The information content of price limit moves.

Belcher, Larry ; Ma, Christopher K. ; Mallett, James E. 等

ABSTRACT

An asset's price in an environment with price limit rules can be replicated by the price of a portfolio consisting of a riskless asset and two synthetic options. A procedure is used to unbundle the unobservable option values imbedded in the actual futures price and impute a theoretical true futures price. Under this framework, evidence from the Treasury Bond futures market suggests that theoretical true futures prices diverge from actual futures prices, on average, three hours prior to the activation of price limit rules, indicating that price limit moves might be predictable. As the magnitude of the difference between the theoretical futures prices and the actual futures prices is significantly larger for limit moves resulting in trading halts for the entire trading day, as compared to limit moves on trading days in which trading resumes, intraday trading halts and consecutive daily limit moves can also be predicted. The reversal of both the actual futures prices and the theoretical futures prices back within the limit range after a limit move provides support for the possibility that traders tend to overreact when market prices are near price limits.

JEL: G12, G13, G14

Keywords: Price limits; Futures markets; Trading halts

I. INTRODUCTION

Financial market crashes, such as those of October 19th, 1987 and October 27th, 1997 have led to discussions regarding the effectiveness of different methods of market discipline (Barro, 1989; The Brady Commission, 1988; Garg, Kim and Swinnerton, 1999). After the 1987 crash, circuit breakers were implemented to foster market discipline, but the 1997 crash cast doubt on their effectiveness (Ip, 1997). This led to modifications of the breakers by the New York Stock Exchange in 1997.

Circuit breakers can take different forms, including trading halts triggered at certain price limits, position limits, and trading halts arising from significant imbalances between buy and sell orders (Moser, 1990). The most commonly mentioned form is the price limit-a trading halt based on either an up or down price movement, to a pre-established level (the limit price).

II. DEBATE ON PRICE LIMITS

One potential benefit ascribed to price limits is that such measures serve to limit credit risk on the part of market participants and aid in mitigating the loss of financial confidence by providing a period to settle-up and ensure solvency. (1) That is, price limits may serve to constrain the daily financial exposure of trading by providing a ceiling on the dollar amount of margin calls as a result of the day's trading. Such measures are often considered to protect the market from overreacting to news events, particularly during periods of significant uncertainty. The overreaction hypothesis is consistent with the notion that prices can move beyond equilibrium values, but will eventually reverse themselves as traders sort through the information. In this sense, price limits may aid the market in the price discovery process by allowing the market to pause and cool off. (2) Further, it has been claimed that price limits counter the illusion that markets are perfectly liquid and can absorb massive one-sided volume. Thus, limits may serve to slow down certain trading strategies, which can be disruptive not only to the institutions employing them, but also to the market. (3)

On the other hand, critics argue that price limits act as a barrier to market clearing. Traders willing to trade may be unable to do so because the true price lies outside the price limit in effect on that day. Market participants may also not be able to liquidate their positions or establish hedge positions at prices advantageous to both sides, due to the imposition of price limits. Therefore, long (short) positions in a downward (upward)-moving market face liquidity problems because of the potential inability of buyers (sellers) to enter the market. An additional problem is that limits can disrupt spot and futures prices co-movements, increasing the price risk for hedgers.

Opponents of price limits further argue that they serve no purpose other than to slow down or delay the ultimate price change. (4) This view regards price limits as impeding, rather than enhancing, the price discovery process. Rather than stabilizing price changes, price movements will continue to move in the direction of the true price as new trading limits are established in subsequent trading periods. Finally, it has been argued that price limits tend to be self-fulfilling. Because fear of illiquidity and being locked-in to a position, traders rush to cover themselves through active trading; volume would be heavy and the price limit would thus serve as a "magnet" drawing the price closer to it. (5)

On this debate, Brennan (1986) argues that futures traders may decide to renege on contracts if the price change allowed results in a loss greater than the required margin. If the price change is limited, however, then traders are uncertain about the true loss, and the probability of default is reduced. In this sense, the price limit serves to create "noise" in the trader's forecast of the true futures price. However, this function is useless if traders can obtain additional information regarding the true futures price, such as through costless arbitrage between the spot and futures markets. Thus, Brennan predicts that price limits in markets that have a high correlation between spot and futures prices are redundant.

III. PRICE LIMITS IN FUTURES MARKETS

Given that serious disagreements concerning the effectiveness of price limit rules exist, there is surprisingly little published research in the area of price limits. We will examine two hypotheses about the effect of price limits in futures markets. Consider a market where the futures price and its volatility are driven by the arrival of information and all traders have access to and process this information in such a way that every trader knows the true or expected true futures price. If a limit is present in the market and the true price falls outside of the current day's price limit, then the futures price moves to the appropriate limit and trading will suspend. Trading then resumes (perhaps the next day) when the true price once again falls within the allowable trading range. In this scenario, the futures price and its volatility should reflect no information during the limit move; the price limit simply delays trading until a time when the true price falls within the allowable trading range. Because the futures price cannot adjust to the new true level during the limit move, the process of price resolution is impaired and traders bear the liquidity and price risk of unhedged positions.

Alternatively, consider the same market except that traders do not process information in an efficient fashion, but rather overreact to new information. In this case, the market presents a noisy estimate of the true price since the market price might react erratically to new information. The activation of the price limit suspends trading and allows market participants additional time to evaluate new information. During this reassessment period, market supply and demand schedules can be revised and trading then resumes at a more equitable true price level. During the limit move, volatility is reduced and the futures price can adjust to the pre-limit price range, thus improving the process of price resolution.

This provides us with two explanations of the effect of imposing price limits, based on the underlying causes of market volatility. The information hypothesis asserts that price levels and their variances are driven by the arrival of public or private information. The implication is that trading will be halted when the limit is reached and the true price, as determined by new information, lies outside of the tradable range. The overreaction hypothesis asserts that price and volatility are affected by trading noise. The limit and trading halt allows noise to clear, resulting in a resumption of trading after the noise clears.

In one early direct test of the two competing hypotheses described above, Ma, Rao, and Sears (1989a, 1989b) found that in the Treasury Bond futures market, (1) futures price levels stabilized following up-limit moves; (2) futures price patterns reversed following down-limit moves; and (3) volatility declined significantly following both types of limit moves. In their analysis of circuit breakers, Greenwald and Stein (1991) provide similar results consistent with the overreaction hypothesis. In contrast, McMillan (1991), in a study of the S&P 500 futures market, found that circuit breakers might instead hinder information flows, lending support to the information hypothesis.

Hall and Kofman (2001) investigated the influence of market microstructure issues with regard to futures markets with price limits by proposing a framework to distinguish between the overreaction and information hypotheses. By comparing observed futures prices with theoretical futures prices (as determined by a cost-of-carry model), they find an S-shaped relationship in one grain market, indicating that price limits may stabilize prices as traders anticipate the potential effects of price limits. Their results in other agricultural markets were not significant.

IV. MODELING FUTURES PRICES IN A PRICE LIMIT ENVIRONMENT

Since trading halts result in significantly higher liquidity costs, it is extremely useful to distinguish the underlying nature of the limit moves as a result of information flow or overreaction. Chance (1994) developed a model to compare futures prices in the presence of price limits with futures prices imputed by a cost-of-carry model. This approach assumes that the futures price eventually converges with the cash price at settlement, i.e. limits are removed at some point. Thus, the difference in theoretical and observed prices was based on the interest amounts due to marking to market. Ackert and Hunter (1994) also compared observed futures prices with theoretical futures prices. Their model viewed the exchange as owning a call option against long positions and a put option against short positions (a straddle) to test the appropriateness of actual price limit levels, based on trading off the benefits of reduced margins with the costs associated with trading interruptions.

The evidence regarding price limits and price overreactions is mixed. Chen (1998) notes that, while not ruling out overreaction by market participants, the direction of price movements on the day after a big price swing is not predictable. Hall and Kofman (2001) find some evidence of a stabilizing effect of price limits. Further research into these competing theories may help further this discussion of the effectiveness of price limits in futures markets.

V. MODEL DEVELOPMENT

A formal model of asset price movements when price limits are present is presented in Holder, Ma, and Mallett (2002). Readers are referred to that paper for more details. In the model, a long position in the futures contract is replicated by a synthetic position consisting of investment in a riskless asset and long a call futures option with strike price equal to [F.sub.c]--daily limit and a short call futures option with strike price equal to [F.sub.c] + daily limit, with [[F.sub.c] the closing price of the previous trading session. This yields a payoff profile that is essentially the same as the futures contract, even on limit move days. If price resolution is functioning fully during limit moves, traders will price the true futures price, regardless of where the price lies, into the options imbedded in the futures price movements. This is restated as follows:

A. Pricing Proposition

In markets with price limit rules, the actual futures price movement is equivalent to that of a portfolio investing in a riskless asset for the amount of the discounted difference between the close price of the previous trading session and the limit size and in a spread position of the synthetic futures options. That is,

[F.sub.t] =([F.sub.c] -L)[e.sup.-rt] + CI([F.sub.t.sup.*], [sigma], K1,t,r)-C2([F.sub.t.sup.*], [sigma], K2, t, r) (1)

where r is the risk-free rate, [F.sub.t] is the futures price at time t with the limit rule in effect, [F.sub.c] is the closing price of the futures contract in the previous trading session, and [F.sub.t.sup.*] is the futures price at time t without the price limit rule. L is the magnitude of the daily limit. C1(.) is the premium of the call option with the strike price, K1, and C2(.) is the call option with the strike price, K2, where K1 = Fc - L, and K2 = Fc + L.

These theoretical futures prices, even though they are not observed, can be imputed even when a trading halt occurs. This allows the testing of the information and overreaction hypotheses when the theoretical futures price is examined around limit moves. One can also test whether the expected futures price can predict the outcome of limit moves.

VI. EMPIRICAL DESIGN

A. Data

The sample used in this study is the same one employed in Holder, Ma, and Mallett. It is tick prices from the Time and Sales File compiled by the CBOT. The sample period used in this study extends from January 1, 1980 to December 31, 1988. Table 1-A presents the sample in detail. On average, there are 253 trading days each year. Around 13 different contracts are traded in a typical year, with about 200 trading days per contract.

B. Identifying the Limit Move Days

Based on Regulation 1008.01 of the Chicago Board of Trade, trading is prohibited in a futures contract when trading on the exchange at a price higher or lower than plus or minus the predefined limit of either (1) the settlement price for such commodity on the previous day, or (2) the average of the opening range of prices or the first trade during the first day of trading in a futures contract. Prior to 1979, the price limit for Treasury Bond futures contracts was 0.75 points; in 1979, it became 1 point. It was raised to 2 points in 1980 and 3 points in September 1988. Furthermore, a variable limit is applied if three or more contracts within a contract year close on the limit bid or on the limit sell, for three successive business days. (6) The limit for the following three business days is expanded to 150 percent of the original limit. Limits are lifted the second business day proceeding the first day of the delivery month. Additionally, it should be noted that accounts at the exchange are marked-to-market each day. While the account is not closed at the end of each day, the marking-to-market process effectively settles each account at the closing price for each trading day.

The imposition of margin requirements can be considered a substitute for price limit rules; however, this complication is reduced by eliminating the day of and days surrounding changes in the margin requirement from the original sample. Days after the lifting of limits in the delivery month are also removed, since limits are not in effect for the delivery month of a contract. Using the above rules, all limit moves for each year in the sample are identified in Table 1-B. There were a total of 111 calendar days when price limits were activated, of which 63 days were up limit moves and 48 days were down limit moves. In total, there were 358 limit moves on the 111 different calendar dates, since multiple limit moves can occur during the same day and across different contract expirations. In years 1980, 1986, and 1987 limit moves occurred frequently, while there were few instances of limit moves in the years 1981 through 1984.

The time series pattern of the daily limit moves is reported in Table 1-C. It appears that while about eighty percent of the limit moves occurred on a single day, there were successive limit moves on 24 of the 111 days. Of those 24 successive limit moves, 20 were two-day limit moves and 4 were three-day limit moves. This suggests that limit moves, in general, do not occur as a cluster.

It has been argued that limit moves often result in trading halts for the rest of the day. This observation requires the examination of the microstructure of the intraday price movements in a typical limit move day. Table 1-D, shows that there are, on average, 500 different price changes in a typical limit move day. The first limit move occurs almost at the middle of these different price changes. This implies that the first limit move does not always result in a trading halt for the day. In fact, only 49 out of the 203 up-limit moves and 42 out of the 155 down-limit moves are associated with complete trading halts for the day.

Although a significant proportion of the limit moves result in trading halts (25%), this observation may be derived from the bias due to the timing of the first limit move, especially if the first limit move occurs later in the day. Therefore, it is necessary to examine the intraday timing of the first limit move. In Table 1-E, it is clear that the first limit move most likely occurs either during the beginning or end of a trading session. Approximately 42 percent of the first limit moves occurred near the end of the day (between 1:00 p.m. and 3:00 p.m. E.S.T.). This percentage is much higher than the proportion of the trading halts in the sample. This high proportion of late occurrences of first limit moves would result in a pattern of trading halts after the limit move, since the trading is closer to ending.

C. Imputing the Theoretical True Futures Prices

For each limit move day identified from the previous section, the Pricing Proposition (equation (1)) can be rewritten as follows:

[F.sub.t] = ([F.sub.c] - L)[e.sup.-rt] + [[F.sub.t.sup.*]N(x1) - ([F.sub.c] - L)[e.sup.-rt] N(x1 - [sigma][square root of t]]

-[[F.sub.t.sup.*]N(x2) - ([F.sub.c] + L)[e.sup.-rt] N(x2 - [sigma][square root of t]] (2)

where

x1 = log [F.sub.t.sup.*] - log {([F.sub.c] - L)[e.sup.rt]}/[sigma] [square root of t] + 1/2 [sigma][square root of t]

x1 = log [F.sub.t.sup.*] - log {([F.sub.c] - L)[e.sup.rt]}/[sigma] [square root of t] + 1/2 [sigma][square root of t]

[F.sup.*.sub.t] is the theoretical true price, [F.sub.t] is the actual futures price, N(.) is the c.d.f. of the normal distribution, and [sigma] is the volatility of the futures return. Equation (2) assumes that both options C1 and C2 are priced by the Black-Scholes Option Pricing Model. In the current context, one useful implication from this formulation is that the theoretical true futures price, [F.sup.*.sub.t], while not directly observable, can be unbundled from the actual futures price, [F.sub.t]. This procedure is similar to the standard methodology of estimating implied volatilities of options.

Holder, Ma, and Mallett (HMM) utilize an iterative search procedure to simulate the average actual and implied theoretical futures price changes. Based on an evaluation of the price data, they found significant differences between the theoretical and actual futures price changes for both up and down limit moves. The implication of this is that the theoretical prices, when limit rules are in effect, provide predictive power as to the occurrence of first limit moves. We wish to address this information content relative to the information and overreaction hypotheses of price limit movements.

D. Overreaction versus Information Flow

The cause of limit moves, either information flow or overreaction, can be examined by observing price movements following the limit moves. If the limit moves are a result of new information arrival, the fact that prices hit the limit does not eliminate the impact of the new information. Therefore, the post-limit move futures price should be beyond the limit at its true level. On the other hand, if limit moves are caused by short-term overreaction, the limit provides a cooling environment and the true theoretical futures prices should revert back to the true level. The following proposition presents a testable hypothesis, based on the implied theoretical futures prices and actual futures prices from the HMM model:

[FIGURE 1 OMITTED]

1. Limit Move Proposition

If the limit moves are caused by traders' short-term overreactions, then post-limit moves of the true theoretical futures price should occur within the limit range. On the other hand, if the limit moves are caused by the arrival of new information, then post-limit moves of the true theoretical futures price should consistently be outside the limit range.

The testing of such differences may not be complete by simply observing the ex post, actual futures price movements following the limit moves, since the actual futures prices are not allowed to move outside the range in the same trading day by design, thus the test may suffer from some sample bias. However, the theoretical true futures price is not restricted by the limit range, thus providing a valid test of the two causes. In Figure 1 and Tables 2-A and 2-B, both the actual and theoretical true futures prices revert back to the limit range after first limit moves. Moreover, even within the limit range, the theoretical futures prices are still significantly lower (higher) than the actual futures prices after the down (up) limit moves, at the 1 percent level. In short, there is evidence to suggest that the majority of the limit moves in the sample are caused by overreaction. Correspondingly, Arak and Cook (1997) and Park (2000) also find data supporting price reversals after limit moves.

E. The Theoretical Price as an Unbiased Predictor of True Prices

The relationship between futures prices and the expected future price of an asset has most often been investigated within the context of market rationality. This view assumes that agents make full use of available information. Thus, the price of an asset or futures contract at any moment should be an unbiased predictor of the price at a future moment. This implies that future price changes are not correlated with current prices. This is a necessary condition for rationality, since if it were not true, agents would learn from past prices and us this to forecast future prices to increase their wealth, a violation of weak-form efficiency (Fama, 1989). Symbolically, the current futures price, [F.sub.t] can be demonstrated by the following:

[F.sub.t] =E([F.sub.t+n] | [[phi].sub.t])+ [[epsilon].sub.t] (3)

where E([F.sub.t+n]) is the rational or efficient forecast, conditional on all information available at t, [[phi].sub.t]. As the rationality condition requires that the information sets [[phi].sub.t], and [[phi].sub.t+n] are not correlated, the expectation of future spot prices is formulated assuming the forecast error, [[epsilon].sub.t], follows a random walk, i.e., identically and independently distributed (i.i.d.). (7)

F. Rationality Proposition

The closing theoretical true futures price on the days of limit moves is an unbiased predictor of the open futures price on the following trading day.

Testing the unbiasedness hypothesis involves estimating the following relationship:

[F.sub.0] = [alpha] + [beta][F.sub.t.sup.*] + [e.sub.t] (4)

where [F.sub.0] is the open price for the day following a limit move day, and [F.sup.*.sub.t] is the closing theoretical true price of the limit move day. The acceptance of the null hypothesis: ([alpha],[beta]) = (0,1) implies rational information content of the theoretical true futures prices.

To test the unbiasedness hypothesis, equation (4) is estimated. The robustness of the information content of the theoretical true futures prices can be further tested by showing that the acceptance of the unbiasedness hypothesis is unique to days of limit moves. It is conceivable that the actual closing price of the current day is an unbiased predictor of the open price of the following day, despite any limit moves. Therefore, the estimation of equation (4) is replicated on several control samples. In addition to the estimation of equation (4), the actual closing price, [F.sub.c], of the limit move day is used as the proxy for the true price. Secondly, the above two tests are repeated using a control sample of days of with no limit moves.

Further testing of the close and the next trading day's open prices indicates the presence of unit roots. However, these series are co-integrated, thus the most appropriate method to test the null hypothesis is OLS. In Table 3, the results of the four tests are presented. For all four tests, the only case where the unbiasedness hypothesis cannot be rejected is when equation (4) is estimated in the limit move sample. In all other cases, the null hypothesis of an unbiased predictor of the opening price on the following day is rejected. The implication is that in days experiencing limit moves, the theoretical true futures price provides sufficient and unbiased information regarding the actual price in the future.

G. Successive Daily Limit Moves

The information content of limit moves in days followed by successive limit moves could be different from that of limit moves in days not followed by successive limit moves. Essentially on a retrospective basis, the difference between the theoretical true futures prices and the actual futures prices should be more pronounced, providing information for the resulting successive limit move days. Therefore, the following proposition can be derived accordingly:

1. Information Proposition 1

The difference between the theoretical true futures prices and the actual futures prices is significantly higher during limit move days that result in successive limit move days than on single limit move days.

For the purpose of distinguishing singular and successive limit moves, the relationship between the theoretical true futures prices and actual futures prices is compared between two such cases. The entire sample is first divided into a subsample of single limit move days and another subsample of successive limit move days. The mean price change is computed as before for both samples and summarized in Table 4. The t-tests in this instance are conducted to compare the magnitude of the price difference between the two subsamples. The comparative pattern is presented in Figure 2.

The pattern is very different between the two samples. From Figure 2, the magnitude of the theoretical futures price changes for the sample of successive daily limit moves are clearly larger than that of the sample of single day limit moves. From Table 4, the price changes are statistically different between the two subsamples at all time points observed. It is also interesting to note that there is a sharp contrast between the two samples in the post-limit move price movements. Apparently for the case of successive up limit moves, the theoretical prices after the first limit move stay outside the range for the rest of the day, while the same pattern for the down limit moves is much weaker. This is consistent with the information hypothesis that if the true prices are outside the limit range, the limit moves cannot eliminate the impact of information, resulting in consecutive daily limit moves.

[FIGURE 2 OMITTED]

H. Intraday Trading Halts

A similar test on the information content of theoretical futures prices, contrasting the information versus overreaction hypothesis is conducted using a second approach. The same argument in the previous section can apply to the often-observed cases of intraday trading suspension after the first limit move. The information hypothesis argues that theoretical futures prices provide information content about the actual prices. Thus, significant deviation of the theoretical price from the actual price would imply that trading would be suspended until the new limit price range is established the next day. However, if the limit moves are caused by overreactions, the significantly different theoretical futures prices provide no information content other than short-term irrationality. Thus, trading would resume for the rest of the day after the first limit move. Therefore, the testable hypothesis is that the mean price difference in days of trading halts should be significantly higher than that of other limit move days. In short, the intraday price movements can be described by the following proposition:

2. Information Proposition 2

The difference between the theoretical true futures prices and the actual futures prices is significantly higher on limit moves days which result in trading halts after the first limit move than on limit move days in which trading resumes after the first limit move.

To make such comparisons, the original sample is again divided into two subsamples. The first sample includes all limit move days in which the first limit move results in trading halts for the rest of the day. The second sample has the days where trading resumes after the first limit move. The comparisons are conducted in a fashion similar to the previous section. Since there are no observations after the first limit move due to the trading halts, only the comparisons to the time point 10 minutes after the first limit move are reported. In Table 5 and Figure 3, the theoretical prices are compared with the actual prices for the sample of the trading halts. The pattern is very similar to those reported in Tables 2-A and 2-13. However, the difference between the two prices appears to be larger in this sample at the time of the first limit move. Therefore, in Table 6, the theoretical prices are compared between the two samples. Figure 4 also demonstrates the theoretical futures price changes for both samples. In the case of up limit moves, theoretical prices prior to the first limit moves are indicative for the occurrence of trading halts, while there is a reversal in the price difference immediately prior to the down limit moves which also end in trading halts. However, especially closer to the time of the limit moves, the magnitude of the imputed price changes is always significantly larger in cases of trading halts, compared with "regular" limit move days.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

VII. CONCLUSIONS

By modeling futures prices under price limits by a synthetic futures options portfolio, a set of implied theoretical futures prices can be "unbundled" from the options prices. Using these prices, it can be shown that the existence of first limit moves can be predicted. The prices can also be used to predict intraday trading halts and successive limit moves, based on the magnitude of the deviation of actual futures prices from the true futures prices. This supports the information hypothesis. Post-limit futures prices were also found to be unbiased predictors of opening futures prices in the following trading session, also lending support to the information hypothesis. Finally, after the first limit move, both the true and actual futures prices were found to revert back to the trading limit range. This result lends credence to the overreaction hypothesis.

APPENDIX I
The payoff pattern of the portfolio of futures contracts under price
limit rules

 Current time

Portfolio Cash Flow [F.sup.*]
 < Kl
Portfolio A

Buy a call at K1 -C1 0

Write a call at K2 +C2 0

Lend amount -([F.sub.c] ([F.sub.c]-
 -limit) limit)
 [e.sup.-rt]

Portfolio B

Long futures +[F.sub.t] -([F.sub.c]-
contract limit)

Total -Cl+C2- 0
 ([F.sub.c]-
 limit)
 [e.sup.-rt]+Ft

Portfolio Value at expiration

Portfolio A K1 < [F.sup.*] K2 < [F.sup.*]
 < K2
Buy a call at K1

 [F.sup.*]- [F.sup.*]-
 ([F.sub.c]- ([F.sub.c]-
 limit) limit)

Write a call at K2 0 -[F.sup.*]-
 ([F.sub.c]+
 limit)

Lend amount ([F.sub.c]- ([F.sub.c]-
 limit) limit)
Portfolio B

Long futures -[F.sup.*] -([F.sub.c]-
contract limit)

Total 0 0

ACKNOWLEDGMENTS

This paper benefited from numerous discussions with Robert Daigler, Marcelle Arak, Gerald Gay, Donald Chance, and Frank Fabozzi. The comments of Franklin Edwards, Gregory Kuserk, Richard Roll, Bruce Lehmann, and Merton Miller in the Regulatory Reform Conference on a number of points raised in this paper and in an earlier paper are especially appreciated.

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NOTES

(1.) Regarding the issue of solvency and protection against credit risk, Brennan (1986) notes there may be some redundancy since margins and price limits in futures markets, to some degree, serve the same purpose. The issue of establishing the proper level of limits in stock index futures markets, relative to existing margin requirements, also caught the attention of Miller, Scholes, Malkiel, and Hawke (1987) and Ackert and Hunter (1994). In any event, if one purpose of price limits is to provide protection against credit risk, it would seem that such measures need to be coordinated with existing margin levels.

(2.) French and Roll (1986) also note that noise is associated with excess volatility and as noise begins to dissipate, volatility should return to its normal level. Proponents of the overreaction hypothesis thus argue that price limits could be beneficial in controlling excessive volatility as well as unwarranted price movements.

(3.) Price limits are also a competitive element among exchanges in the design of contracts (Silber, 1981).

(4.) This view has many proponents in the academic community (e.g., Fama (1989), Meltzuer (1989), Chance (1994), and Telser (1981)). In related evidence on this view, Roll (1989) did a comparative study for the cash equity markets of 23 countries and found that, after controlling for differences in price volatilities, price limits had no differential effect on the rate of decline in prices during a market crash.

(5.) The magnet concept is discussed by Arak and Cook (1997) and Fama (1989).

(6.) Recently, the CBOT has dropped price limits for Treasury futures contracts, but still maintains price limits in agricultural contracts.

(7.) Using a typical development of sigma-fields and conditional expectations, we can test for a rationality hypothesis by comparing the price relationship with respect to different information priors at various points of time. For a detailed discussion of probability theory and sigma fields see Baxter and Rennie (1996).

Larry Belcher (a)

Christopher K. Ma (b)

James E. Mallett (c)

(a) Chairman of the Department Finance and Director of the George Investments Institute, Stetson University

(b) Roland & Sarah George Chair of Investments, Stetson University, and Principal of KCM Asset Management, Inc.

(c) Professor of Finance, College of Business Administration, Stetson University

Table 1-A
Sample descriptions of the Treasury bond futures contracts

 Number of Number of Trading Days
 Year Trading Days Contracts Per Contract

 1980 252 14 188
 1981 253 13 207
 1982 251 13 191
 1983 253 14 181
 1984 253 14 194
 1985 253 14 194
 1986 253 14 193
 1987 253 14 194
 1988 253 12 207

 Total Daily
 Daily Volume Volume Open
 Year Per Contract Per Day Interest

 1980 2602 25492 109412
 1981 6493 54640 247769
 1982 8983 66632 177510
 1983 9781 77574 159315
 1984 14643 118935 195907
 1985 21093 160539 238051
 1986 30469 209404 242579
 1987 34354 259496 294328
 1988 42791 267406 396173

Note: This table reports descriptive information for the CBOT Treasury
Bond futures contracts studied. Items include; the number of different
contracts traded in a given year, the average number of trading days
for a typical contract in a given year, the average daily trading
volume for a typical contract in a given year, the total trading volume
per day for all the contracts in a given year, and the average daily
open interest in a given year.

Table 1-B
Sample description of price limit moves

 Days of
 Number of Days of Up Down
Year Limit Days Limits Limits

1980 40 23 17
1981 12 6 6
1982 5 4 1
1983 - - -
1984 3 2 1
1985 19 18 1
1986 24 7 17
1987 7 2 5
1988 1 1 -

Total 111 63 48

 Number of
 Number of Number of Down
Year Limit Moves Up Limits Limits

1980 103 58 45
1981 15 10 5
1982 14 14 -
1983 - - -
1984 3 2 1
1985 4 3 1
1986 168 85 83
1987 45 25 20
1988 6 6 -

Total 358 203 155

Note: This table reports the occurrences of limit moves in the
contracts studied. There were a total of 111 days with limit moves.
The total number of limit moves was 358, since multiple limit moves
can occur in a single day.

Table 1-C
Sample description of successive daily limit moves

 Number of Number of Number of
 Limits Single Limits Succ. Limits

Up Limit 63 49 14
Down Limit 48 38 10

Total 111 87 24

Up Limit Occurring in Occurring in
Down Limit Pairs Triplets

Total 13 1
 7 3

 20 4

Note: This table provides information on the number of single-day
versus multiple-day limit moves. Of the total of 111 limit days,
87 were single days and 24 were multiple day moves.

Table 1-D
Intraday price changes during the limit day

 Changes Changes Trade Halts
 Number of Before After After 1st
 Changes 1st Limit 1st Limit Limit

Up Limit 495 252(51%) 242(49%) 49(24%)
Down Limit 508 296(58%) 211(42%) 42(27%)

Total 503 273(54%) 453(46%) 91(25%)

Note: This table looks at the number of price changes during limit
move days. The average number of price changes on limit move days
is reported in the first column, with the number of price changes
before the limit move and after the limit move reported in the
following columns. The number of trading halts after the first limit
move is also reported.

Table 1-E
Timing of limit move occurrences

 Time of Day (Eastern) of Limit Moves

 8-9 9-10 10-11 11-12 12-1 1-2 2-3

Up Limit 59 17 16 26 9 55 16
Down Limit 40 11 14 6 9 57 23

Total 99 28 30 32 18 112 39

Note: The time of day for each occurrence of a limit move is provided.
Some clustering of limit moves towards the beginning and end of the
trading day is seen. Limit moves occurring near the end of a trading
day may have a higher chance of resulting in a trading halt, due to
the shorter amount of trading time left in the day.

TABLE 2-A
Actual and implied futures price changes around down limit moves

Time (1) ACT (2) IMP (2) t (3) N (4) Obs (%) (5)

 -280 -0.412 -0.438 0.79 46 861 (40%)
 -270 -0.337 -0.341 0.11 38 818 (41%)
 -260 -0.335 -0.381 0.76 44 843 (43%)
 -250 -0.515 -0.528 0.51 49 1215 (44%)
 -240 -0.736 -0.792 2.26 ** 46 1051 (56%)
 -230 -0.736 -0.759 0.93 48 1032 (61%)
 -220 -0.647 -0.638 -0.37 52 1202 (55%)
 -210 -0.638 -0.635 -0.09 53 1168 (61%)
 -200 -0.694 -0.699 0.19 55 1002 (61%)
 -190 -0.717 -0.715 -0.07 55 1100 (62%)
 -180 -0.857 -0.925 2.34 ** 51 1072 (68%)
 -170 -0.832 -0.913 3.02 * 52 1194 (74%)
 -160 -0.888 -0.972 3.13 * 52 1234 (85%)
 -150 -0.929 -1.044 4.20 * 53 1233 (80%)
 -140 -1.015 -1.122 3.62 * 63 1278 (83%)
 -130 -0.903 -1.018 3.77 * 61 1127 (85%)
 -120 -0.795 -0.987 5.56 * 69 1172 (77%)
 -110 -0.820 -0.985 4.64 * 58 1251 (79%)
 -100 -0.912 -1.111 6.05 * 70 1304 (82%)
 -90 -1.005 -1.182 5.66 * 73 1481 (85%)
 -80 -1.117 -1.355 5.94 * 73 1580 (89%)
 -70 -1.005 -1.217 7.49 * 71 1488 (87%)
 -60 -1.146 -1.315 5.80 * 74 1514 (88%)
 -50 -1.165 -1.416 9.26 * 80 1590 (86%)
 -40 -1.275 -1.526 9.97 * 78 1633 (89%)
 -30 -1.342 -1.600 9.93 * 88 1817 (89%)
 -20 -1.323 -1.602 10.03 * 95 1998 (88%)
 -10 -1.380 -1.823 25.33 * 118 4745 (93%)
 0 -2.000 -2.627 6.36 * 114 134 (81%)
 +10 -1.440 -2.095 28.83 * 115 3734 (89%)
 +20 -1.465 -1.926 15.14 * 85 1609 (89%)
 +30 -1.446 -1.935 18.33 * 82 1705 (91%)
 +40 -1.516 -2.047 18.84 * 82 1544 (92%)
 +50 -1.454 -1.956 15.06 * 69 1230 (92%)
 +60 -1.457 -1.997 16.05 * 60 1168 (93%)
 +70 -1.419 -1.940 14.19 * 57 1034 (94%)
 +80 -1.374 -1.767 10.97 * 59 989 (92%)
 +90 -1.307 -1.668 10.13 * 56 1105 (91%)
 +100 -1.322 -1.552 6.84 * 54 1083 (92%)
 +110 -1.413 -1.651 7.18 * 54 969 (93%)
 +120 -1.280 -1.516 6.26 * 49 945 (91%)
 +130 -1.043 -1.291 5.31 * 48 860 (84%)
 +140 -1.068 -1.291 4.67 * 48 741 (83%)
 +150 -1.117 -1.245 3.05 * 42 756 (84%)
 +160 -1.194 -1.455 5.94 * 42 789 (87%)
 +170 -1.308 -1.595 7.17 * 38 732 (91%)
 +180 -1.254 -1.554 6.60 * 42 708 (93%)
 +190 -1.372 -1.751 8.93 * 43 796 (92%)
 +200 -1.441 -1.852 8.76 * 39 684 (94%)
 +210 -1.409 -1.737 7.05 * 41 709 (94%)
 +220 -1.321 -1.601 5.61 * 38 652 (96%)
 +230 -1.296 -1.677 7.49 * 38 650 (93%)
 +240 -1.457 -1.991 11.61 * 37 617 (96%)
 +250 -1.438 -1.820 9.85 * 34 670 (98%)
 +260 -1.437 -1.934 10.80 * 34 594 (100%)
 +270 -1.420 -1.792 9.40 * 33 489 (100%)
 +280 -1.348 -1.846 13.20 * 29 517 (100%)

Note: (1.) The time is measured by the number of minutes, plus or
minus, from the first limit move (time 0).

(2.) ACT is the mean actual price change and IMP is the mean
implied true price change, both from the closing price of the
previous day.

(3.) The t-test is for equivalence between ACT and IMP; *-indicates
significance at the 1 percent level, **-indicates significance at the
5 percent level.

(4.) N is the number of days with limit moves during time interval t.

(5.) Obs is the total number of observations in the time period, where
the percentages are the proportion of the cases where IMP is lower than
the ACT in down limit moves.

Table 2-B
Actual and implied true futures price changes around up limit moves

Time (1) ACT (2) IMP (2) t (3) N (4) Obs (%) (5)

 -280 0.817 1.086 -7.79 * 36 1085 (85%)
 -270 0.985 1.282 -7.96 * 36 1158 (89%)
 -260 1.028 1.293 -6.88 * 39 1154 (91%)
 -250 0.991 1.180 -4.92 * 37 993 (92%)
 -240 0.904 1.272 -10.29 * 39 979 (91%)
 -230 0.927 1.146 -6.41 * 40 982 (90%)
 -220 0.958 1.285 -8.11 * 41 936 (89%)
 -210 0.885 1.207 -7.56 * 43 919 (89%)
 -200 0.970 1.298 -6.95 * 47 867 (88%)
 -190 0.950 1.219 -7.50 * 55 1158 (81%)
 -180 0.907 1.212 -7.50 * 56 1300 (75%)
 -170 0.840 1.074 -5.73 * 59 1164 (72%)
 -160 0.891 1.126 -6.24 * 63 1356 (77%)
 -150 0.867 1.082 -5.94 * 61 1254 (77%)
 -140 1.007 1.218 -5.63 * 61 1166 (79%)
 -130 0.988 1.241 -6.14 * 60 1110 (79%)
 -120 0.949 1.199 -6.12 * 73 1240 (82%)
 -110 1.113 1.373 -8.22 * 69 1500 (84%)
 -100 1.191 1.436 -8.04 * 72 1468 (86%)
 -90 1.165 1.487 -10.30 * 77 1284 (85%)
 -80 1.224 1.514 -11.12 * 74 1567 (89%)
 -70 1.236 1.565 -12.37 * 74 1486 (92%)
 -60 1.286 1.609 -11.99 * 77 1483 (92%)
 -50 1.346 1.643 -11.66 * 78 1504 (96%)
 -40 1.330 1.564 -9.20 * 93 1681 (94%)
 -30 1.402 1.718 -12.74 * 92 1992 (93%)
 -20 1.361 1.777 -16.13 * 93 1874 (94%)
 -10 1.517 2.065 -32.92 * 135 5180 (93%)
 0 2.000 3.427 -10.53 * 112 129 (86%)
 +10 1.653 2.583 -46.98 * 127 4849 (97%)
 +20 1.656 2.427 -25.63 * 105 1984 (96%)
 +30 1.532 2.295 -26.92 * 109 2120 (98%)
 +40 1.451 2.132 -21.06 * 92 1865 (97%)
 +50 1.478 2.150 -21.08 * 92 1829 (99%)
 +60 1.466 2.130 -18.75 * 80 1551 (98%)
 +70 1.380 1.913 -14.93 * 79 1556 (94%)
 +80 1.266 1.707 -13.63 * 81 1588 (95%)
 +90 1.167 1.600 -12.09 * 78 1535 (94%)
 +100 0.977 1.390 -9.85 * 84 1493 (85%)
 +110 0.888 1.252 -8.48 * 75 1377 (76%)
 +120 0.930 1.279 -8.43 * 64 1230 (84%)
 +130 1.155 1.610 -10.71 * 71 1261 (88%)
 +140 1.095 1.462 -9.23 * 70 1231 (92%)
 +150 1.095 1.473 -8.76 * 73 1194 (91%)
 +160 1.100 1.555 -9.81 * 72 1259 (88%)
 +170 1.089 1.466 -8.48 * 58 1109 (90%)
 +180 1.039 1.411 -7.89 * 63 1104 (86%)
 +190 1.037 1.408 -7.49 * 57 1074 (86%)
 +200 1.024 1.523 -8.60 * 58 954 (87%)
 +210 0.996 1.329 -6.04 * 58 932 (77%)
 +220 0.965 1.204 -4.81 * 59 938 (81%)
 +230 0.929 1.241 -6.49 * 56 1031 (75%)
 +240 0.797 1.008 -4.62 * 58 888 (71%)
 +250 0.868 1.079 -4.71 * 53 894 (74%)
 +260 0.857 1.098 -5.17 * 53 821 (78%)
 +270 0.865 1.175 -5.96 * 46 664 (82%)
 +280 0.781 1.015 -4.60 * 44 796 (77%)

Note: (1.) The time is measured by the number of minutes, plus or
minus, from the first limit move (time 0).

(2.) ACT is the mean actual price change and IMP is the mean implied
true price change, both from the closing price of the previous day.

(3.) The t-test is for equivalence between ACT and IMP; *-indicates
significance at the 1 percent level, **-indicates significance at the
5 percent level.

(4.) N is the number of days with limit moves during time interval t.

(5.) Obs is the total number of observations in the time period, where
the percentages are the proportion of the cases where IMP is higher
than the ACT in up limit moves.

Table 3
Tests of the unbiasedness hypothesis

Parameter Days with limit moves (1)

Independent variable Theoretical Observed
 closing price closing price

Intercept coefficient -0.118 0.086
(t-statistic) (0.219) (0.247)

Beta coefficient 1.0017 1.003
(t-statistic) (0.003) * (0.003) *

Adj. R-Squared 0.996 0.993

F-Ratio test of Null 0.578 6.1377 *
(significance) (0.5612) (0.0023)

Null Hypothesis
Ho: ([alpha], Cannot reject Reject
[beta]) = (0,1)

Parameter Days without limit moves (2)

Independent variable Theoretical Observed
 closing price closing price

Intercept coefficient 0.292 0.281
(t-statistic) (0.036) (0.035)

Beta coefficient 0.9966 0.9966
(t-statistic) (0.0005) * (0.0004) *

Adj. R-Squared 0.994 0.994

F-Ratio test of Null 39.62 * 32.09 *
(significance) (0.001) (0.0001)

Null Hypothesis
Ho: ([alpha], Reject Reject
[beta]) = (0,1)

Note: This table provides results for OLS regressions (equation 4) of
the futures open price and the theoretical true futures prices on days
with and without limit moves. The actual futures prices are also used
to determine which variable provides unbiased information with regard
to the following day's futures open price. The unbiased hypothesis
cannot be rejected only for the case of limit move days using the
theoretical true futures prices as an unbiased predictor of the next
trading day's open price.

* Significant at the one percent level.

Table 4
The implied true futures price changes for single day limit
moves versus successive daily limit moves

 Down Limit Moves

 Time (1) IMPM (2) IMPS (2) t (3)

 -280 -0.493 -0.245 -5.07 *
 -270 -0.365 -0.250 -2.50 *
 -260 -0.415 -0.277 -3.16 *
 -250 -0.564 -0.316 -6.03 *
 -240 -0.844 -0.430 -8.79 *
 -230 -0.839 -0.493 -9.03 *
 -220 -0.734 -0.408 -9.61 *
 -210 -0.746 -0.312 -12.39 *
 -200 -0.811 -0.384 -11.25 *
 -190 -0.832 -0.333 -12.32 *
 -180 -1.069 -0.460 -14.20 *
 -170 -1.053 -0.499 -14.51 *
 -160 -1.074 -0.633 -11.72 *
 -150 -1.216 -0.530 -18.16 *
 -140 -1.207 -0.903 -5.38 *
 -130 -1.126 -0.704 -6.60 *
 -120 -1.041 -0.842 -2.87 *
 -110 -1.032 -0.836 -3.59 *
 -100 -1.108 -1.121 0.18
 -90 -1.231 -1.001 -3.78 *
 -80 -1.453 -0.956 -8.25 *
 -70 -1.285 -1.001 -5.48 *
 -60 -1.337 -1.242 -2.09 **
 -50 -1.421 -1.398 -0.44
 -40 -1.584 -1.338 -5.95 *
 -30 -1.709 -1.151 -11.77 *
 -20 -1.830 -1.084 -16.18 *
 -10 -2.019 -1.410 -20.08 *
 0 -2.647 -2.587 -2.80 *
 +10 -2.252 -1.713 -12.20 *
 +20 -2.021 -1.685 -5.60 *
 +30 -1.975 -1.830 -2.63 *
 +40 -2.154 -1.798 -6.69 *
 +50 -2.104 -1.589 -7.99 *
 +60 -2.178 -1.528 -10.17 *
 +70 -2.136 -1.432 -10.84 *
 +80 -1.923 -1.374 -8.66 *
 +90 -1.768 -1.442 -5.14 *
 +100 -1.647 -1.373 -4.71 *
 +110 -1.751 -1.386 -5.60 *
 +120 -1.712 -1.062 -9.51 *
 +130 -1.404 -1.078 -3.91 *
 +140 -1.486 -0.943 -7.26 *
 +150 -1.436 -0.920 -9.01 *
 +160 -1.702 -0.884 -13.67 *
 +170 -1.837 -0.825 -18.73 *
 +180 -1.737 -1.124 -8.71 *
 +190 -2.003 -1.147 -13.07 *
 +200 -2.168 -1.172 -13.42 *
 +210 -1.972 -1.190 -11.41 *
 +220 -1.918 -1.042 -12.68 *
 +230 -1.860 -1.253 -8.14 *
 +240 -2.224 -1.442 -10.12 *
 +250 -2.104 -1.398 -12.07 *
 +260 -2.265 -1.471 -11.16 *
 +270 -1.763 -1.823 0.96 *
 +280 -1.737 -1.912 3.05 *

 Up Limit Moves

 Time (1) IMPM (2) IMPS (2) t (3)

 -280 1.670 0.919 16.17 *
 -270 1.882 1.113 18.77 *
 -260 2.182 0.975 26.92 *
 -250 2.018 0.946 27.43 *
 -240 1.891 1.108 20.51 *
 -230 1.650 1.050 10.07 *
 -220 1.872 1.162 12.82 *
 -210 1.946 1.044 12.84 *
 -200 1.976 1.086 17.86 *
 -190 1.965 1.088 21.81 *
 -180 2.030 1.091 22.49 *
 -170 2.015 0.954 23.56 *
 -160 1.941 1.024 15.30 *
 -150 1.911 0.971 20.96 *
 -140 1.992 0.102 15.65 *
 -130 2.090 1.128 12.93 *
 -120 2.273 1.036 17.24 *
 -110 2.076 1.179 18.31 *
 -100 2.181 1.238 21.09 *
 -90 2.116 1.356 17.65 *
 -80 2.137 1.402 18.32 *
 -70 2.363 1.433 20.92 *
 -60 2.108 1.516 14.70 *
 -50 2.456 1.519 18.40 *
 -40 2.451 1.419 21.02 *
 -30 2.185 1.643 13.82 *
 -20 2.382 1.695 13.36 *
 -10 1.832 2.120 -8.07 *
 0 3.848 3.282 2.93 *
 +10 2.759 2.543 4.98 *
 +20 2.897 2.323 6.44 *
 +30 2.833 2.149 12.92 *
 +40 3.021 1.920 15.76 *
 +50 3.151 1.968 15.54 *
 +60 3.274 1.912 7.06 *
 +70 3.169 1.671 19.01 *
 +80 2.924 1.499 20.64 *
 +90 3.548 1.369 19.22 *
 +100 3.352 1.217 13.85 *
 +110 3.696 1.039 16.38 *
 +120 2.776 1.167 8.93 *
 +130 3.824 1.434 13.77 *
 +140 3.187 1.265 13.01 *
 +150 2.766 1.344 10.33 *
 +160 3.168 1.352 14.89 *
 +170 2.986 1.328 10.70 *
 +180 3.380 1.188 13.83 *
 +190 3.128 1.167 13.60 *
 +200 3.563 1.087 19.78 *
 +210 3.409 1.010 14.87 *
 +220 3.435 1.016 13.45 *
 +230 3.448 1.022 19.43 *
 +240 3.110 0.825 14.25 *
 +250 2.811 0.909 16.06 *
 +260 3.008 0.854 20.42 *
 +270 2.941 0.915 16.65 *
 +280 2.606 0.712 20.59 *

Note: (1.) The time is measured by the number of minutes, plus or
minus, from the first limit move (time 0).

(2.) IMPM (IMPS) is the implied true futures price changes for the
successive (single) day limit moves.

(3.) The t-test compares the difference between IMPM and IMPS;

*-Indicates significance at the 1 percent level.

**-indicates significance at the 5 percent level.

Table 4
The implied true futures price changes for single day limit
moves versus successive daily limit moves

 Down Limit Moves

 Time (1) IMPM (2) IMPS (2) t (3)

 -280 -0.493 -0.245 -5.07 *
 -270 -0.365 -0.250 -2.50 *
 -260 -0.415 -0.277 -3.16 *
 -250 -0.564 -0.316 -6.03 *
 -240 -0.844 -0.430 -8.79 *
 -230 -0.839 -0.493 -9.03 *
 -220 -0.734 -0.408 -9.61 *
 -210 -0.746 -0.312 -12.39 *
 -200 -0.811 -0.384 -11.25 *
 -190 -0.832 -0.333 -12.32 *
 -180 -1.069 -0.460 -14.20 *
 -170 -1.053 -0.499 -14.51 *
 -160 -1.074 -0.633 -11.72 *
 -150 -1.216 -0.530 -18.16 *
 -140 -1.207 -0.903 -5.38 *
 -130 -1.126 -0.704 -6.60 *
 -120 -1.041 -0.842 -2.87 *
 -110 -1.032 -0.836 -3.59 *
 -100 -1.108 -1.121 0.18
 -90 -1.231 -1.001 -3.78 *
 -80 -1.453 -0.956 -8.25 *
 -70 -1.285 -1.001 -5.48 *
 -60 -1.337 -1.242 -2.09 **
 -50 -1.421 -1.398 -0.44
 -40 -1.584 -1.338 -5.95 *
 -30 -1.709 -1.151 -11.77 *
 -20 -1.830 -1.084 -16.18 *
 -10 -2.019 -1.410 -20.08 *
 0 -2.647 -2.587 -2.80 *
 +10 -2.252 -1.713 -12.20 *
 +20 -2.021 -1.685 -5.60 *
 +30 -1.975 -1.830 -2.63 *
 +40 -2.154 -1.798 -6.69 *
 +50 -2.104 -1.589 -7.99 *
 +60 -2.178 -1.528 -10.17 *
 +70 -2.136 -1.432 -10.84 *
 +80 -1.923 -1.374 -8.66 *
 +90 -1.768 -1.442 -5.14 *
 +100 -1.647 -1.373 -4.71 *
 +110 -1.751 -1.386 -5.60 *
 +120 -1.712 -1.062 -9.51 *
 +130 -1.404 -1.078 -3.91 *
 +140 -1.486 -0.943 -7.26 *
 +150 -1.436 -0.920 -9.01 *
 +160 -1.702 -0.884 -13.67 *
 +170 -1.837 -0.825 -18.73 *
 +180 -1.737 -1.124 -8.71 *
 +190 -2.003 -1.147 -13.07 *
 +200 -2.168 -1.172 -13.42 *
 +210 -1.972 -1.190 -11.41 *
 +220 -1.918 -1.042 -12.68 *
 +230 -1.860 -1.253 -8.14 *
 +240 -2.224 -1.442 -10.12 *
 +250 -2.104 -1.398 -12.07 *
 +260 -2.265 -1.471 -11.16 *
 +270 -1.763 -1.823 0.96 *
 +280 -1.737 -1.912 3.05 *

 Up Limit Moves

 Time (1) IMPM (2) IMPS (2) t (3)

 -280 1.670 0.919 16.17 *
 -270 1.882 1.113 18.77 *
 -260 2.182 0.975 26.92 *
 -250 2.018 0.946 27.43 *
 -240 1.891 1.108 20.51 *
 -230 1.650 1.050 10.07 *
 -220 1.872 1.162 12.82 *
 -210 1.946 1.044 12.84 *
 -200 1.976 1.086 17.86 *
 -190 1.965 1.088 21.81 *
 -180 2.030 1.091 22.49 *
 -170 2.015 0.954 23.56 *
 -160 1.941 1.024 15.30 *
 -150 1.911 0.971 20.96 *
 -140 1.992 0.102 15.65 *
 -130 2.090 1.128 12.93 *
 -120 2.273 1.036 17.24 *
 -110 2.076 1.179 18.31 *
 -100 2.181 1.238 21.09 *
 -90 2.116 1.356 17.65 *
 -80 2.137 1.402 18.32 *
 -70 2.363 1.433 20.92 *
 -60 2.108 1.516 14.70 *
 -50 2.456 1.519 18.40 *
 -40 2.451 1.419 21.02 *
 -30 2.185 1.643 13.82 *
 -20 2.382 1.695 13.36 *
 -10 1.832 2.120 -8.07 *
 0 3.848 3.282 2.93 *
 +10 2.759 2.543 4.98 *
 +20 2.897 2.323 6.44 *
 +30 2.833 2.149 12.92 *
 +40 3.021 1.920 15.76 *
 +50 3.151 1.968 15.54 *
 +60 3.274 1.912 7.06 *
 +70 3.169 1.671 19.01 *
 +80 2.924 1.499 20.64 *
 +90 3.548 1.369 19.22 *
 +100 3.352 1.217 13.85 *
 +110 3.696 1.039 16.38 *
 +120 2.776 1.167 8.93 *
 +130 3.824 1.434 13.77 *
 +140 3.187 1.265 13.01 *
 +150 2.766 1.344 10.33 *
 +160 3.168 1.352 14.89 *
 +170 2.986 1.328 10.70 *
 +180 3.380 1.188 13.83 *
 +190 3.128 1.167 13.60 *
 +200 3.563 1.087 19.78 *
 +210 3.409 1.010 14.87 *
 +220 3.435 1.016 13.45 *
 +230 3.448 1.022 19.43 *
 +240 3.110 0.825 14.25 *
 +250 2.811 0.909 16.06 *
 +260 3.008 0.854 20.42 *
 +270 2.941 0.915 16.65 *
 +280 2.606 0.712 20.59 *

Note: (1.) The time is measured by the number of minutes, plus or
minus, from the first limit move (time 0).

(2.) IMPM (IMPS) is the implied true futures price changes for the
successive (single) day limit moves.

(3.) The t-test compares the difference between IMPM and IMPS;

*-Indicates significance at the 1 percent level.

**-indicates significance at the 5 percent level.

Table 5
Actual and implied true futures price changes before trading halts

 Down Limit Moves

Time (1) ACT (2) IMP (2) t (3) N (4) Obs (%) (5)

 -280 -0.316 -0.328 -6.95 16 164 (34%)
 -270 -0.232 -0.191 -7.50 8 120 (31%)
 -260 -0.409 -0.390 -7.50 * 12 107 (32%)
 -250 -0.286 -0.316 -5.73 * 13 132 (28%)
 -240 -0.512 -0.532 -6.24 * 11 101 (46%)
 -230 -0.388 -0.401 -5.94 * 10 77 (48%)
 -220 -0.213 -0.027 -5.63 * 12 112 (30%)
 -210 0.079 0.097 -6.14 * 12 112 (30%)
 -200 -0.011 -0.011 -6.12 * 17 95 (41%)
 -190 0.036 0.091 -8.22 * 16 163 (26%)
 -180 -0.442 -0.341 -8.04 * 11 108 (30%)
 -170 -0.311 -0.335 -10.30 * 10 125 (29%)
 -160 -0.342 -0.373 -11.12 * 13 164 (74%)
 -150 -0.509 -0.773 -12.37 * 10 128 (90%)
 -140 -0.442 -0.449 -11.99 * 13 161 (91%)
 -130 -0.254 -0.263 -11.66 * 14 119 (93%)
 -120 -0.643 -0.662 -9.20 * 14 104 (66%)
 -110 -0.619 -0.631 -12.74 * 9 84 (91%)
 -100 -0.789 -0.919 -16.13 * 17 100 (82%)
 -90 -1.024 -1.093 -32.92 * 16 191 (89%)
 -80 -1.048 -1.288 -46.98 * 16 167 (92%)
 -70 -1.058 -1.244 -25.63 * 13 172 (85%)
 -60 -1.081 -1.130 -26.92 * 14 147 (91%)
 -50 -1.109 -1.160 -21.06 * 16 119 (93%)
 -40 -1.087 -1.494 -21.08 * 12 176 (100%)
 -30 -1.095 -1.435 -18.75 * 19 138 (99%)
 -20 -1.416 -1.863 -14.93 * 18 202 (100%)
 -10 -1.633 -2.132 -13.63 * 25 265 (100%)
 0 -2.000 -3.048 -12.09 * 33 41 (95%)

 Up Limit Moves

Time (1) ACT (2) IMP (2) t (3) N (4) Obs (%) (5)

 -280 1.201 1.650 0.19 9 40 (95%)
 -270 1.001 1.386 -0.46 9 31 (100%)
 -260 1.370 2.247 -0.24 9 57 (98%)
 -250 1.216 1.326 0.39 8 37 (100%)
 -240 1.181 1.273 0.25 8 33 (91%)
 -230 1.127 1.454 0.14 8 40 (98%)
 -220 1.144 1.812 -1.90 8 21 (95%)
 -210 1.182 2.381 -0.14 8 45 (100%)
 -200 1.246 1.735 -0.00 10 49 (100%)
 -190 1.001 1.355 -0.52 13 48 (94%)
 -180 1.330 1.945 -0.93 10 70 (99%)
 -170 0.977 1.581 0.34 11 38 (95%)
 -160 1.197 1.915 0.42 14 46 (91%)
 -150 1.127 1.689 3.11 * 13 32 (91%)
 -140 1.489 2.148 0.09 10 47 (98%)
 -130 1.501 2.568 0.12 9 30 (97%)
 -120 1.525 1.843 0.20 17 57 (93%)
 -110 1.163 1.650 0.10 14 43 (81%)
 -100 1.295 1.479 1.22 14 44 (86%)
 -90 1.351 1.622 2.00 ** 14 37 (89%)
 -80 1.633 1.982 5.80 * 12 52 (96%)
 -70 1.603 2.083 4.11 * 13 48 (100%)
 -60 1.685 2.344 1.33 11 35 (97%)
 -50 1.440 1.999 1.17 13 34 (91%)
 -40 1.487 1.757 9.45 * 15 41 (93%)
 -30 1.473 2.090 6.80 * 9 34 (100%)
 -20 1.469 1.864 8.63 * 13 66 (100%)
 -10 1.484 2.187 7.95 * 25 145 (99%)
 0 2.000 3.938 13.51 * 31 32 (97%)

Note: (1.) The time is measured by the number of minutes, plus or
minus, from the first limit move (time 0). (2.) ACT is the actual
price change and IMP is the implied true price change, both from
the closing price of the previous day. (3.) The t-test is for
equivalence between ACT and IMP; *-indicates significance at the 1
percent level; **-indicates significance at the 5 percent level.
(4.) N is the number of days with limit moves at the time interval,
t. (5.) Obs is the total number of observations in the time period,
where the percentages are the proportion of the cases where IMP is
higher (lower) than the ACT in up (down) limit moves.

Table 6
Implied true futures price changes in regular limit move days and in
days of trading halts

 Down Limit Moves

Time (1) IMPR (2) IMPH (2) t (3)

 -280 -0.438 -0.328 -2.50 **
 -270 -0.341 -0.191 -2.28 **
 -260 -0.381 -0.390 0.16
 -250 -0.529 -0.315 -3.95 *
 -240 -0.792 -0.532 -4.76 *
 -230 -0.759 -0.401 -5.55 *
 -220 -0.638 -0.027 -8.52 *
 -210 -0.635 0.097 -8.59 *
 -200 -0.699 -0.011 -7.44 *
 -190 0.715 0.091 -11.87 *
 -180 -0.925 -0.341 -7.66 *
 -170 -0.913 -0.335 -11.36 *
 -160 -0.972 -0.373 -12.06 *
 -150 -1.044 -0.773 -4.70 *
 -140 -1.122 -0.449 -13.07 *
 -130 -1.018 -0.263 -14.67 *
 -120 -0.987 -0.662 -4.56 *
 -110 -0.985 -0.631 -4.15 *
 -100 -1.111 -0.919 -2.67 *
 -90 -1.182 -1.093 -2.76 *
 -80 -1.355 -1.288 -1.63 **
 -70 -1.217 -1.244 0.70 **
 -60 -1.315 -1.130 -5.57 *
 -50 -1.416 -1.160 -6.84 *
 -40 -1.526 -1.494 -1.53 **
 -30 -1.600 -1.435 -1.61 *
 -20 -1.602 -1.862 6.37 *
 -10 -1.823 -2.132 5.72 *
 0 -2.627 -3.048 3.93 *
 +10 -2.094 -2.998 8.57 *

 Up Limit Moves

Time (1) IMPR (2) IMPH (2) t (3)

 -280 1.086 1.650 -4.34 *
 -270 1.282 1.386 -0.94
 -260 1.293 2.247 -6.36 *
 -250 1.180 1.326 -1.26
 -240 1.272 1.273 -0.01
 -230 1.146 1.454 -1.97
 -220 1.285 1.812 -1.90
 -210 1.207 2.381 -6.92 *
 -200 1.298 1.735 -4.48 *
 -190 1.219 1.355 -1.18
 -180 1.212 1.945 -9.96 *
 -170 1.074 1.581 -3.15 *
 -160 1.126 1.915 -4.11 *
 -150 1.082 1.689 -3.74 *
 -140 1.218 2.148 -6.27 *
 -130 1.241 2.568 -5.18 *
 -120 1.199 1.843 -5.45 *
 -110 1.373 1.650 -1.58
 -100 1.436 1.479 -0.29
 -90 1.487 1.622 -0.93
 -80 1.514 1.982 -3.68 *
 -70 1.565 2.083 -4.59 *
 -60 1.609 2.344 -3.91 *
 -50 1.643 1.999 -1.72
 -40 1.564 1.757 -3.07 *
 -30 1.718 2.090 -0.79
 -20 1.777 1.864 -1.59
 -10 2.065 2.187 -2.74 *
 0 3.427 3.938 -8.00 *
 +10 2.583 4.002 -0.97

Note: (1.) There are no observations 10 minutes after the first limit
move for the subsample of trading halts.
(2.) IMPR (IMPH) is the implied true futures price changes for the
regular (trading halt) sample of limit moves
(3.) The t-test is to compare the difference between IMPR and IMPH;
*-indicates significance at the 1 percent level; **-indicates
significance at the 5 percent level.