The ex ante effects of trade halting rules on informed trading strategies and market liquidity

Review of Financial Economics 1997, Vol. 6. No. 1, l-14

The Ex Ante Effects of Trade Halting Rules on Informed Trading Strategies and Market Liquidity Avanidhar Subrahmanyam Universio of California at Los Angeles

In this paper, we investigate strategic informed trading in a regime with rule-based market closures. Closure rules can be designed to reduce the ex ante trading costs of liquidity traders but cause informed traders to scale back their trading in order to reduce the chance of the closure being triggered. In the Stackelberg equilibrium where the informed act as leaders and market makers as followers, this phenomenon causes trading costs for small investors to increase. Thus, ex ante strategic behavior by informed traders in response to closure rules can result in increased trading costs for precisely the individuals whom the closures are intended to benefit: namely, the uninformed retail investors who trade small orders. We show, however, that this effect can be mitigated by randomizing the halting rule and thus reducing the degree of predictability of the halt from the perspective of informed traders.

A variety of trading halts are in place on exchanges around the world. ’ An important issue is the effect of such impediments to trade on market liquidity and trading costs. Thus, market closures have been widely researched by both theorists and empiricists in recent years. For example, on the empirical side, Lee, Ready, and Seguin (1994) and Ring, Pownall, and Waymire (199 1) investigate volume and price behavior around NYSE trading halts, while in an early paper, Kryzanowski (1979) investigates halts on the Toronto Stock Exchange. Theoretical analyses of trading halts include Slezak (1994) and Greenwald and Stein (1991). In these theoretical models all traders are assumed to be competitive and thus do not react to the closure strategically. In contrast, the goal of our paper is to explore the ex ante response of strategic informed traders to closure boundaries, and consequently the ex ante effect of closures on market liquidity. The paper, in essence, fills a gap in the literature in that much of microstructure in general Direct all correspondence to: Avanidhar Subrahmanyam of California at Los Angeles, Los Angeles, CA.

, Anderson Graduate School of Management,

Copyright 0 1997 by JAI Press Inc. 1058-3300

1

University

2

SUBRAHMANYAM

focuses on strategic informed trading, but the closure literature generally assumes competitive agents.2 Our analysis accords with Kryzanowski (1979), who finds significant abnormal returns both prior to and following firm-specific closures. This is suggestive of the notion that much informed trading takes place before closures are triggered. It is often claimed that closures disallow informed traders from profiting in times of extreme information asymmetry, and thereby protect small investors from incurring large losses during times of extreme information asymmetry. Thus, Greenwald and Stein (1991) state that “the primary function of a circuit breaker should be to reinform market participants,” while Lee, Ready, and Seguin (1994) argue that “by lowering informational asymmetries between traders, halts could permit the orderly emergence of a new consensus price.” In the spirit of the above statements, we assume that the policy goal of the exchange is to reduce trading costs for uninformed market participants during periods of extreme information asymmetry. This can be accomplished by instituting a closure rule if the bid-ask spread (or the price move) is higher than some threshold. We analyze the Stackelberg equilibrium in which informed traders act as leaders and market makers as followers.3 Following introduction of a closure rule, informed traders strategically choose their order size based on whether the price will hit the closure threshold if they trade a specific quantity. As in Easley and O’Hara (1987), we allow the informed to trade either large or small quantities. Easley and O’Hara show that informed agents generally prefer to trade large quantities in an unrestricted environment, as trading larger order sizes generally leads to higher profits. Consider, however, the effect of the closure rule on the informed agents’ trading strategies. Since an informed trader knows that trading large quantities will cause the closure bound to be crossed and make him lose his profit potential, he scales back his trading in response to the closure. What this ends up doing is increasing the spread on the small side (from zero to a positive number)! Thus, in the new equilibrium after the imposition of the closure rule, there is no spread on the large side of the market, but a positive spread on the small side of the market. A similar situation arises if one begins at a pooling equilibrium. In such an equilibrium, informed traders randomize between the small and the large quantities prior to the imposition of the closure. After imposition of the closure, they separate at the small quantities, thus again increasing the spread for small quantities and decreasing the spread for large quantities.4 It seems to us that our analysis has important policy implications. We show that the ex ante effect of trading halts can be (indeed, is likely to be) to reduce the welfare of the agents which they were intended to benefit, such as, the small retail investors. The analysis underscores the importance of explicitly considering the ex ante response of strategic agents to closure boundaries, as opposed to merely modeling them as competitive agents. It also is worth noting that the problems with trading halts that we outline are not easily addressed. If the closure bound is set too low, then the market always remains shut both for small and large orders. If it is set too high, the market never closes and the closure is ineffective. If it is set to be in an intermediate range, we have the scaling back effect discussed above, which increases the spread for retail investors. We show, however, that the probREVIEW OF FINANCIAL

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lem can be mitigated somewhat if the closure is discretionary, so that the closure boundary is not perfectly predictable from the perspective of the informed trader. This paper is organized as follows. Section I presents the analysis with a separating equilibrium. Section II analyzes the pooling equilibrium. Section III discusses randomized halting rules, while Section IV concludes. Proofs appear in the Appendix. I. A.

The Model and Its Separating Equilibrium

Separating Equilibrium

Without a Closure Rule

The model closely follows that of Easley and O’Hara (1987). Only the bid side of the market is analyzed; the analysis for the ask size is symmetric. There is a competitive, risk neutral market maker who executes exactly one order, which can either arise from a liquidity trader or an informed trader. All liquidity traders wish to trade one of two trade sizes S 1 and S2, with S lVL. Also, V*=~VL+(~-~)VH is the unconditional mean of the asset’s value, where 6 is the unconditional probability of the asset being worth VL. The robability of a small liquidity order is Xi while that of a large liquidity order is X,.!Z The probability of informational event occurring and thus revealing VH or VL to informed traders is a, and /.t is the market maker’s expectation of the fraction of trades made by the informed. We will assume that prior to the round of trade we analyze, there is no adverse selection, so the price is V*. Thus, all references to “price moves” in the paper are price changes with respect to V*. It is evident that the informed traders will trade either Sl or S2, as to trade any other quantity will identify them as informed traders. After the order, the market maker updates his belief about 6. We modify the equilibrium concept adopted by Easley and O’Hara (1987), in that in equilibria throughout our paper, we assume that market makers change their strategies in a consistent fashion in response to a deviation by an informed agent. That is, we assume that when an informed trader changes his strategy, the market maker realizes this and responds by changing his price to conform to the new probability of dealing with the informed. The informed trader takes this response into account while evaluating alternative trading strategies. This assumption is equivalent to postulating that the informed agents behave as Stackelberg leaders, while the market makers behave as Stackelberg followers. This is unlike Easley and O’Hara (1987), who analyze Nash equilibria, and thus assume that any deviation by the informed does not take into account the market maker’s response to the deviation. Our equilibrium notion is identical to that used by Admati and Plleiderer (1988) (see their discussion of equilibria with endogenous information acquisition), and that in Subrahmanyam (1994). Easley and O’Hara characterize two kinds of equilibria in the model. The first is a separating equilibrium in which the informed traders choose to trade large quantities. In this case, by a simple application of Bayes’ theorem, the updated 6 is given by: REVIEW OF FINANCIAL ECONOMICS, VOL. 6, NO. 1,1997

SUBRAHMANYAM

Using this, and the zero profit condition which requires that the market maker break even on each trade, one finds that the bid price if the informed trader trades S2 is given by:

b s2

=

E(V(S,)

=

v*-

SC1-WV,-

V&P (1)

Xs’(

1 - ap) + 6al.l

Note that in a separating equilibrium with the informed trading large orders, both the bid and ask prices for small orders equal V*. However, the bid price if the informed trader trades S 1 is given by S(l-6)(ri,b sl = E(VIS,)

V,)ap

= v* -

(2) X&l-cLp)+&xp

The market is in a separating equilibrium if and only if the profit to an informed trader from a small trade is smaller than that from a large trade, for instance,

Wbs2-Vd 1 Sl(b,l - V,) Substituting for proposition. 5

bs2 and b,l

(3)

from (1) and (2) above leads us to the following

Pl: Consider the game where the informed trader acts as a Stackelberg leader and the market maker acts as a Stackelberg follower. In a separating equilibrium without a closure, informed traders prefer to trade large quantities if and only if $2 >

q-

(Iap) + 6apl 2 XdXi (I- ap) + 6apl xkx:

(4)

is satisfied. B.

Trading Halts

Current “circuit breakers” on the NYSE mandate trading halts for a stipulated period of time if the Dow Jones Industrial Average (DJIA) moves by more than a certain amount over the previous day’s close.6 Further, trading halts in indiREVIEW

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vidual stocks are typically triggered by extreme order flow imbalances or the imminence of firm-specific informational events. While we define a mandated trading halt as one that is triggered on extreme price movements, the model may also be construed as applying to trading halts initiated by extreme order flow imbalances or extreme informational events. This is because extreme order flow realizations and the imminence of informational events both are typically accompanied by large price moves (see Lee, Ready, and Seguin, 1994), and floor brokers and specialists who maintain a continuous presence on the floor will have an accurate estimate of the degree of the order flow imbalance or required to trigger a halt. We discuss the implications of allowing exchange officials to randomize the halting rule in Section III. A Rationale for a ‘hading Halt. We consider a trading halt which is triggered if the price move (i.e., V* less the bid price) is greater than an exogenous bound C. That is, if a price move exceeds C, the market remains closed until the true value of V is revealed. Note that V* less the bid price measures the adverse price impact suffered by liquidity traders as a consequence of asymmetric information. Thus the rule we postulate, in effect, forces market closure when the degree of asymmetric information is high. Our closure mechanism thus is consistent with the view of several proponents of trading halts (e.g., Mann and Sofianos, 1990, and Schwert, 1990) that halts should help reduce trading costs for lessinformed market participants. The question naturally arises as to whether this closure rule is consistent with a rational trading decision by the liquidity trader. We now show that the answer to the above question is in the affirmative in a scenario where the liquidity trader faces execution price uncertainty. Such a situation is plausible insofar as monitoring market prices continously is prohibitively costly for liquidity traders. Thus, consider the trading decision of a liquidity trader wishing to trade X: and for whom the execution price is random. In a stylized sense, one can assume that the liquidity trader trades because he believes that the expected cost of trading is outweighed by the benefit from trading. If bs2 is the bid price and B the benefit from being able to trade, the liquidity trader will trade if:

[v* -

E( bszl L)] X: < B

where L denotes the information set possessed by the liquidity trader. Now, by instituting the closure bound to correspond to the condition V*-bt
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B=4.5. In this case, the average price impact faced by the liquidity trader is 4, which is less than the benefit B, so the liquidity trader will trade, though, if the execution price were known to him with certainty, he would prefer not to do so in the scenario in which the bid price turns out to be 5. Now suppose the exchange institutes a closure which states that the market will close any time the price move is greater than 4. In this case, when the bid price is 5, the market closes, while if bid price is 3, it remains open. It is thus evident that introduction of the closure prevents the liquidity trader from bearing extreme adverse selection losses in states of the world in which, ex post, the trader would find it suboptimal to do so. This illustrates how closures triggered on the basis of price moves (or bid-ask spreads) can be beneficial to liquidity traders. The above example illustrates that the closure rule is most effective if the benefit from trading B and the other exogenous parameters associated with informed trading are known. In reality, closure bounds are set ex ante and these parameters will not be known. Thus, in this situation, C has to be set based on probability distributions for these quantities. The tradeoffs are straightforward: if the closure bound is set too low, the market closes too often and the liquidity trader is precluded from trading at times when he would have found it optimal to do so, and if it set too high, the liquidity trader is made to trade at times when he would not have wished to do so, had he known the execution price with certainty. We do not explicitly model the choice of a closure bound by the exchange; this exercise adds little to the central issues we address in this paper. The goal of the above discussion is simply to rationalize the existence of trading halts based on their ability to reduce the ex ante trading costs of liquidity traders under execution price uncertainty. Equilibrium After Institution of the Halting Rule. Having discussed a rationale for rule-based closures, we now proceed to analyze their effects on the trading strategies of informed traders. After institution of the closure, each informed trader will strategically choose his quantity taking into account the possibility of the price crossing the closure bound if he trades a specific quantity. As pointed out earlier, in deriving equilibria with closure rules, we assume that when an informed trader considers switching his trade size in response to a closure rule, he takes the response of the market maker into account. Thus, if there are relatively more liquidity traders on the small side of the market, and the spread for large orders is sufficiently high, the informed trader will switch to trading the small quantity after the closure rule is imposed. This can be formalized in the following proposition.

P2: Consider the game where the informed trader acts as a Stackelberg leader and the market maker acts as a Stackelberg follower. Suppose that, without a closure, (4) holds, so that the market is in a separating equilibrium with the informed trading the large quantity Sp Then the following statements hold. REVIEW OF FINANCIAL

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1.

If

~(l-8)~~(VH-VL)>c>B(l-~)~~(vH-vL) X$.1 - ap) + 6ap

Xs'(l

(5)

- al.t) + 6ap

then, after imposition of the closure, the quantity traded by the informed trader in equilibrium becomes S 1. 2.

If C is greater than the LHS of (5), then the market never closes (the closure is ineffective), while if C is less than the RI-IS of (5), the market never opens.

3.

After imposition of the closure, if (5) is satisfied, then (i) the price impact of large orders goes to zero, (ii) the price impact of small orders increases (though the market remains open).

Thus the closure results in an increase in the spread for small quantites, which is exactly the opposite of what it is supposed to accomplish, for instance, “protect” the small, retail investors. Also, as Proposition (2) suggests, this effect is not easily addressed. If the closure bound is set below b,1, then the market always remains shut both for small and large orders. If it is set above bs2, the market never closes and the closure is ineffective. If it is set between bsl and bs2 then we have the scaling back effect discussed above, which increases the spread for small investors (i.e., the liquidity traders wanting to trade small orders). We have assumed the Stackelberg equilibrium concept here, as to us it seems conceptually most appealing. It is easy to verify that if V-b,z>C>V*-b,,, where bs2 and bsl denote the Nash equilibrium values of bs2 and b,1 (the analog of (5) for the Nash case), no Nash equilibrium in pure strategies exists when the closure is imposed.7 This is because trading small quantities is not a Nash equilibrium, as trading large quantities will yield a zero spread and hence higher profits (taking the market maker’s strategy as given). Further, trading large quantities is also not a Nash equilibrium, because the market will close and therefore the informed trader would like to switch to small quantities to capture at least some trading profits. For these reasons, we believe that the Nash concept is not the most desirable one for analyzing the issues addressed in this paper. The effects described above are distinct from the gravitational or magnet effects of trading halts that are often discussed in the literature (see, for example, Greenwald and Stein, 1991; Subrahmanyam, 1994; and Mann and Sofianos, 1990). A gravitational effect refers to individuals accelerating their trading in order to beat the trading halt, and thereby exacerbating volatility. In contrast, in the Stackelberg equilibrium we describe, informed traders scale buck their orders in order to prevent the price move from exceeding the closure bound, which increases trading costs for liquidity traders wishing to trade small orders. Indeed, in the gravitational effect model of Subrahmanyam (1994), traders’ strategic REVIEW OF FINANCIAL

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response to the closure results in increased ex ante liquidity, whereas in our model it decreases liquidity for small order sizes.8 II.

The Pooling Equilibrium

In our model, there also may exist a pooling equilibrium in which informed traders randomize between trading S2 and St. In such an equilibrium, the informed trader is indifferent between trading large and small orders, so that:

Wbpl - Vd = S2@p2 - V,)

(6)

where b,, and bp2 represent the bid price for the small and large quantities, respectively. Let w be the probability that the informed trader trades the small order. Then, using the zero profit condition and Bayes’ theorem, it can easily be shown that the spreads in the pooling equilibrium are:

$,I= &s,)v, + (1 - &s,>) v,

(7)

+ (1 - 6(s2))vH

(8)

bp2 = s(g,)v,

where S(S,) = G(ylpa + (1 - aj.t)X,l>/ (GcZ~/,l + (1 - ap) X,‘)

s = 6[( l-I@ua

+ (1 - wx;1/

mu - yap+(1 -

ap)X,21

The pooling equilibrium will exist if it is possible to choose a w between 0 and 1 such that (6) is satisfied. This leads us to the following proposition. P3:

Consider the game where the informed trader acts as a Stackelberg leader and the market maker acts as a Stackelberg follower. Suppose that one can choose a w between 0 and Z such that (6) is satisfted.9 It is possible to do this if

(9)

is satisfied. In this situation, the economy will be in a pooling equilibrium without a closure, with bid-ask spreads being given by (7) and (8). REVIEW OF FINANCIAL ECONOMICS, VOL. 6, NO. 1, 1997

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In this case, the following necessary conditions for the informed trader to separate at small quantities can be stated. P4:

Consider again the game where the informed trader acts as a Stackelberg leader and the market maker acts as a Stackelberg follower.

-

1. ZJfor a given equilibrium value of w, v*- b,_,z> C (without a closure rule), i.e.,

Gw,>-

a(1- wv(V,

Xi(1 - ap)

- V,) + 6al.l

aw,-

>C>

v;, >c

(10)

S(l - mwL(v~- V,)

(11)

Xi( 1 - czp) + Sal.4

then the informed trader switches to trading small quantities in a separating equilibrium when the closure is imposed. 2.

Zf the informed trader switches to small quantities, then the price impact for large orders goes to zero and the price impact for small orders increases afer imposition of the closure.

In other words, the informed traders switches from pooling to separating by trading small quantities if (i) the large quantity price move with pooling exceeds the closure bound and (ii) the closure bound is less than the price move caused by separating at the large quantity and greater than that caused by separating at the small quantity. Numerical Example: Consider the following parameter values: VL=O, VH=l, 6eo.2, S1=500, S2=1000, a=l, p~O.8, Xg =0.9, and Xz =0.4. Under these parameter values, (9) holds, and a pooling equilibrium exists. The (unique) equilibrium value of \v betwten 0 an2 1 )” this eq$li!tium is 0.:7. It is easy to ve,“fy that in this situation, V =0.8, V -b,=O.15, V -b, =0.48, V -b,l=O.38, and V -b&.53. This implies that for any closure bound greater than 0.53, the market never closes. For a closure bound less than 0.38, the market never opens, because neither the separating nor the pooling equilibrium can be sustained. The pooling equilibrium prevails if the closure bound is such that 0.48
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It can thus be seen that the closure has the same effect even when a pooling equilibrium prevails in the absence of the closure. Provided liquidity trading is more intense on the small side of the market, it benefits the informed trader to switch to trading small orders to avoid the closure bound being crossed. This widens the bid-ask spread for small investors. Thus again the imposition of the closure ends up hurting small investors. Note that this result is obtained ex ante, for instance, regardless of whether the closure is actually triggered. It is the strategic behavior of the informed to closure boundaries that causes the effect to obtain. Further, we reiterate that the closure bound cannot be adjusted effectively to prevent this effect from occurring. Either the closure bound is such that the market never opens, or it is such that the closure is not effective at all, or it is such that the spread for small orders actually widens. In the next section, we discuss one way to mitigate the above problem. Specifically, we argue that randomizing the halting triggger can reduce the parameter space under which the informed trader scales back his trades and causes bid-spreads for small orders to increase. III.

Rules Versus Randomization

A trading halt can either be rule-based (e.g., stipulating that the market will close when the price crosses a certain preannounced bound) or it can occur at the discretion of exchange officials. Thus far, our closure has been treated as rulebased, for instance, it was assumed that the trading halt occurs as soon as a publicly observable variable (the price move) exceeds a known bound. However, quite a few halts (e.g., the firm specific halts on the NYSE) occur at the discretion of exchange officials and the market makers. The issue we address in this section is the effect of introducing discretion (interpreted here as randomness) into the halting rule on informed trader strategies. To implement the discretion, the exchange can, for example, make diffuse statements about the imminence of a trading halt and vary the definitiveness of such statements. We find that introducing discretion actually reduces the parameter space under which the phenomenon of the spread increase for small orders occurs. For simplicity, we restrict ourselves to the separating equilibrium of Section I. Thus, assume that (4) is satisfied. We introduce discretion into the halting rule by assuming that the closure bound is random from the perspective of the informed trader. Thus, suppose that the closure bound can take on the value C1 and C2 with equal probability. Further, the mean closure bound e satisfies (5) with C replaced by e. Finally, assume that Cl satisfies (5) with C replaced by Cl, while C2 is greater than the LHS of (5). It is evident in the above scenario that if the closure bound were known to be e with certainty, from Proposition 2, the informed trader will choose to switch from trading the large quantity S2 to the small quantity Sl after the closure is imposed. However, if the closure bound is random with the above distribution, then the informed trader knows that if he continues to trade Sz, then, with a probability of l/2, he will realize a profit of S2(bsrVL), while with a probability of l/2, he will realize no profits. lo REVIEW OF FINANCIAL ECONOMICS, VOL. 6, NO. 1, 1997

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Thus, in this case, the informed trader will switch to trading Sr if and only if the condition Sl(b,I-VL) > 0.5Sz(b,rVL), which is equivalent to the condition

> 0.5s,

(1-6)(V,[

1

S( 1 - 6)ap

s,[ (1 -w+V,)-

x;(l-CXp)+6al.t

S( 1 - 6)al.t

vL)-

X,‘(l -al.t)+

6ap

1

Thus, as can be seen, with an appropriately randomized rule, the spread for retail investors increases under a smaller parameter space after the closure is imposed. This implies that discretionary, or more appropriately, randomized, trading halts have a potential advantage in that they may result in increased market liquidity under certain conditions. Of course, in the above example, introducing randomness reduces the efficacy of the closure in that with a probability of l/2, the closure is ineffective, so that the market does not close when the adverse selection is high. Ultimately, whether to introduce randomness into the halting rule depends on the relative weights placed by policy-makers on the reduced efficacy of the closure in shutting down the market when adverse selection is high, and the reduced parameter space under which informed traders scale back his trades and cause the spread for small orders to increase ex ante. Before we conclude, we briefly discuss some empirical implications of our analysis. Our model can be tested by extending the study of Lee, Ready, and Seguin (1994) to examine not just volume and volatility, but also spreads and the distribution of trade sizes around firm-specific trading halts. We predict a decrease in the effective spread (alternatively, the price impact) for large orders and an increase in the same for small orders. Our analysis also suggests a general shift of the distribution of trade sizes towards small orders prior to trading halts. Finally, our analysis also predicts a widening of quoted bid-ask spreads prior to closures, as such quotes are typically valid only for small orders. IV.

Conclusion

In a survey conducted by the New York Stock Exchange (1990), 50% of the investors surveyed regarded the stock market as being “erratic,” “unstable,” or “volatile.” The New York Stock Exchange thus took the position that “circuit breakers should help restore investor confidence during periods of large price swings by allowing time for a broader range of investors to participate in the market.” A particularly relevant statement is provided by Mann and Sofianos (1990) in support of trading halts “individual investors do not continuously monitor stock prices, so they are likely to react slowly when prices move fast. Slowing down trading may help reassure them.” Finally, Kryzanowski (1979) states that the goal of trading halts on the Toronto Stock Exchange is to “improve investor REVIEW OF FINANCIAL

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equity by ensuring that all investors have equal access to all relevant information about stocks they own or contemplate purchasing or selling.” In this paper, we have shed light on the efficacy of closures in attaining the above policy objectives, by analyzing the effect of closures on trading costs for small, retail investors. In the context of a Stackelberg game in which informed agents act as leaders and market makers as followers, we have argued that explicit consideration of strategic behavior by informed traders near closure boundaries has important regulatory implications. Thus, we show that closures may have the ex ante effect of increasing the spread for small retail investors, which is exactly the opposite of what policy-makers would like to have happen. It is important to note that closures do not have to be actually triggered for the effects of this paper to obtain. The mere fact of an impending closure can cause the informed trader to scale back his order and cause the spread for small orders to increase. We show, however, that this effect can be mitigated somewhat by having randomized closure boundaries, which cause the informed trader to scale back his trades under a smaller parameter space. Appendix Proof of Proposition 1: This proposition follows by substituting for bs2 andb,l (3).

from (1) and (2) into

Proof of Proposition 2: Now, note that the condition

is equivalent to the condition V*-bs2 >CV*-b,l . If this condition holds, the market does not close if the informed trades S,, but closes if he trades S2. Thus, if the informed trades S2 he makes zero profits, but he makes a positive profit if he trades Sl. Since the informed trader acts as a Stackelberg leader and therefore takes the market maker’s response to his strategy into account, the optimal strategy for the informed trader is to trade Sl if (5) holds. Also, in this scenario, since there is no informed trading on the large trade side of the market, the spread is zero for a trade size of S2, while the spread on the small side is given by (2). The other portions of the Proposition follow in a straightforward fashion. That is, if C is greater than the LHS of (5), the closure bound is never triggered, because no matter what quantity the informed trades, the price move is smaller than that required to close the market, while if C is less than the RHS of (5), no matter what quantity the informed chooses to traded, the closure is triggered. REVIEW OF FINANCIAL

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Proof of Proposition 3 This proposition follows by subsitituting for b 1 and b 2 from (7) and (8) into

(6) and deriving the condition for w to be between 8 and 1 for the resulting equality to be satisfied. Proof of Proposition 4

If (10) is satisfied, the price move on the large side in a pooling equilibrium triggers the closure and thus causes the informed trader to make zero profits on the large side but positive profits on the small side. In this case, (6) does not hold, thus destroying the pooling equilibrium. Further if (11) (which is identical to (5)) is satisfied, the closure is not triggered if the informed trades S1, but is triggered if the informed trades S2. Thus, the optimal strategy for the informed is to switch to separating at S 1. Acknowledgment:

I thank two anonymous referees for useful comments, and Matthias Schaefer for research assistance. All errors are my sole responsibility. +< 8

Notes 1. For example, the NYSE has both market-wide and firm specific closures in place. Further, the NASDAQ, the London Stock Exchange, and Toronto Stock Exchange all have firm-specific suspension rules in place. Subrahmanyam (1994) analyzes the strategic response of liquidity traders to closure bounds, whereas our focus is on the strategic behavior of informed traders around closure boundaries. Part of the reason for using this equilibrium concept is that, in many instances, a Nash equilibrium in pure strategies does not exist when a market closure rule is in place. We assume here that uninformed traders do not monitor the market continuously. Allowing some uninformed to strategically choose their order sizes in response to market closure rules will not affect our intuition, which mainly focuses on the hitherto unaddressed issue of the strategic response of infomed traders to closure boundaries. 5 Note that in inequality (3). we use b,, instead of V* on the right-hand side to conform to the Stackelberg assumption. Under the Nash assumption, the informed trader would not take the market maker’s response to his deviation into account, so that the, right-hand side would be St(V*-V,) (as in Easley and O’Hara, 1987). 6. The current rule involves halting trade in all stocks for one or two hours if the DJIA moves by more than 250 or 400 points respectively over the previous day’s close. See Mann and Sofianos (1990) for a detailed description of this and other proposed circuit breakers. 7. Consideration of mixed strategies does not alter the intuition behind the arguments. Mixed strategies are addressed in the next section. 8. It is also worth discussing whether our results are robust to alternative model specifications. While the issues we have addressed are difficult to model in Kyle (1985)-type frameworks (because of the truncated distribution of the price which causes one to lose linearity), the intuition appears to be robust. Thus, in the Admati and Pfleiderer (1988) adaptation of the Kyle model, competing informed traders would scale back their trading towards the quantity traded by the monopolist to reduce the probability of the closure bound being crossed, which would cause market liquidity to decrease, just as in our model. Of course, unlike Admati and Pfleiderer (1988), we do not analyze discretionary liquidity traders. However, it is unlikely that retail investors have the discretion to switch to trading large orders (e.g., blocks); our goal is precisely to consider the effect of closures on such nondiscretionary traders. REVIEW OF FINANCIAL

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9. It is easy to verify that such a t+r.if it exists, is unique. 10. We assume here that the closure is triggered after the bid price for the trade is established, and if V* less bid price exceeds the closure bound, the trade is canceled, and the market is closed till the value of V is revealed. Alterations in this specific mechanism will not affect the basic nature of our results. 11. In a significant paper, Gerety and Mulherin (1992) show that closing volume on the NYSE is positively related to expected overnight volatility and argue that the desire of investors to accelerate trade before risk periods indicates a cost of mandated circuit breakers. Our analysis is not readily linked to theirs, however, because we address the issue of how informed traders can influence rule-based halts by strategically changing their order sizes. Our intuition thus does not apply to halts whose timing is fixed and not influenced by endogenous variables such as order flows or price moves.

References Admati, A., and P. Pfleiderer. 1988. A Theory of Intraday Patterns: Volume and Price Variability, Review of Fintincial Studies, 1: 3-40. Easley, D., and M. O’Hara. 1987. Price, Trade Size, and Information in Securities Markets, Journal of Financial Economics,l9: 69-90. Gerety, M., and J. Mulherin. 1992. Trading Halts and Market Activity: An Analysis of Volume at the Open and the Close, Journal of Finance.47: 1765-1784. Greenwald, B., and J. Stein. 1991. Transactional Risk, Market Crashes, and the Role of Circuit Breakers, Journal of Business,64: 443-462. King, R., G. Pownall, and G. Waymire. 1992. Corporate Disclosure and Price Discovery Associated with NYSE Temporary Trading Halts, Contemporary Accounting Research,8: 509-53 1. Kyle, A.S., 1985, Continuous Auctions and Insider Trading, Econometrica,53: 1315-1335. Kryzanowski, L. 1979. The Efficacy of Trading Suspension: A Regulatory Action Designed to Prevent the Exploitation of Monopoly Information, Journal of Finance,34: 1187-1200. Lee, C., M. Ready, and P. Seguin. 1994. Volume, Volatility, and NYSE Trading Halts, Journal of Finance,49: 183-2 14. Mann, R., and G. Sofianos. 1990. “Circuit Breakers” for Equity Markets. In Market Volatility and Investor Confidence, El-E34. New York: New York Stock Exchange. New York Stock Exchange. 1990. Market Volatility and Investor Confidence. New York: New York Stock Exchange. Schwert, G.W. 1990. Stock Market Volatility. In Market Volatility and Investor Confidence, Cl-C24. New York: New York Stock Exchange. Slezak, S. 1994. A Theory of the Dynamics of Security Returns Around Market Closures, Journal of Finance,49: 1163-1211. Subrahmanyam, A. 1994. Circuit Breakers and Market Volatility: A Theoretical Perspective, Journal of Finance,491 237-254.

REVIEW OF FINANCIAL

ECONOMICS,

VOL. 6, NO. 1,1997