grant from the Krueger Center
the participants of the finance seminars at Yale University. Special thanks are due to remaining errors are mine. This work has for Finance.
NOTES
1. This might be an overly strict interpretation of Cootner and of SamuelsonII. Nevertheless, it helps focus the analysis. 2. The assumption of no-dividends is not essential. It is assumed for simplicity of exposition. 3. Let (3, : t 2 0) be a filtration-an increasing family of sub o-algebras of 3-that describes the development of the information structure in time. (A special case of a filtration is a description of the information structure by an ‘event tree’.) The stochastic process {P(o, t): t 2 0} is adapted to the filtration, which means that the random variable P(o, t) can depend only on the information revealed by the filtration up to-and including-time z, but not any later. 4. The trades are carried out according to predictable right-continuous trading strategies; i.e., by strategies that at any given time can depend at most on information received up to-but exclusive of-that time. (Right-continuity does not exclude discontinuities in the strategies, however.) Non-predictable strategies admit arbitrage opportunities. 5. A point belongs to the support of a random variable if and only if all its neighborhoods contain realizations of the random variable with positive probability. Formally, let F(.) denote the distribution function of a random variable X. Let x be such a point that, for every E > 0, we have F(x + E) - F(x - E) > 0. The set of all such x is called the support of X. 6. Actually, given information at time 0, we can let the discount factor be stochastic as long as it is bounded with probability 1. We keep it deterministic to streamline the exposition. 7. A constant known interest rate is a special case of our setup. 8. A non-positive discount factor is inconsistent with equilibrium, since it admits arbitrage opportunities. 9. In other words, the event E belongs to 3,t (to < t,).
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10. All probabilities
here are conditioned
on information
11. To see that, denote by Cc the complement Pr(G) = 1. Now, A = (AnG)
at time 0
to G in R, then Pr(GC) = 0 because
u (AnGC), where the two sets that form the union are dis-
joint. Therefore, 0 < Pr(A) = Pr[(AnG) u (AnGC)] = Pr(AnG) + Pr(AnGc) = Pr(AnG). 12. This is the Wright-Fisher gene frequency model, slightly modified to suit our purpose. 13. More precisely, this process is a regular diffusion on the interval [0, b], where the two end-points of that interval are classified as entrance boundaries. A diffusion on an interval I is said to be regular, if starting from any point in the interior of I, any other point in the interior of I may be reached with positive probability. The meaning of 0 and b being entrance boundaries is that if the process is started within (0, b), it cannot attain values outside that interval, but it can enter from the outside. Intuitively, this follows from the fact that when S(t) = b (respectively, = 0), the instantaneous variance is zero, and the drift is negative (positive). So, neither the variance nor the drift can drive the process to reach beyond b (0). For a definitive account of boundary classification for diffusion processes, see Karlin and Taylor (198 1, pp. 226-26 1). 14. This price of one numeraire unit per share in the money market fund results from interest-rate sample paths that stay at zero level almost all the time. Such a path has a nonzero probability in the CIR example.
REFERENCES
Cootner, P. (1964). Stock prices: Random vs. Systematic changes. In: Paul Cootner (Ed.), The random character of stock market prices. Cambridge, MA: MIT PRess. Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985). An intertemporal general equilibrium model of asset prices. Econometrica, 53, 363-384. Cox, J.C. & Ross, S.A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics 3, 145-166. Harrison, J.M. & Kreps, D. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20, 381-408. Karlin, S. & Taylor, H.M. (1981). A second course in stochastic processes. New York: Academic Press. Kim, I.J. & Yu. G.G. (1996). Valuation of American options and applications. Review of Derivatives Research, 1, 6 1-85. Lucas, R.E. (1978). Asset prices in an exchange economy. Econometrica 46, 1429-1445. Samuelson, P.A. (1981). Paul Cootner’s reconciliation of economic law with chance. In: William F. Sharpe & Catharyn M. Cootner, (Eds.), Financial economics: Essays in honor of Paul Cootner. Englewood Cliffs, N.J.: Prentice-Hall.
Asset Price Ranges
YAACOV Z. BERGMAN
The purpose of this paper is to ascertain what statements can be made about price ranges in equilibrium. This paper shows that equilibrium conditions impose restrictions on ranges of price processes. Specifically, it shows that the upper and the lower barriers on the price of a risky asset at any given time are themselves bounded by the discounted upper and lower barriers, respectively, on the price at any later time. It is also shown that, as a result, in the presence of a positive riskless yield, a constant finite barrier on the price of a risky asset is inconsistent with equilibrium, with an analogous result for the lower barrier. It is shown, on the other hand, that if there is a finite upper barrier on prices, then the barrier must grow at least as fast as the interest rate. The paper stresses that these restrictions must be obeyed when modeling asset prices.
I. INTRODUCTION With regard to the laws that govern the dynamics of asset prices, several possibilities exist with respect to the statements that differ in their scope. One possibility is to completely specify the stochastic process of equilibrium asset prices in terms of exogenous stochastic processes of other observables; e.g., Lucas (1978) and Cox, Ingersoll and Ross (1985). Another possibility is to derive relationships between moments of asset prices distributions that must hold in equilibrium; e.g., Sharpe-Lintner CAPM. Yet another possibility is to consider the ranges in which asset prices may move. An example for the latter is Cootner (1964):
Prices will behave as a random walk with reflecting tend to move like a random walk.
barriers. Prices within those upper and lower limits will
When discussing Cootner’s suggestion, Samuelson cations of some diffusion price processes.
(1981) expresses discontent
with impli-
From now on I can stick to Cootner’s belief that blind chance is held in by walls of economic law. The Brownian dance of Bechelier, Einstein, and Wiener implies that the price of a Cadillac can become anything relative to that of a pea or a share of IBM. Yaacov Z. Bergman - School of Business and the Center for Rationality and Interactive Decision Theory, The Hebrew University, Mount Scopus, .Ierusalem*91905, Israel; E-Mail: [email protected]; Tel: +972-2-X38-31 16; Fax: +972-2-588-1341. International Review of Financial Analysis, Vol. 5, No. 3, 1996, pp. 161-169 Coptyright 0 1996 by JAI PRESS Inc., All Rights of reproduction in any form reserved.
ISSN: 10574219
162
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ANALYSIS
/Vol.
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According to Samuelson, the fact that the Brownian diffusion implies an infinite price range is somewhat troublesome.1 The purpose of this paper is to see what statements can be made about price ranges in equilibrium. In particular, we will investigate whether it is feasible to have, as Samuelson puts it, blind chance being held in by walls of economic law. Our main result is stated and proved in Section II. This proposition asserts that the upper and the lower barriers-in a sense to be made precise-on the price of a risky asset at a given time are themselves bounded by the discounted upper and lower barriers, respectively, on the price at any later time. The rationale for this statement relies on the following general observation. In an arbitragefree, frictionless price system there should be no positive probability event in which the returns on one asset dominate those of the other. But this is exactly what happens if the stated relationship between the barriers at different times is violated. In Section III, we show that this result implies that, if the (real) riskless yield to some finite maturity is positive (negative), then the price of a dividendless risky asset cannot have a constant upper (lower) barrier during any time interval that spans that maturity. On the other hand, nonexistence of a finite upper barrier on the price is consistent with equilibrium. Further applications are discussed in Section III. Conclusions are summarized in Section IV.
II. RELATIONSHIPS
BETWEEN
PRICE BOUNDS AT DIFFERENT
TIMES
In order to cast the results in general terms, we will use a general model of information revelation. Let R be the set of all possible states of the world, which agents commonly believe could materialize in a given economy. A “state of the world’ w E Q is a full description of one possible sequence of circumstances from time 0 onward. The o-algebra % describes the collection of events, subsets of Q to which agents can commonly assign probabilities, which, in turn, are described by the measure f’r(.). Assume that it is known at time 0 that a frictionless asset market will be open at least at two subsequent times, tl and t2 (0 < tl < t2). Two goods, which do not pay dividends2 will then be traded; a numeraire, and a risky asset (e.g., gold and silver). Let {P(o, t): t IO} denote a stochastic process3 on the probability space, so that if the market is open at time 7, let the price of the risky asset-in terms of the numeraire-be equal to P(o,T).~ This formulation of the price process is quite general, and it includes diffusion, jump, diffusion-jump, and binomial processes as special cases. Using the support5 of the random variable P(o,r), define P(z) := sup [support P(0, r)] fi(z) := inf [support P(w, r)]. Intuitively, the support is the set of probable realizations of a random variable, and P(Z) and P(z) are, respectively, the upper and the lower barriers on the price of the risky asset at time z. These barriers need not be finite. For example, if the price process is a geometric Brownian motion, then for all z > 0: f(z) = 0, and P(T) = 00. What is important for our purpose is to note that the price at time z falls inside [e(z), P(T)] with probability 1. We also assume that at time 0 it is known that later on, when the asset market is open at time t1, a default-free zero-coupon bond, which guarantees a payoff of one unit of the
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numeraire at maturity factor of D(tl, t2). Pl:
time t2, will be traded, implying
In a frictionless
a deterministi&
positive8 discount
asset market, the following is a necessary condition for equilibrium
D(t1, t2) e(n)
5 e
Proposition 1 asserts that the upper and the lower barriers on the price of a risky asset at a given time are themselves bounded by the discounted upper and lower barriers, respectively, on the price at any later time. For the proof, we need the following two lemmas. LEMMA 1. rium: Given the t1,9 such that for on usset i (i = 1,
In a frictionless asset market, the following is inconsistent with equilibinformation at time 0, let there exist a non-zero probability event E at time all w CITE:R,(o) > R2(0), where R,(o) is the (tl, t2) holding-period return 2).
Proofi Compare the following two investment strategies. Strategy I: Do nothing from time 0 until time tl. If event E does not materialize at time tl, then remain idle until time t2. But if it does materialize at time tl, then buy asset 1 and hold it until time 12.Strategy 2: Same as 1, but substitute asset 2 for 1. Clearly, strategy 1 dominates strategy 2 because, for every state w in Q, the (to, t2) holding-period-return on strategy 1 is either larger than (for o E E ) or equal to (for o E Q/E) that of strategy 2. In a frictionless market, this is inconsistent with equi1ibrium.B LEMMA 2.
In a frictionless
market, equilibrium
implies:
Pr{o: D(t,, t2) e(t2) 5 P(0, tl) I D(t,, t2) P(t2)) = 1. Proofi Denote: A := {w : D(t,, t2) P(t2) < P(o, t,)}, B := {o : D(t,, t2) F’(t2) 5 P(w, tl) C D(t,, t2) P(t2)}, c := {w : P(0, t,) < D(t,, tz) _P(t2). We want to prove that in equilibrium Pr(B) = 1. 10 Suppose that this is not the case, so Pr(B) < 1. Since A, B, and C partition R, it must be either that Pr(A) > 0 or Pr(C) > 0 (or both). Suppose that Pr(A) > 0. We will show that there exists a non-zero probability event (at time t,) in which the safe bond dominates the risky asset. Denote: G := {o : P(o, t2) 2 P(Q)_ Obviously, Pr(G) = 1. It followst t that Pr(AnG) > 0. By definition of A, if o E A then l/D(t,, t2) > p(t,)lP(w, t,). By definition of G, if o E G then P(w, t2) I P(t2). Hence, it follows from both last inequalities that forallwc
AnG:
1 ~>------. Wt,, t2)
P(m tr) P(W t, 1
But the left side of the last inequality is just the (t,, t2) holding-period return on the safe asset, and the right side is the (tl, t2) holding-period return on the risky asset. We now have the antecedent of Lemma 1, where the role of the set E of that lemma is played by AnG. Consequently, the assumption Pr(A) > 0 is inconsistent with equilibrium. The proof, that Pr(C) > 0 is also inconsistent with equilibrium, is analogous. Hence, only Pr(B) = 1 remains consistent with equilibrium.=
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ProafofProposition I: We want to prove first that in equilibrium (tt) 5 D(tt, t2) (t*). Suppose the opposite, that &tt) > D(tt, t2) P(Q. Then there exists an E > 0 such that D(tt, t2) P(t2) < &tt) - E. Denote H := ((0 : p(tl> - E < P(o, tl) I @tl) + E}. By definition of the support and its supremum, Pr(H) > 0. On the other hand, H L A (A is the set from Lemma 2). Hence, 0 < Pr(H) 2 Pr(A), or 0 < Pr(A), which by Lemma 2 is inconsistent with equilibrium. The left inequality of Proposition 1 is proved along the same lines. The cases, where any of the bounds is infinite, can be treated separately using similar ideas, and are, therefore, 0mitted.m
III. APPLICATIONS
Constant Barriers Proposition 1 is now brought to bear on the question raised in the introduction, namely, whether there can exist barriers on a price of a risky asset. We find that, in general, if such barriers exist, they cannot be constant. COROLLARY 1. Suppose that the riskless yield to somefinite maturity is positive (negative), then the price of a risky asset that does not pay dividends cannot have a constant upper (lower) barrier during any time interval that contains that maturity. The intuition behind ruling out a constant barrier on the risky asset’s price is that there is a non-zero probability for the latter to reach close enough to the barrier, where it becomes dominated by the risk-free asset. Proof Suppose that riskless yield at time tl for maturity at time t-2is positive. This implies that the corresponding discount factor is less than unity, i.e. D
[$(b-S)-F,S]dt+J/S(b-S)ldW(t) 0 I t,
0 < S(0) < b,
0 2 6,, 6,
(1)
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Equilibrium Asset Price Ranges
where W(t) is a standard Wiener process. This specification of the price system in (1) seems innocuous at first sight. However, a closer examination reveals that the risky asset’s price is confined within the interval [0, b] for any time after 0. 13 Consequently, this specification is ruled out by Corollary 1 as inconsistent with equilibrium. To see the arbitrage opportunity that is admitted by (1) consider the strategy of waiting until the first time t that the risky asset’s price hits the value b.exp[-r(TT)], the discounted barrier. According to the specification of the price process in (l), this event happens with positive probability. At such time the strategy calls for selling the risky asset short and using the proceeds to buy b units of the bond, then holding them until maturity T. At that time, the portfolio is worth b - S(T) units of the numeraire-a quantity that is non-negative with probability one, and that is positive with positive probability. If the risky asset’s price is already above the discounted barrier, then the trade should be executed right away. If it never hits the discounted barrier before maturity, then the strategy winds up with a zero value at time T. The rationale behind the arbitrage strategy is straightforward. An upper bound on the risky price process serves to limit the potential loss on a short position in that asset. Harrison and Kreps’s (1979) characterization of viable price systems can also be used to detect the inconsistency introduced by the prices in our example. They have shown that a necessary condition for the viability of a price system is that the original probability measure and the risk-neutral transformed measure are equivalent: i.e., that the two measures assign zero probability to the same events. But in our example the measures are not equivalent. This is because the probability-under the original measure-for the price to reach values above the barrier is zero, whereas it is positive under the risk-neutral measure. To see that, note that the risk-neutral transformed price process of (1) is. dS*(t)
= [6,(b-S*)-&S*]dt-dmdW(t),for
t2O.But
in this
transformed
process the drift at the upper barrier (S* = b) is no longer zero, as is the case with the original process. Rather, it is positive. Consequently, the transformed process pierces through that barrier and travels beyond with non-zero probability. In a recent paper on valuation of American options, Kim and Yu (1996, p. 72) discuss an example in which they postulate a price process that is in direct conflict with Corollary 1 in the present paper. They write: We consider the case in which the [risky] security price follows a lognormal diffusion but with an absorbing barrier at a constant positive number A. This price behavior of this process is that it follows [a geometric Brownian motion] as long as the price is greater than A. Once it hits the barrier A from above, the process is absorbed at state A permanently.
According to this specification, conditioned on the risky price process hitting from above the absorbing barrier at A-an event of positive probability-it cannot rise again. Thereafter, A becomes a constant upper barrier for that price. But this is ruled out by Corollary 1. The arbitrage opportunity that is introduced by this specification is as follows. At the first time that the risky security price hits the absorption barrier at A, the risky security is sold short for A dollars and the proceeds are invested in the risk-free asset. These two positions are held for a finite time interval, and then both positions are closed. Since the price of the risky asset was absorbed at A dollars, this amount is needed to buy back the risky security. But the investment in the riskless asset generates the principal of A dollars plus the interest. The latter part is the arbitrage profit.
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To stress our point, let us summon Samuelson’s metaphor from the Introduction. Suppose that, in an economy with a positive-yield riskless bond that matures in one day from now, we try to put a cap on the price of a pea at a constant upper barrier of 101ut) Cadillacs for the duration of that day only. We find that this is impossible; an arbitrage opportunity creeps in. On the other hand ‘the Brownian dance of Bechelier, Einstein, and Wiener [that] implies that the price of a pea can become anything relative to that of a Cadillac’--e.g., a geometric Brownian motion with no upper bound-would not suffer from a similar affliction.
Non-Constant Price Supports Corollary 1 does not rule out non-constant supports for risky asset’s prices. As a matter of fact, it clearly follows from Proposition 1 that, in the case of constant positive interest rate, it is possible for a risky asset’s price to be bounded from above as long as the bound increases at least as fast as the interest rate (with an analogous statement holding for negative interest rates.) As an example, consider a European put option on a dividendless risky underlying asset in the framework of the Black-Scholes option pricing model. It is straightforward to verify that, in that case, the rate of increase of the upper barrier on a put option price is exactly equal to the interest rate. An observations which is not captured by Corollary 1, but which is true for similar reasons, is the following. Even if the interest rate is zero (D(tt , t2) = I), a constant upper (lower) barrier on the risky asset’s price, which is a reflecting barrier rather than absorbing, is inconsistent with equilibrium for similar reasons. When the risky asset’s price hits the reflecting upper (lower) barrier, it becomes dominated by (it dominates) the riskless asset.
Detection of Inconsistencies in Bond Pricing by Arbitrage Cox, Ingersoll and Ross (CIR) (1985) demonstrate the advantage of bond pricing in a fullequilibrium framework rather than by arbitrage methods, where certain parameters have to be specified exogenously. They show that, when the latter method is followed, arbitrage opportunities may be inadvertently embedded in the economy, whereas this peril is avoided when using the former. We refer specifically to CIR (1985, p. 398) and use their notation. The instantaneous interest rate in their example follows a diffusion described by dr =
K(e
-r)dt+
o&dz,,
(2)
where dz,, is a one-dimensional Wiener process. Also, if o2 > 20, then zero is an accessible reflecting barrier for the interest rate; it can reach zero but not below. CIR show that, in their economy, a bond maturing at time T paying then one unit of the numeraire has at time t a price of
n(r, t, T) = [A(t,
T)](K@mW”)‘K’ exp [-rB(t,
T)],
whereA(r, 7) < 1 for r< T. CIR stress, that a choice of Yu # 0 in (3), which would seem innocuous from an arbitragepricing point of view, is rejected as inviable in a full-equilibrium analysis. This is because at a time when the instantaneous riskless interest rate hits zero, the rate of return on the bond, which instantaneously also becomes riskless, is then equal to -Y&t, 7), which is not zero.
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It is interesting to observe how Proposition 1 detects the arbitrage inducing specifications of the CIR example. To that end, we will assign the role of the risky asset in our framework to a share in a money-market type mutual fund in CIR’s economy, which accumulates gains at that stochastic riskless interest rate. If a share in that fund has a unit value at time 0, then the lower barrier on its price at any later time is also unity.14 Formally, P(T) = P(T) = 1 for all t > 0.
(4)
In this framework, the discount factor D(t, 7) is given by a the concurrent bond price II(0, t, 7). Proposition 1 implies, then, that a necessary condition for equilibrium is D(t, r)E(7’) SE(t), which using (4) can be rewritten in the current notation as: II(O,t,r)<
1.
(5)
In view of the fact that A(t, n < 1 for t < T, inequality (5) can be true only if Yo < K@. In other words, if Yu > KCO,then the price system specification contains an arbitrage opportunity, which presents itself whenever the instantaneous interest rate becomes zero. In that situation, the price of a bond with a promised payment of one unit of the numeraire is more than one. Hence buying and holding the bond until maturity is dominated by investment in the money market fund in which a share’s price never declines, because of the sure non-negativity of the interest rate in (2). Notice, though, that, whereas CIR observe an arbitrage opportunity for any specification of Yo that is different from zero, our Proposition 1 detects such opportunities only for Y() > K@. The following is an explanation for this dissimilarity. CIR’s interest rate process (2) has the following intriguing property. Consider any finite time interval-as short as desired-commencing at a time when the interest rate hits zero level. Then there is zero probability that it will cctntinuousZ~ sojourn at that level for the duration of that time interval. Nevertheless, the set of times when the interest rate again hits zero level in that interval has a positive measure. As mentioned earlier, if Yo f ~0 and ‘l’u I K@ the CIR prescription for an arbitrage strategy is to wait until the instantaneous interest rate hits zero, then to open opposite offsetting positions in the interest rate instrument and in the bond. Now, if this strategy is carried out at one point in time only, then the size of the arbitrage profits that are produced is infinitisimal-not finite. But infinitisimal profits are not inconsistent with equilibrium; therefore, strategies that produce them should not be recognized as arbitrage. Indeed, Proposition 1 in the paper recognizes as arbitrage only such strategies that produce finite-sized profits; it does not recognize infinitesimal profits as arbitrage profits. It may still be argued, though, that a finite-sized profit can be produced by the following strategy. During a finite time interval, whenever the instantaneous interest rate is zero,$&e opposite offsetting positions in the interest rate instrument and in the bond are opened; whenever the interest rate is positive-the positions are closed. But, as mentioned above, since the interest rate does not continuously stay at zero level for any finite duration, it follows that the suggested strategy is discontinuous at every point in the interval with probability one. Consequently, such strategy is not right-continuous either; thus, it is ruled out in the Duffie and Huang (1985) type economy, which we postulate in the current paper. (A strategy that calls for infinite offsetting positions at a single happenstance of a zero interest rate is ruled out for similar reasons. It is not even well defined in our framework.) It should be stressed, however, that the CIR example does demonstrate an arbitrage opportunity in our framework for any non-zero Yu that is larger than K@.
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IV. SUMMARY We have show that equilibrium conditions impose restrictions on ranges of price processes. Specifically, we have shown that the upper and the lower barriers on the price of a risky asset at any given time are themselves bounded by the discounted upper and lower barriers, respectively, on the price at any later time. We also showed that, as a result, in the presence of a positive riskless yield, a constant finite barrier on the price of a risky asset is inconsistent with equilibrium (with an analogous result for the lower barrier). We showed, on the other hand, that, if there is a finite upper barrier on prices, then the barrier must grow at least as fast as the interest rate. Our findings stress that these restrictions must be obeyed when modeling asset prices.
ACKNOWLEDGMENTS
For their helpful comments, I would like to thank the Hebrew University, New York University, and Chi-Fu Huang, John B. Long, and Steve Ross. All been supported by a grant from the Krueger Center
the participants of the finance seminars at Yale University. Special thanks are due to remaining errors are mine. This work has for Finance.
NOTES
1. This might be an overly strict interpretation of Cootner and of SamuelsonII. Nevertheless, it helps focus the analysis. 2. The assumption of no-dividends is not essential. It is assumed for simplicity of exposition. 3. Let (3, : t 2 0) be a filtration-an increasing family of sub o-algebras of 3-that describes the development of the information structure in time. (A special case of a filtration is a description of the information structure by an ‘event tree’.) The stochastic process {P(o, t): t 2 0} is adapted to the filtration, which means that the random variable P(o, t) can depend only on the information revealed by the filtration up to-and including-time z, but not any later. 4. The trades are carried out according to predictable right-continuous trading strategies; i.e., by strategies that at any given time can depend at most on information received up to-but exclusive of-that time. (Right-continuity does not exclude discontinuities in the strategies, however.) Non-predictable strategies admit arbitrage opportunities. 5. A point belongs to the support of a random variable if and only if all its neighborhoods contain realizations of the random variable with positive probability. Formally, let F(.) denote the distribution function of a random variable X. Let x be such a point that, for every E > 0, we have F(x + E) - F(x - E) > 0. The set of all such x is called the support of X. 6. Actually, given information at time 0, we can let the discount factor be stochastic as long as it is bounded with probability 1. We keep it deterministic to streamline the exposition. 7. A constant known interest rate is a special case of our setup. 8. A non-positive discount factor is inconsistent with equilibrium, since it admits arbitrage opportunities. 9. In other words, the event E belongs to 3,t (to < t,).
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10. All probabilities
here are conditioned
on information
11. To see that, denote by Cc the complement Pr(G) = 1. Now, A = (AnG)
at time 0
to G in R, then Pr(GC) = 0 because
u (AnGC), where the two sets that form the union are dis-
joint. Therefore, 0 < Pr(A) = Pr[(AnG) u (AnGC)] = Pr(AnG) + Pr(AnGc) = Pr(AnG). 12. This is the Wright-Fisher gene frequency model, slightly modified to suit our purpose. 13. More precisely, this process is a regular diffusion on the interval [0, b], where the two end-points of that interval are classified as entrance boundaries. A diffusion on an interval I is said to be regular, if starting from any point in the interior of I, any other point in the interior of I may be reached with positive probability. The meaning of 0 and b being entrance boundaries is that if the process is started within (0, b), it cannot attain values outside that interval, but it can enter from the outside. Intuitively, this follows from the fact that when S(t) = b (respectively, = 0), the instantaneous variance is zero, and the drift is negative (positive). So, neither the variance nor the drift can drive the process to reach beyond b (0). For a definitive account of boundary classification for diffusion processes, see Karlin and Taylor (198 1, pp. 226-26 1). 14. This price of one numeraire unit per share in the money market fund results from interest-rate sample paths that stay at zero level almost all the time. Such a path has a nonzero probability in the CIR example.
REFERENCES
Cootner, P. (1964). Stock prices: Random vs. Systematic changes. In: Paul Cootner (Ed.), The random character of stock market prices. Cambridge, MA: MIT PRess. Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985). An intertemporal general equilibrium model of asset prices. Econometrica, 53, 363-384. Cox, J.C. & Ross, S.A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics 3, 145-166. Harrison, J.M. & Kreps, D. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20, 381-408. Karlin, S. & Taylor, H.M. (1981). A second course in stochastic processes. New York: Academic Press. Kim, I.J. & Yu. G.G. (1996). Valuation of American options and applications. Review of Derivatives Research, 1, 6 1-85. Lucas, R.E. (1978). Asset prices in an exchange economy. Econometrica 46, 1429-1445. Samuelson, P.A. (1981). Paul Cootner’s reconciliation of economic law with chance. In: William F. Sharpe & Catharyn M. Cootner, (Eds.), Financial economics: Essays in honor of Paul Cootner. Englewood Cliffs, N.J.: Prentice-Hall.