High-Frequency Trading Invariants for Equity-Index Futures

High-Frequency Trading Invariants for Equity-Index Futures Torben G. Andersen, Oleg Bondarenko Albert S. Kyle and Anna A. Obizhaeva First Draft: July, 14 This Draft: January 6, 15 The high-frequency trading patterns of the S&P5 E-mini futures contracts between January 8 and November 11 are consistent with the following invariance relationship: the number of transactions is proportional to a product of dollar volume and volatility in /3 power. Equivalently, the return variation per transaction is loglinearly related to trade size, with a slope coefficient of. This factor of proportionality deviates sharply from those associated with prior hypotheses relating volatility to the transactions count or trading volume. High-frequency trading invariance is, a priori, motivated by the notion of market microstructure invariance introduced by Kyle and Obizhaeva (13), though it does not follow from it directly. Preliminary Draft Not for External Circulation Andersen: Kellogg School of Management, Northwestern University, 1 Sheridan Road, Evanston, IL 68, NBER, and CREATES, t-andersen@kellogg.northwestern.edu. Bondarenko: Department of Finance (MC 168), University of Illinois at Chicago, 61 S. Morgan St., Chicago, IL 667, olegb@uic.edu. Kyle: Robert H. Smith School of Business, University of Maryland, College Park, MD 74, akyle@rhsmith.umd.edu. Obizhaeva: New Economic School, Moscow, obizhaeva@nes.ru. Andersen acknowledges support from CREATES, Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation. Kyle works as a director for an asset management company which trades international stocks and has also worked as a consultant for various U.S. government agencies, including the SEC and CFTC. We appreciate comments from participants at the Workshop on Recent Advances in High-Frequency Statistics at Humboldt University, Berlin, November 14, the Market Microstructure from Many Perspectives Conference in Paris, December 14, and the High- Frequency Financial Data Conference, Montréal, December 14. 1

I. Introduction An extensive literature has documented a pronounced, systematic and highly correlated intraday variation in financial market activity variables such as the trading volume, the number of transactions, the bid-ask spread, and the return volatility. These basic qualitative relations remain in effect across different time periods, asset classes, and market structures. In fact, it is natural to view this common variation as universal attributes, arising endogenously in liquid and competitive financial markets. In this paper, we look for a quantitative relationship between these variables that is consistent with their intraday and intertemporal variation. Our specification of a precise relationship between return volatility, average trade size, and transaction frequency is motivated by the market microstructure invariance developed by Kyle and Obizhaeva (13). This theory stipulates that markets for different financial securities operate similarly when viewed from the perspective of an appropriate business time clock. The markets are analogous in terms of the dollar amounts expected to be at stake, the scale of risk transferred, the magnitude of transaction costs, and the size of profits, measured in business time. Translated into the time series context, the idea suggests that such market structure features remain invariant, also for a given security observed across time. We use this intuition to guide our exploration of the interdependencies of the market activity variables across both the intraday pattern and trading days. Motivated by invariance, we hypothesize that the number of transactions is proportional to the product of volume and volatility in /3 power when studying the data over shorter intraday intervals. We use the tick-by-tick data on the E-mini S&P 5 futures contract the most actively traded equity contract in financial markets from January 8 to November 11 and, indeed, estimate the corresponding exponent from a regression over the respective intraday activity patterns to be.678. Moreover, invariance explains the vast majority of the variation in the number of transactions across time. Similarly, the theory account well for the average trade size across the daily trading cycle, with the most striking successful prediction being that the transaction size shrinks in a specific fashion relative to the return volatility, but rises with trading intensity. We further verify that intraday trading invariance holds up well in times of extreme market activity, such as the periods following the release of scheduled macroeconomic announcements or episodes like the 1 flash crash. However, as the market collapsed at the end of the flash crash and its tight linkages with related financial markets vanished, the invariance relation also failed, further suggesting that invariance is tied to inherent features of properly functioning markets. We also explore the implications of time series invariance for the return volatility and trading volume relation over time and contrast the quality of fit with earlier theories of the volatility-volume interaction. Invariance performs vastly better than alternative explanations associated with either the number of transactions or volume serving as the directing process for volatility, see, e.g., Mandelbrot and Taylor (1967) or Clark (1973).

Finally, we document that, although intraday trading invariance provides an excellent first order approximation to the dynamic interaction among return volatility and trading intensity, it fails systematically during the market opening phase. This feature may potentially more generally allow invariance violations to serve as a contemporaneous indicator of the state of the market. 3 II. Trading Invariance We first informally review the ideas underlying the notion of market microstructure invariance in Section II.A. Motivated by the encouraging empirical results obtained in prior studies based on this framework, in Section II.B, we move beyond the purview of the theory to develop hypotheses that predict the contemporaneous relation between the number of transactions, (average) trade size, trading volume, and return volatility. Section II.C outlines how we may specialize the invariance hypothesis to analyze the performance of the theory during bouts of extreme trading activity. Such instances occur in the aftermath of regularly scheduled macroeconomic announcements or when the markets undergo exceptionally stressful episodes, such as the flash crash on May 6, 1. Finally, in Section II.D, we review alternative specifications for the intraday patterns in trading activity and volatility inspired by prior theories in the literature. This sets the stage for a direct comparison of their relative performance along the intraday dimension. A. Market Microstructure Invariance The market microstructure invariance hypothesis of Kyle and Obizhaeva (13) arises from the notion that investors acquire information and explore market opportunities with the objective of balancing risk and reward in business time. The progression of business time is given by the expected number of bets arriving to the marketplace per unit of calendar time. Bets denote meta orders, reflecting the desire of investors to change their exposure to an underlying asset by a sizeable magnitude. Bets are typically shredded into smaller orders that are submitted strategically to minimize price impact and related transactions costs. The strategic implementation of bets by intermediaries, such as brokers and trading firms, generate equilibrium conditions that govern the relation between key features of the trading process and the price variability, both over time and across assets. Specifically, in more active markets, bets arrive more frequently and the (business) clock runs faster. This lowers trading costs, market become more efficient, and the average distance between prices and (latent) fundamental values decrease in proportion to the square root of the bet arrival rate. Traders sustain their expected dollar profit per bet when trading costs fall and price efficiency increases by scaling up the dollar bet size proportionally. Thus, holding volatility constant, invariance implies that the dollar size of bets increases in proportion to the square root of the number of bets: as trading volume grows, the number of bets increases

twice as fast as the bet size. In other words, 1/3 of variation in volume stems from variation in the size of bets and /3 comes from variation in their number. Formally, market microstructure invariance stipulates that the dollar-risk transferred per bet measured in business time, I, is invariant, i.e., I is independently and identically distributed over time and assets, (1) I = P Q B σ N 1/ B, where the asset price is P, the bet size is Q B, the expected return volatility is σ, and the expected arrival rate of bets (the speed of business time) is N B. For empirical work, it is often convenient to employ the corresponding linear representation that arises from taking the logarithm on both sides in equation (1). We have that log I also is i.i.d. and, () log I = p + q B + 1 s 1 n B, where lower case letters indicate the logarithm of the corresponding upper case letters, while s = log σ. Empirical analysis of this invariance relation is complicated by the fact that business time the bet arrival rate is not observable. However, invoking the auxiliary hypothesis that the bet volume is proportional to the total trading volume, Kyle and Obizhaeva (13) generate a number of testable restrictions that center on the notion of total risk transfer or market activity. Letting V denote the trading volume, the risk transfer (in units of calendar time), or market activity, is defined as, W = P V σ. They then obtain, defining w = log W, (3) n B 3 w. Likewise, shifts in the expected bet size may be related to market activity, (4) q B 1 3 w [ p + 1 s ]. Equations (3)-(4) formalize the decomposition of market activity into the proportion /3 stemming from variation in the rate of risk transfers (speed of business time) and 1/3 from the magnitude of bets, for fixed s and p. This relation between the size and number of bets lies at the heart market microstructure invariance. 1 Several empirical hypotheses inspired by this type of invariance find support in the data. Kyle and Obizhaeva (13) document invariance relationships for the size distributions of portfolio transition orders, while Kyle, Obizhaeva and Tuzun (1) document them for the size distribution of transactions versus trading activity in the Trade and Quote data set (TAQ). Kyle et al. (14) study invariance relations for the number of monthly news articles. Bae et al. (14) discuss an 1 Several additional invariance relations between trading activity and market characteristics such as bid-ask spreads, price impact, and price efficiency, may be derived. However, we focus solely on the interaction between trading activity and asset volatility. 4

invariance relationship for the number of buy-sell switching points in the South Korean market. As noted, some of these relations require auxiliary identifying assumptions. We now turn to an exploration of invariance for high-frequency trading patterns which, similarly, is not a direct implication of the underlying theory. B. An Intraday Trading Invariance Hypothesis Motivated by market microstructure invariance, we stipulate a relation between average trade size, Q, the expected number of transactions (increment to the business clock), N, expected volatility, σ, and asset price, P. It effectively extrapolates the invariance relation in equation (1) or () to more readily observable intraday variables. The analogues to equations (1) and () become, (5) I dt = P dt Q dt σ dt / N 1/ dt and log I dt = p dt + q dt + 1 s dt 1 n dt. The specification replaces bet size, Q B, and arrival rate, N B, from market microstructure invariance with the average trade size, Q dt, and expected number of transactions, N dt, for the intraday interval (d, t), i.e., interval t on trading day d, where t = 1,..., T and d = 1,..., D. Likewise, σ dt refers to the expected volatility and P dt the (average) price over the given interval. As for the original invariance hypothesis, we conjecture that strategic interaction among market participants renders (log)i dt identically and independently distributed across time, i.e., i.i.d. over both d and t. In effect, this moves the invariance relationship into the realm of intraday transactions, implying the distribution of nominal risk exposure for individual trades, measured in business time, is equated over time. If the expected trade size and asset price are constant, intraday trading invariance implies that the expected return variation is proportional to both the number of transactions and trading volume, σ N and σ V. In contrast, if there is a systematic relation between trade size and return variation then, following equation (5), the trading activity is no longer proportional to volatility. Hence, invariance is either consistent with both of these existing hypotheses or incompatible with both. Invariance predicts the latter is likely to occur due to endogeneity of the trade size, stemming from traders active management of the risk exposure in business time. There is an extreme diurnal pattern in the trading of S&P 5 E-mini futures, as illustrated by Figure 1. It depicts the share volume V t, volatility σ t, number of trades N t, and trade size Q t for each minute across the trading day, where the observations are averaged across all days in the sample. Calendar time is displayed on the horizontal axis and three distinct trading regimes are separated by dashed vertical lines. The first regime spans -6:45 (17:15 of the prior day) to : CT (Chicago Time) and covers the period just before and during regular trading in Asia, the second regime reflects European trading hours, and the third regime ranging from 8:3 to 15:15 CT coincides with active trading in North America. We provide a detailed description of the data later. For now, Figure 1 is purely illustrative. 5

6 x 1 4 3.5 Volume V.8.7.6 Volatility σ.5 1.5.4 1.5.3..1 5 5 1 15 5 5 1 15 Number of Trades N Trade Size Q 1 5 1 8 6 15 4 1 5 5 5 1 15 5 5 1 15 Figure 1. The figure shows averages across all days for share volume V t (per minute), annualized realized volatility σ t, trade size Q t, and number of trades, N t. The averages are computed at a granularity of t = 1-minute. The dashed vertical lines separate the three trading regimes. Invariance predicts that the intraday oscillations in the individual series occur in specific proportions relative to each other. Moreover, since the relationship is highly nonlinear, it is masked and potentially undetectable in time-aggregated data. Therefore, we collect the basic data over periods for which these variables are expected to be (nearly) unchanged. The latter clearly does not hold for daily observations: the average trading volume increases more than fifty-fold and the transaction count twenty five-fold between regime 1 to 3. In contrast, volatility and trade size roughly double. Consequently, we rely on short, say, one-minute, intervals over which the expected variation in the trading characteristics is negligible. Finally, we note that the average return is close to zero and displays no significant within-day variation. In fact, the typical daily high-low range for price is about 1-% thousand-fold less than for trading intensity. In terms of testing for systematic intraday fluctuations, the price level is immaterial. Our primary focus is to test for trading invariance across the pronounced intraday, or diurnal, pattern in the equity-index futures market. Given the empirical regularities discussed above, we henceforth simplify the exposition by ignoring the (expected) variation in the price level.

The intraday trading invariance hypothesis (5) involves only variables whose realizations are directly observable or readily approximated. Nonetheless, we still must confront measurement issues. The theory treats Q as arising from endogenous choices of market participants, given their trading needs and perception of the market, as summarized by the expected speed of business time and volatility. However, volatility and trading intensity display a large degree of idiosyncratic variation on top of the diurnal pattern, so their a priori expected values often differ greatly from their ex post realizations. Later in this section, we explain how we deal with this potential errors-in-variables problem. Our initial test for intraday trading invariance is motivated by the original approach of Kyle and Obizhaeva (13). 3 Taking conditional expectations given the information available during interval (d, t), while recognizing that the trade intensity, n dt, and volatility, s dt, already reflect expected values and ignoring the price level, equation (5) implies the following regression-style representation, (6) n dt = c + q dt + s dt + u dt, where c denotes a generic real constant whose value will differ across equations, and u dt indicates an uncorrelated error term. To construct a test based on equation (6), we need estimators, or proxies, for the expected values. For the transaction count, trading volume, and volatility, we expect active trading participants to acquire real-time information on the state of the market and therefore assume them to form corresponding unbiased expectations over the next instant. Hence, we treat their ex-post realizations as unbiased, albeit noisy, estimators of ex-ante expectations. In implementing tests based on equation (6), we aggregate a large number of suitably transformed one-minute observations to alleviate the impact of sampling variation and measurement error. Moreover, to enhance test power, we seek a large degree of variation in the resulting aggregate units. We accomplish this by testing for invariance in the cross-section, i.e., across the diurnal pattern. A different possibility is to apply the test in the time series dimension, i.e., across trading days. We reserve this approach for our robustness checks. Specifically, we define the transaction count variable for intraday interval t, n t, by averaging the observations for this interval across all the days in the sample, (7) n t = 1 D D n dt, t = 1,..., T. d=1 Likewise, averaging across all intraday intervals on trading day d, we obtain, (8) n d = 1 T T n dt, d = 1,..., D. t=1 3 Kyle and Obizhaeva (13) employ different variables as they focus on invariance relations induced by meta orders or bets, and they investigate the relations at a much lower frequency. 7

Notice the theory implies that these variables are obtained as the mean of the logarithmic values, and not the logarithm of the mean of the underlying variables. That is, we do not require invariance to hold for time-aggregated data. Obviously, we may construct s t, q t, v t, w t, s d, q d, v d and w d similarly. Then, exploiting v dt = n dt + q dt and the definition of w dt, the relations in equations (7) and (8) imply the following regression relation, (9) n j = c + 3 w j + u n j, where j refers to distinct intraday intervals, i.e., j = t, t = 1,..., T, or different trading days, i.e., j = d, with d = 1,..., D, while u n j represents the regression residuals. Since our primary focus is the intraday variation, we shall mostly be concerned with the relationship (9) for the former case, i.e., j = t. Similarly, intraday invariance can be stated in terms of the association of the average trading size to market activity, (1) q j = c + 1 3 w j 1 s j + u q j. Equations (9) and (1) represent our intraday trading invariance versions of (3) and (4) in the original invariance relation of Kyle and Obizhaeva (13). For fixed s, the variation in trading volume arises in proportion /3 from fluctuations in the number of transactions and 1/3 from the trade size. However, as volatility varies, these relationships are modified in specific ways. C. Testing Invariance During Specific Episodes Particular market events create dramatic intraday fluctuations in trading activity and asset volatility. For example, the release of macroeconomic announcements typically induces an immediate price jump and a subsequent surge in trading volume and return volatility, see, e.g., Andersen and Bollerslev (1998). Similar extreme oscillations occur during stressful episodes associated with financial crises or market disruptions such as the May 6, 1, flash crash. Nonetheless, if the market is operative and processing a vast number of trades, this may simply be ascribed to a sharp elevation in the speed of the business clock. Hence, we seek to test whether the usual regularities involving the trading activity, return volatility, and transaction size remain in effect during these extreme scenarios. Similarly, markets may function differently in the transition from one trading regime to another, i.e., at the open and close of the market segments. Since the null hypothesis implies that the I variable is i.i.d. across all intraday intervals, we may test the implications for any exogenously specified time periods by focusing exclusively on the relevant trading periods. To formally characterize the observations involved in the different scenarios, we denote the full set of trading days by D = {1,..., D} and the set of all intraday intervals by T = {1,..., T }. Then, as an illustration, testing our invariance relation 8

for the subset of days, D k, where a given macroeconomic announcement is released, involves observations in the set of intervals {(d, t) : (d, t) O k = D k T }. Similarly, if we seek to test the invariance relation only for a set of intraday intervals, T k, around such announcements, then O k = D k T k. D. Alternative Empirical Hypotheses There is a long history of theories characterizing the relationship between trading activity and return volatility, going back at least to Clark (1973), who builds on the notion of subordination, or a stochastic business time clock, introduced into the modeling of financial returns by Mandelbrot and Taylor (1967). Clark (1973) argues that trading volume accounts for the increments to business time, thus generating a direct proportionality between expected volume and return variation, i.e., σdt V dt, or equivalently, s dt = c + v dt = c + n dt + q dt. In contrast, Jones, Kaul and Lipson (1994) find the daily transactions count to be better aligned with daily volatility, and Ané and Geman () assert that intraday returns become i.i.d. Gaussian when normalized by the (stochastic) transaction count. 4 These hypotheses imply, σdt N dt, or s dt = c + n dt. In this section, we contrast these representations with our intraday trading invariance hypothesis. First, we develop the modifications of equation (9) to ensure a similar /3 power representation for the auxiliary specifications. Since the expression implies N W /3 = (P V σ) /3 = (P QNσ) /3, we obtain, upon following the procedure of Section II.B, that the Clark (1973) specification takes the form, (11) n j = c + [ w j 3 ] 3 q j + u n j. Likewise, the Ané and Geman () formulation implies, (1) n j = c + 3 [ w j q j ] + u n j. We now develop specifications of these alternative hypotheses that are more aligned with traditional representations. Both Clark (1973) and Ané and Geman () view volatility as evolving at a uniform rate in business time which, in turn, is governed by their respective notion of the trading activity. In the context of the intraday trading invariance hypothesis, it is also natural to associate the transaction count with a measure of business time. This allows us to highlight the key distinction between the alternative theories in a simple regression set-up, namely the role of trade size in the trading activity-volatility interaction. Specifically, all hypotheses imply a link between the volatility per transaction and the average trade size, (13) s j n j = c + β q j + u q j. 4 For further refinement and testing of the trading activity-volatility relation, see, e.g., Tauchen and Pitts (1983), Andersen (1996), Bollerslev and Jubinski (1999), and Liesenfeld (1). 9

where the slope coefficient differs sharply across the theories, i.e., β = 1 (Clark), β = (Ané-Geman), or β = (invariance). It is straightforward to explore the performance of the different specification in this regression setting, and we do so in Section IV. However, we first provide a description of the data used in the empirical work. III. The S&P 5 E-mini Futures Data We use the best bid and offer (BBO) files for the E-mini S&P 5 futures contract from CME DataMine. These top-of-the-book files provide tick-by-tick information regarding the best quotes, order book depth, trade prices, and trade sizes, time stamped to the second. Since the contract is traded exclusively on the CME GLOBEX electronic platform, our data contain all transactions executed during our sample, covering nearly four years from January 4, 8, to November 4, 11. The procedure behind the recording of trades and trade sizes is critical for our empirical tests. The exchange reports a market order, filled at the identical price across distinct limit orders, as a single transaction. Thus, the trade size reflects the number of contracts traded when a marketable order crosses one or more standing limit orders at the top of the book. Since the trade size and associated risk exposure for active investors is a key feature behind the intraday trading invariance, this convention is in line with the underlying theoretical motivation: the transaction size is classified from the perspective of the party actively demanding liquidity. 5 The notional value of the E-mini S&P 5 futures contract is $5 times the value of the S&P 5 stock index, and it has a tick size of.5 index points ($1.5) or approximately basis points of notional value. The E-mini contract has four expiration months per year. We exploit data for the front month contract until it reaches eight days to expiration, at which point we switch into the next contract. This ensures we use the most actively traded contract throughout our analysis. The E-mini contract trades essentially twenty-four hours a day, five days a week. From Monday to Thursday, trading is from 15:3 to 15:15 (Chicago Time) of the following day, with a half-hour maintenance halt from 16:3 to 17:. On Sunday, trading is from 17: to 15:15 of the following Monday. We define three distinct trading regimes: 17:15 through :, : through 8:3, and 8:3 through 15:15. These regimes roughly correspond to regular trading hours in Asia, Europe, and North America, respectively. Due to the unusually low trading volume and short trading hours around holidays, we discarded eleven days from our analysis, resulting in a total of D = 959 trading days. Each trading day is partitioned into T = 1, 3 intervals of length t equal to one minute. The three trading regimes consist of T 1 = 55, T = 39, and T 3 = 45 intervals, respectively; T = T 1 +T +T 3. For future convenience, we define the set of one-minute observations belonging 5 Hence, the method of recording transactions can complicate testing for intraday trading invariance. For example, if the exchange instead counts each limit order involved in trading as a separate trade, the transaction count will reflect the order flow intermediated by the supply side. This inflates the number of transactions relative to the procedure used in the current paper. 1

to each trading regime using the notation established in Section II.C. Letting T 1 = {1,..., T 1 }, T = {T 1 + 1,..., T } and T 3 = {T + 1,..., T 3 }, regime 1 involves the data within the set O 1 = D T 1, while regime and 3 correspond to the sets O = D T and O 3 = D T 3, respectively. Our analysis focuses on the share volume V, number of trades N, average trade size Q, and realized return volatility σ (computed from 1-second returns). The intraday variation of the four characteristics are depicted in Figure 1. The corresponding time series, obtained by averaging the intraday observations daily within each regime, are displayed in Figure. The series clearly display a fair degree of commonality in the dynamic features. Trading volume, volatility, and the number of trades increase substantially during the Société Générale scandal in January 8 and the collapse of Bear Stearns in March 8, rise once more during the financial crisis from September 8 to February 9, and spike during the Flash Crash in May 1 as well as the second half of 11. In contrast, the average trade size drops during the same periods. Finally, all these variables appear to be slightly subdued towards the end of each year. 11 Volume V, Three regimes Volatility σ, Three regimes 14 1 1 8 6 4 8 9 1 11 1 1.5 1.5 8 9 1 11 1 16 Number of Trades N, Three regimes Trade Size Q, Three regimes 14 1 15 1 8 1 6 4 5 8 9 1 11 1 8 9 1 11 1 Figure. This figure plots the times series of the average statistics for each regime on each trading day: volume V (per minute), volatility σ (per minute, annualized), number of transactions N (per minute), and average trade size Q (during one minute). Regime 1 figures are plotted in blue, regime in green, and regime 3 in red. The sample period is January 4, 8, to November 4, 11.

Although the series exhibit common features, there are also systematic differences. In particular, as discussed in Section II.B, all activity variables are much lower in regime 1 than regime 3. Table 1 provides detailed summary statistics. 1 Table 1 Descriptive Statistics for the S&P 5 E-mini Futures. Regime 1 Regime Regime 3 All Volatility.16.5.4.6 Volume 9.1 6.73 475.56 1663.97 # Trades 13.87 66.74 36.4 135.7 Notional Value, $Mln 5.5 34.37 65.84 94. Trade Size 6.55 8.59 1.99 9.13 Market Depth 54.6 65.1 984.7 41.76 Bid-Ask Spread 6.54 5.69 5.13 5.86 Business Time 6.13 5.43 1..65 The statistics are reported separately for the three regimes and the entire day. Volatility is annualized. The volume, notional value, and number of trades are one-minute averages. The market depth is the average number of contracts at the best bid and ask. The bid-ask Spread is measured in index units times 1. Business Time is proportional to W /3 and, for ease of comparison, it is normalized to be unity in Regime 3. Before engaging in the empirical analysis of intraday trading invariance, we provide a back-of-the-envelope assessment. Substituting the number of transactions, N, for the speed of business time, N B, in equation (3), as discussed in Section II.B, we find the ratio of the average number of trades divided by the product of the average dollar volume and volatility in the /3 power to be 15.6, 15.9 and 16. for the three regimes, respectively. The stability of this ratio suggests that further analysis along the lines of intraday trading invariance is warranted. The market microstructure invariance principle has additional implications regarding the bid-ask spread and price efficiency. As an empirical matter, these predictions depend on the tick and contract size not being excessively constraining. For the E-mini futures, however, we clearly encounter frictions along these dimensions. Table 1 shows that the average bid-ask spread in all three regimes is almost identical to the minimal value of.5 index points. In other words, the spread is binding and equals a single tick almost always. Similarly, Figure 3 demonstrates that a disproportional number of trades across all three regimes occur at the minimum contract size of one unit. Hence, while the E-mini futures market provides a near ideal setting for exploring the interaction between trading intensity and volatility due to the integrated electronic trading platform and uniformly high degree of transaction activity, it does not constitute a natural environment for exploring invariance predictions regarding bid-ask spreads and trading costs. Finally, we reiterate that the summary statistics in this section involve the observed one-minute trade and volatility data, while the theoretical relations in equations (9) and (11)-(13) concern the expected values for the logarithm of those variables. The one-minute sampling frequency represents a specific trade-off between bias and variance. If we sample at a lower frequency, we exploit more data in the

13 1 Proportion of Trades with Q=1 and Q.14 Proportion of Trades with Q and Q 5.8.1.1.6.8.4.6..4. 5 5 1 15 5 5 1 15 Figure 3. The figure plots the proportion of trades with sizes Q = 1 and Q contracts (the left panel) and Q and Q 5 contracts (the right panel). The proportions are computed at t = 1-minute granularity. The dashed vertical lines indicate the three trading regimes. computation of any given expectation and obtain a less noisy measure, reducing the variance of the estimator. On the other hand, as the time interval grows, the underlying variables vary more substantially across the interval, inducing an increasing bias in the estimator for the nonlinear mapping implied by the various hypotheses. We deliberately err on the side of reducing the bias, and then mitigate the imprecision of our estimators, stemming from sampling and measurement errors, by averaging the noisy log-transformed variables over a large set of oneminute intervals. That is, we do not test whether the relations are satisfied for all distinct one-minute observations, but rather for specific subset of observations relative to others. As discussed previously, for the trade intensity, N, and asset price volatility, σ, we assume that active traders are attuned to the market developments and form unbiased forecasts for the relevant combination of variables over short intervals. Hence, for the transaction count, we use the observed values directly in our analysis. For the latent volatility, due to the liquidity of the E-mini contract, we are able to use six consecutive squared ten-second returns to compute a realized volatility measure without any major detrimental effects from microstructure noise. While this yields a very noisy estimator for the true local volatility, the subsequent aggregation across numerous one-minute intervals implies that we benefit from the same error diversification principle that accounts for the accuracy of standard high-frequency based realized volatility estimators. Nonetheless, the log transform presents a practical problem. During quiet market conditions, we may not observe a single trade over a given minute or the latest trade prices may be identical at the end of each of the six ten-second interval within a given minute, resulting in zero realized volatility measure. This renders the log transformation infeasible. Thus, we delete all one-minute intervals that feature zero realized volatility. Although the intraday trading invariance is valid

for any exogenously chosen subset of observations, the elimination of observations conditional on zero observed price changes will, in theory, induce an upward bias to the estimated volatilities of the remaining intervals. However, the filtered observations stem from periods where overall market activity also is extremely low, if not entirely devoid of trading, implying that various constraints may be binding and further complicate any meaningful test for invariance over these isolated intervals. As a robustness check, we also test for invariance at the substantially lower fiveminute sampling frequency, which enables us to retain almost all observations. 6 However, we stick with the one-minute sampling in our main analysis because it provides a more challenging test and allows us to enhance the granularity of our tests around key transition points in the market when the characteristics of the trading process shift abruptly. 14 IV. Testing Invariance for High-Frequency Data This section tests for invariance across the intraday market activity pattern using alternative specifications from Section II. A. Transactions and Market Activity We first depict the regression fit provided by equations (9), (11) and (1). The invariance relation in (9) implies that the transaction count over short time intervals, for which expected volatility and trading intensity is constant, varies systematically with the nominal dollar risk transfer executed through the market. Figure 4 plots the estimated fit for the intraday pattern, where each one-minute observation is obtained by averaging the stipulated relation across all trading days in the sample. Table Intraday OLS Regression of log N: 3 Models c β se(c) se(β) R Model 1.4119.9757.31.16.9965 Model 1.7538.849.18.6.9993 Model 3.8579.678.33.7.9986 This table reports on intraday OLS regressions of log N onto log W Q 3/ (Clark), log W Q (Ané & Geman), and log W (Invariance). All models predict β = /3. For each regression there are T = 1, 3 observations. The results provide strong qualitative support for the intraday trading invariance hypothesis as the estimated slope coefficient, depicted in the bottom panel 6 For the five-minute intervals, we filter 1.63%,.3% and.9 % of all observations due to zero realized variance in the three regimes. For the one-minute intervals, the corresponding numbers are 9.48%, 8.65% and 1.4%. These periods are effectively inactive markets periods and excluded from the tests.

15 logn vs logw/q 3/ logn vs logw/q 7 6 5 4 3 7 6 5 4 3 4 6 4 6 8 logn vs logw 7 6 5 4 3 4 6 8 1 Figure 4. This figure shows scatter plots of log N onto log W (Clark), log W (Ané & Geman), and Q 3/ Q log W (invariance). Also shown are OLS Regression lines (solid) and model predicted lines (dashed). of Figure 4, is very close to /3, while the alternative specifications in the top row generate slopes that deviate substantially from their benchmark values of /3. Additional details regarding the regressions are provided in Table. Figure 4 displays points corresponding to the Asian, European and North American segment in blue, green and red, i.e., the colors represent observations generated from the intervals contained in sets O 1, O and O 3, respectively. While the data are pooled in the regression, it is evident that the patterns for each region are consistent with the invariance predictions. Moreover, the cluster of 16 points indicated by (red) crosses, falling below the regression line for the invariance relation, indicates observations from 3:-3:15pm CT, where the cash market for equities is closed, but the futures market operates until the scheduled break at 3:15. In this segment, large trading firms tend to clear residual positions among themselves. Hence, the composition of traders changes and the average trade size increases significantly, as documented in Figure 3. Apparently, this shift in composition and transaction motive modifies the constant term but, strikingly, the slope generated by the crosses is again, for all practical purposes, identical to /3.

16 B. Trading Intensity and Trade Size While the analysis in Section IV.A favors intraday trading invariance over the volume or transaction clock hypotheses, it is striking that all specifications empirically generate a log-linear relationship and an extremely high degree of explanatory power in excess of 99% for the intraday variation in the transaction count. Indeed, the discrepancy between the alternative hypotheses is not the explanatory power but rather the regression slope relative to theoretical predictions. The strong log-linear relations are likely a consequence of the transaction count appearing both as a dependent and, implicitly, an explanatory variable. Due to our translation of market microstructure invariance to intraday trading invariance, the transaction count enters in lieu of the number of bets on the left hand side of the regressions, while market activity on the right hand side involves volume, i.e., the product of the number of transactions and trade size. Hence, the trading intensity is partially regressed upon itself, generating a strong association and rendering the degree of explanatory power invalid as a diagnostic. Nonetheless, the test is not flawed: the slope estimate generally attains the correct value only if the underlying theory is valid, as evidenced by the failure of the alternative hypotheses. The fundamental difference between the intraday trading invariance and existing theories is the role of the trade size, as brought out cleanly by equation (13). The theories offer dramatically different predictions regarding the association of volatility per transaction with trade size, ranging from a positive relation (Clark, β = 1), no relation (Ané-Geman, β = ), to a strongly negative relation (β =, invariance). This renders the regression a particularly striking test of the relative explanatory power of the alternative hypotheses. As in Section IV, we perform a cross-sectional regression analysis. Thus, we estimate the regression lines across the intraday pattern, where the logarithmic values of the relevant variables for each one-minute observation are averaged across all days in the sample. The results are presented in Figure 5 and Table 3. Table 3 Indraday OLS Regression of log σ N c β se(c) se(β) R -.675 -..199.99.9687 onto log Q This table reports the results of the intraday OLS regression, s t n t = c + β q t + u t. Clark, Ané & Geman and invariance predict β = 1, β =, and β =, respectively. The regression exploits T = 1, 3 observations. The left panel of Figure 5 shows that the distribution of the intraday observations now is less tightly centered on the regression line, even if the relation again is close to log-linear and generates an overall adjusted R of 96.7%. We also note that the estimated regression line (dashed) literally lies on top of the full line (theoretical

17 logσ /N vs logq logσ /N vs logq 5 5 6 6 7 7 8 8 9 9 1 1.5.5 3 3.5 1 1.5.5 3 3.5 Figure 5. The figures provide intraday scatter plots of log σ versus log Q N t t. Also shown are OLS Regression line (solid) and the predicted line according to invariance (dashed). The right panel is the same as the left panel, except it removes observations corresponding to 3 minutes around 1:, 3 minutes around : (start of the European regime), 3+3 minutes around 8:3 (start of Regime 3 and the 9: announcements), and 16 minutes at the end of trading. slope), so the intraday trading invariance relation again holds up remarkably well. In contrast, the fact that the regression line is very far from being either flat or upward sloping provides a dramatic rebuttal of the existing theories predicting a proportional relation between the transaction count or volume and return volatility. Upon inspection, we find that the most of the outliers in the left panel of Figure 5 stem from periods during which the market activity transitions from one trading center to the next. For illustration, if we eliminate observations corresponding to three minutes around 1: and : (the opening of trade in Frankfurt and London), three minutes before 8:3 (start of North American trading), plus the next 3 minutes until the 9: announcements, and the last 16 minutes of U.S. trading (when the cash market is closed), we obtain the right panel of Figure 5. Once more, the invariance prediction is validated in the sense that the regression slope equals the theoretical prediction. But equally striking, almost all outliers have been removed, suggesting the invariance relation is particularly accurate during trading originating from a single stable trading zone. In summary, we find strong evidence in favor of intraday trading invariance relative to existing theories which fail dramatically when confronted with the interaction of the market activity variables across the trading day. Evidently, as stipulated by invariance, the trade size is actively adjusted by the market participants to roughly equate the expected size of the risk transfers across time. C. Invariance during Macroeconomic Announcements We now explore whether intraday trading invariance provides a reasonable characterization of the dynamic market interactions during more extreme conditions. This section considers the release of scheduled macroeconomic announcements. The most influential announcements occur at 7:3, e.g., the Employment report

is generally released at 7:3 CT on the first Friday of the month, and the Consumer Price Index is typically released at 7:3 CT on the second Friday of the month. Other releases at this time-of-day include the Producer Price Index, the Employment Cost Index, the US Import/Export Price Indexes, Real Earnings, etc. These announcement often induce a distinct spike in market activity immediately after the release, and they constitute a challenging test for the invariance relationship. To ensure we truly are studying extreme market environments, we identify the days in our sample with the largest increase in trading activity after 7:3. That is, we compute the ratio of the number of trades for the 1 minutes after 7:3 versus the 6 minutes before 7:3, and we select the 1% of trading days with the highest ratios. In practice, this roughly leads us to pick up the two most influential 7:3 announcements each month. In the terminology of Section II.C, we denote this subset of 97 trading days by D A1, and refer to them simply as the 7:3 announcement days. Figure 6 depicts the same intraday market activity statistics as in Figure 1, but now averaged solely across the 7:3 announcement days, D A1. The upward spikes at 7:3 for the volume, number of trades, and volatility are very pronounced, while closer inspection reveals a simultaneous and smaller, yet clearly distinctive, downward move in the trade size. To assess the invariance hypothesis during these erratic episodes, we further restrict attention to the three minutes prior to and following 7:3, denoted T A1. Figure 7 displays the relevant points, corresponding to equation (9), for T A1. In the left panel, we average across all days, so we exploit all intervals in D T A1, while the right panel averages only across announcement days, i.e., intervals in D A1 T A1. As a benchmark, we also display the regression line from the bottom panel of Figure 4. The left panel of Figure 7 shows that invariance generally provides an accurate depiction of the trading-volatility interaction around 7:3. In the right panel, the observations for the three minutes following 7:3, indicated by crosses, are positioned further towards the upper right corner, reflecting the heightened market activity following announcements, but they still fall close to the regression line. Hence, the invariance hypothesis continues to provide a good characterization of the dynamic interactions during these more turbulent market conditions. Figures 8-9 report on the similar exercise, but for the 9: announcement days. The releases at that time include New Home Sales, Existing Home Sales, the Housing Market Index, Consumer Sentiment, Consumer Confidence, Business Inventories, and several others. On average, the 9: announcements are less influential in terms of their effect on trading intensity. Nonetheless, the spikes at 9: for the volume, number of trades, and volatility are dramatic. As before, we focus on 1% largest increases in trading activity, or 97 trading days. The results confirm that the invariance relationship holds up well in this frantic market environment. 18

19 x 1 4 3 Volume V.5 Volatility σ.5 1.5 1.5 1 1.5.5 5 5 1 15 5 5 1 15 Number of Trades N Trade Size Q 5 5 15 15 1 1 5 5 5 5 1 15 5 5 1 15 Figure 6. The figure shows averages across the days for which the 7:3 announcement resulted in biggest subsequent increase in trading activity. The statistics include share volume V t (per minute), volatility σ t, average trade size Q t, and number of trades. The averages are computed at t = 1-minute granularity. The dashed vertical lines indicate the three trading regimes. logn vs logw, 7:3, All Days logn vs logw, 7:3, Announcement Days 8 7 6 5 4 3 8 7 6 5 4 3 4 6 8 1 4 6 8 1 Figure 7. This figure shows a scatter plot of log N t versus log W t, but only for the 3 minutes before 7:3 (dots) and 3 minutes after 7:3 (crosses). The left panel plots the averages across all days, while the right panel plots the averages across the 7:3 announcement days. The solid line is the OLS regression line from the bottom panel of Figure 4.

x 1 4 3.5 Volume V 1.4 Volatility σ 3 1..5 1.8 1.5.6 1.4.5. 5 5 1 15 5 5 1 15 Number of Trades N Trade Size Q 3 5 5 15 15 1 1 5 5 5 5 1 15 5 5 1 15 Figure 8. The figure shows averages across the days for which the 9: announcement resulted in biggest subsequent increase in trading activity. The statistics include share volume V t (per minute), volatility σ t, average trade size Q t, and number of trades. The averages are computed at t = 1-minute granularity. The dashed vertical lines indicate the three trading regimes. logn vs logw, 9:, All Days logn vs logw, 9:, Announcement Days 8 7 6 5 4 3 8 7 6 5 4 3 4 6 8 1 4 6 8 1 Figure 9. This figure shows a scatter plot of log N t versus log W t, but only for the 3 minutes before 9: (dots) and 3 minutes after 9: (crosses). The left panel plots the averages across all days, while the right panel plots the averages across the 9: announcement days. The solid line is the OLS regression line from the bottom panel of Figure 4.

1 D. Invariance during the Flash Crash A qualitatively different challenge for the intraday trading invariance hypothesis is the chaotic market conditions during the so-called flash crash on May 6, 1. The dramatic developments are captured by the top row and left panel of the bottom row in Figure 1. The vertical lines indicate the official timing of the crash from the joint CFTC-SEC report on the incident. We note that the price only truly enters a free fall during the very last minute of the event window. Moreover, following the brief pause at the end of the crash, prices recover almost as sharply as they dropped. The event is accompanied by record trading volume and extreme realized volatility. The episode is of particular interest as there was a genuine breakdown in liquidity at the market trough, triggering the five-second CME Stop-Logic Functionality halt, and then an exceptional market turn-around. The question is whether the invariance relation is operative prior and during the flash crash period, how it holds up as the market collapses, and whether it displays unusual features during the subsequent recovery. 1 Price P x 1 4 Volume V 7 115 6 5 4 11 3 1 15 9 1 11 1 13 14 15 9 1 11 1 13 14 15 1 Volatility σ 4 3 Standardized logi 8 6 1 4 1 3 9 1 11 1 13 14 15 4 9 1 11 1 13 14 15 Figure 1. The figure shows price P, volume V, volatility σ, and standardized log invariant log I on May 6, 1. Standardized log invariant is computed at 4-minute frequency and is expressed in standard deviations. The solid vertical lines indicate the timing of the Flash Crash.