The Microstructure of Currency Markets: an Empirical Model of Intra-day Return and Bid-Ask Spread Behavior

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1 The Microstructure of Currency Marets: an Empirical Model of Intra-day Return and Bid-As Spread Behavior Paolo Pasquariello Stern School of Business New Yor University Revision July 9 th 00 Abstract This paper analyzes the intra-day relationship between bid-as spreads and maret return volatility for quotes posted in a truly lobal around-the-cloc maret settin, the one for the U.S. Dollar/Deutschemar exchane rate. We are able to identify a statistically and economically sinificant Reverse U-shaped pattern in the bid-offer spread in 996. Tests of the stability and orderin of maret volatility, performed across several different fractions of the day, reveal that variances of intra-day returns are heteroeneous and ordered, declinin around the Asian lunch brea, increasin steadily durin the London mornin tradin hours, peain at the openin of New Yor to subsequently fall with the closin of the European marets. Results also indicate that maret volatility is sinificantly hiher durin intra-day versus overniht periods. Then, we introduce a structural model that attempts to explain those empirical reularities by capturin some currencyspecific features of the data: possibly asymmetric and stochastic tradin cost structure, discrete directional updates and parameters temporal heteroeneity, and by relatin the bid-as spread to two different sources of variation: tradin costs and maret ris. We evaluate these components via GMM usin a set of convenient unconditional intra-day moments implied by the basic confiuration of the model. Analysis of the resultin estimated patterns reveals that tradin costs play a sinificant role in explainin the intra-day variability of bid and offer currency returns. Inventory considerations appear to be more relevant in the tradin mornin and by London s closin, while the perceived ris of arrival of informed trades seems more liely to affect the dealers cost structure durin the late mornin/early afternoon in London, when the bid-as spread is hihest. The contribution of the true currency ris to the total variability of the posted bid and as quotes returns is not surprisinly hihest at the openin of the European marets. Jumps are more liely to occur durin the London mornin hours and less liely by the end of the day. Additionally, the estimated upward ump size is hiher than the estimated discrete downward adustment. This evidence suests that durin 996 central bans unsuccessfully attempted to resist the devaluation trend of the Deutschemar versus the dollar. JEL Classification: C5, F3 Keywords: Maret Microstructure; Exchane Rate; Bid-As Spread; GMM Ph.D. candidate at the Stern School of Business. Please address comments to the author at the Leonard N. Stern School of Business, New Yor University, Kaufman Manaement Education Center, Suite 9-80, 44 West 4th Street, New Yor, NY 00-6, or throuh ppasquar@stern.nyu.edu. This paper has benefited from the comments of Viral Acharya, Stephen Filewsi, Richard Lyons, Arun Muralidhar, Latha Ramchand, Alex Shapiro, Amit Sinha, Marti Subrahmanyam, and the seminar participants at the 00 MFA Annual Conference, and at the 00 EFMA Annual Conference. I am especially rateful to Anelo Ranaldo for many insihtful discussions on the topic and to Joel Hasbrouc for introducin me to the discipline of Maret Microstructure and main available the data on which this research is based. The usual disclaimer applies.

2 . Introduction Microstructure theory often views the evolution of asset prices over short intervals of time as affected by systematic or unsystematic lon-run updates and transient cost components related to the particular maret in which tradin occurs. These costs are alternatively interpreted as arisin from inventory control considerations, asymmetric information and order processin. Schwartz 988 identifies four classes of variables the literature focuses on as determinants of bid-as spreads in financial marets: activity, ris, information and competition. Ho and Stoll 980, 98, 983 show that uncertainty in the order flow limits dealers ability to maintain their optimal inventory position. Consequently, as in Amihud and Mendelson 980, increasin order arrival variability would increase the bid-as spread. Tinic 97, Stoll 979 and Hamilton 978 hypothesize instead a direct relationship between the bid-as spread and the intrinsic ris of holdin a security. Several more recent studies relate information asymmetries between informed and liquidity traders to tradin costs in securities marets. Glosten and Milrom 985 and Hasbrouc 988 reveal that if dealers perceived exposure to private information rises, the bid-as spread would then widen. Few researchers e.. Hamilton 978 have also focused on the intensity of competition amon traders as a source of downward pressure on bid and offer quotes spreads. Data access constraints limited the empirical studies on assessin the statistical sinificance of these issues predominantly to equity marets. However, recent availability of hih-frequency observations has made it possible to apply tests of microstructure tradin theories to the vast forein exchane maret as in Lyons 995, 996. This study loos at the U.S. Dollar/Deutschemar DEM/$ as a currency traded in a truly lobal around-the-cloc maret settin, and identifies a statistically and economically sinificant Reverse U-shaped pattern in the intra-day bid-offer spread from quotes there posted in 996. Tests of the stability and orderin of maret volatility, performed across several different fractions of the day, reveal that intra-day returns variances are heteroeneous and ordered, risin from after the lunch brea in Asian marets until the early afternoon, in correspondence with the simultaneous openin of New Yor and the post-meridian phase of tradin in London, and subsequently declinin. Results also indicate that maret volatility is sinificantly hiher durin intra-day versus overniht periods. To address the issue of whether the behavior of the bid-as spread is related to the temporal profile of return volatility, we model bid and as quote-return dynamics. The chosen structure, similar in principle to Hasbrouc 999, incorporates not only some of the ey microstructure effects that are prominent in the literature stochastic, temporally heteroeneous and possibly asymmetric costs of maret-main that include clearin, information and inventory factors but also several specific For a comprehensive uide to the most influential theoretical wor in Maret Microstructure, we direct the reader to Maureen O Hara s 995 popular compendium.

3 features of currency marets, in particular the existence of discrete directional umps affectin the true expected price process. GMM estimates of intra-day parameters evolution and size show that, consistently with Lyons 995 findins, both tradin costs and intrinsic shocs are important in explainin the time-varyin variability of bid and offer currency returns. Nonetheless, their relative sinificance appears to fluctuate durin the day, with inventory considerations bein more relevant in the Asian mornin and London s closin tradin hours, and the true currency return volatility affectin the dealers cost structure in the early afternoon. Jumps appear to be more liely to occur durin the London mornin and less liely by the end of the day, while the estimated upward ump size is hiher than the estimated discrete downward adustment, suestin that durin 996 central bans unsuccessfully attempted to resist the devaluation trend of the DEM versus the dollar. The paper is oranized as follows. The next section characterizes statistically a sample of half-hourly DEM/$ exchane rate quotes and presents relevant empirical evidence on bid-as spreads and noon mid-quote return variance that motivates the modelin effort of this study. A parsimonious theoretical structure is then presented in Section 3, and estimated in Section 4. A discussion of the results ensues. A brief summary concludes in Section 5.. A Statistical Analysis of the Bid-As Spread in the DEM/$ Maret - The Forein Exchane Marets The forein exchane maret is probably the most active financial maret in the world. Amounts traded are hue, with over one trillion dollars in transactions executed each day, so is the frequency and intensity of tradin. 3 This paper examines bid and as quotes for the DEM/$ in 996. The DEM/$ exchane rate is a truly lobal 4-hours-a-day maret settin, where quotes posted by London-based traders compete with prices in New Yor or Franfurt, and so we treat it in this paper, rather than as a sequence of reional marets, as suested instead by Hsieh and Kleidon 996. Nonetheless, reional patterns are still oin to emere and matter in our analysis, as we will show in reater detail in the next sections. The structure of the DEM/$ now Euro/$ maret, as of any other actively traded currency, stronly differs from the ones of most equity and futures marets. Followin Bessembinder 994, we briefly focus on its three main aspects: oranization, the specific Lyons analysis, althouh includin quantities and transaction prices, is nevertheless limited to quotes observed on ust five tradin days, on the wee of Auust , from 8:30 a.m. to :30 p.m. Eastern Standard Time, too short a period to eneralize his results. Moreover, by introducin artificial openin and closin tradin hours, Lyons study does not address the tradin continuity issues we deal with in this paper. 3 The web site of Olsen and Associates, reports this and other interestin facts about currency marets that we use in this paper. 3

4 characteristics of the traded asset, and the nature of the relevant information flows. As compared with the centralized, order-driven specialist system used on the New Yor Stoc Exchane NYSE, for example, the wholesale currency marets are typically decentralized, quote-driven continuous dealer marets. This decentralized structure implies the absence of a centralized transaction reportin system for executed trades and of a centralized Clearin-House. Althouh many currency trades occur directly between dealers, still a portion of them is executed throuh broers that effectively serve as de facto information clearin houses. Althouh most equity issues and futures contracts are traded on marets with routinely daily openins and closins, forein exchane marets remain practically always active, except for weeends and few ban holidays, when a smaller amount of quotes is usually posted by dealers. While NYSE specialists enae in maret-main activities as an institutional feature of their role, currency dealers are enerally lare or medium commercial and investment bans whose other business activities miht also require extensive currency transactions. The most prominent currency maret maers are located in primary financial centers, lie London, Zurich, New Yor, Toyo and Hon Kon. They operate as dealers, tradin with each other as well as with non-ban customers, includin institutional investors, hede funds, bi and small corporations and multinational firms. Dealers indicate their willinness to trade by postin quotes on electronic systems provided by outside vendors, as Reuters or Telerate. However, those quotes are ust indicative, i.e. do not commit the dealer to execution. Actual transactions are completed privately or throuh electronic communication, as in the Reuters 000 Dealin System. Whereas equity valuations depend on a wide rane of both macro-economic and firm or industry-specific information, currency valuation arises mainly from macro-economic considerations. The resultin nature and frequency of the information flow in the currency marets miht reduce maret-maers lielihood of sufferin a loss to better-informed traders. Nonetheless, forein exchane marets are characterized by the activity of additional players, i.e. central bans and overnments. The monetary authorities of a country periodically intervene in domestic and forein currencies with the clear intent of directin the current and future transaction prices, in order to serve several political or economic aendas. Inflation pressures, competitiveness of local industries, flows of trade and international currency commitments are amon the most liely motivations for routine interventions by central baners. Althouh central bans do not have a hede in the size of the transactions for some of the most actively traded currencies as in the past, their power of moral suasion still appears to condition the activity and expectations of traders in the lobal currency arena. In any case, the presence of these players adds a layer of strateic interaction, not usually faced by equity maret maers, to the opportunity set of currency dealers. While stoc specialists treat their net equity position as inventory, with the domestic currency as the numeraire, forex dealers have an alternative in choosin which currency if either should represent the numeraire and which should instead 4

5 measure the inventory. Moreover, equity prices and futures are always quoted in specific units of currency per equity share or tic respectively. On the contrary, terms of quotation in exchane rate marets are a matter of convention, which varies across currencies, and which, as suested by Bessembinder 994, may even affect the distributional properties of the resultin bid-as spreads. In particular, althouh most maor currencies are quoted aainst the U.S. dollar, i.e. prices are the quantity of the forein currency per dollar, few exceptions are the Euro and some British Commonwealth currencies that are typically quoted in quantity of U.S. dollars per unit of the forein currency. Finally, currency marets do not impose a pre-specified minimum tic to the quoted prices, as in equity and futures marets. Maret maers can choose the desired fineness of the posted quotes, dependin on the thinness of the order flow and the cost structure they face. This maes the implyin transaction costs fully endoenous. Detailed data on intra-day transactions in this maret were not easily available in the past, as bans and other dealers have no obliation to report their private transactions. However, the development of computer and communications technoloy over the past few years has made available live data at shorter and loner time intervals durin the tradin hours. This availability is particularly interestin in the 4-hour-a-day currency marets. It beins every business mornin Greenwich Mean Time, GMT in the Asian marets, and increases rapidly, as Sinapore and Hon Kon oin Toyo and Sidney. There is then an abrupt decline of activity at 4:00 a.m. as these marets brea for lunch. Tradin pics up aain at 5:00 a.m. and remains stron throuhout the mornin, as London and the rest of Europe replace the Asian centers. The overlap of the New Yor and European marets in the afternoon produces the pea of daily volume activity. A steady decline then follows as the U.S. marets close durin the evenin. - Data Characteristics and Bid-As Spread Analysis This section performs a statistical analysis of the intra-day dynamics of the bid-as spread in the DEM/$ exchane rate maret that motivates the development of the model we present and estimate in the remainder of the paper. The sample we use in this study consists of bid and as quotes prevailin on the DEM/$ exchane rate on 6 reular U.K. business days in 996, reported in Deutschemar DM per U.S. dollar. The data are collected from Reuters on a half-hour frequency by Olsen & Associates in the HFDF II data set. Olsen & Associates utilizes a filterin process to remove false prices, or noise, due to mis-typin errors by traders. Althouh those quotes are not necessarily posted by the same dealer, competition amon maret maers, investors desire to trade at the displayed rates, and nowlede of transaction prices allow us to interpret the available bids and ass as bein formulated by a sinle dealer. Fiure shows the behavior of the DEM/$ extracted daily series computed as noon GMT mid-quote, that corresponds rouhly to the middle of the London tradin day. 5

6 DEM/$ Exchane Rate Mid-Quote at :00 Noon GMT DEM/$ //96 //96 3//96 4//96 5//96 6//96 7//96 8//96 9//96 0//96 //96 //96 Fi.. Daily DEM/$ Exchane Rate Mid-quote as of noon, GMT, 996. The Deutschemar experienced a devaluatin pressure aainst the dollar durin the year, althouh throuh some wide fluctuations around the depreciation trend. Prices are quoted here to four decimal places, i.e. with a tic size of 0.000, e DEM per one dollar. Incidentally we observe that the tic size reported here is practically imposed by the particular information system collectin the data. This observation is important, insofar as it implies that the clusterin phenomenon in spreads and quotes reported by the literature miht be instead less intense. 4 In fact for actively traded currencies, as in the case of DEM/$ in 996, traders increase the finesse of the quotes on their screens to the fifth or sixth decimal fiure. 5 As already emphasized in the previous sub-section, the bid and as quotes reistered from the Reuters terminals are ust indicative, i.e. do not necessarily represent prices at which transactions really occurred at or around the time the quotes were recorded. Moreover, these quotes are not necessarily the best bid and as on the maret at that particular time. 6 Nonetheless, Goodhart, Ito, and Payne 996 and Evans 998, 999 compare 4 See for example Bessembinder 994 and, more recently, Hasbrouc 999. From an economic perspective, clusterin arises when maret participants aree explicitly or implicitly to a price increment that is coarser than any technically mandated minimum. It has also been suested Christie and Schultz, 994 that quote clusterin miht reflect a collusive behavior amon dealers. 5 For example, in the case of DEM/$ exchane rate, dealers screens would report, for levels around.55, bidas quotes lie , meanin , thus attenuatin the supposed clusterin around 5 and 0 tics of the bid-as spread. 6 Hasbrouc 999 adds that these quotes are usually dominated by bids and ass appearin on the screens of the inter-dealer maret and on electronic tradin systems such as the Reuters D000- platform. 6

7 indicative quotes collected by Olsen & Associates with short samples of spot transaction prices, and find that the resultin indicative spread overestimates the manitude of the transaction bid-as spread and the relevance of clusterin, but, and most importantly for our study, appears to be a ood proxy for the intraday dynamics of transaction prices and spreads. Additionally, Bollerslev and Melvin 994 observe that reputation effects prevent the postin of quotes at which a ban would not subsequently be willin to trade. In order to analyze the intra-day behavior for the bid-as spread of DEM/$, we partition each day into 4 consecutive 60-minutes intervals and then calculate mean bid-as spreads. Followin Chan, Chun and Johnson 995, we at first define the DEM/$ volatility as the absolute loarithmic return resultin from the bid-as midpoint series. Finally, all variables are transformed into standardized deviates, by subtractin from each the observed mean for the day and dividin by the standard deviation for the day for the spread and the return, respectively. Table reports the mean values for the standardized spread SSPREAD and volatility SVOL for each 60-minutes interval of the day. Table Mean Values of the Standardized Volatility and Bid-As Spread for DEM/$ Exchane Rate from January 996 to December GMT SVOL^ SSPREAD^ 0:00 : :00 : :00 3: :00 4: :00 5: :00 6: :00 7: :00 8: :00 9: :00 0: :00 : :00 : :00 3: :00 4: :00 5: :00 6: :00 7: :00 8: :00 9: :00 0: :00 : :00 : :00 3: :00 0: The sample consists of 688 observations on DEM/$ exchane rate on one-hour frequency for reular U.K. business days in 996. ^The Standardized variable is defined as X it - µ i/s i, where X it is the raw variable, µ i is the mean for the day, and S i is the standard deviation for the day. The bid-as spread reveals a statistically sinificant intra-day pattern. It is lowest at the beinnin of the day, radually increasin after the Asian lunch brea until London s early afternoon before finally 7

8 declinin durin the New Yor evenin. As evident from Fiure, this pattern contradicts the U- shape in spreads typically documented for stocs, for example in Broc and Kleidon 99 and Chan, Chun and Johnson 995 amon others, in oranized exchanes, lie the NYSE, that are characterized by formal openin and closin hours. Standardized Bid-As Spread DEM/$ - January December Standardized DEM/$ Volatility - January December SSpread -0. SVol :00 :4 4:48 7: 9:36 :00 4:4 6:48 9: :36 0:00 0:00 :4 4:48 7: 9:36 :00 4:4 6:48 9: :36 0:00 Fi.. Intra-day DEM/$ SSPREAD and SVOL, 996, one-hour partition. The lines are smoothed interpolations of the variables estimates. Interestinly enouh, the intra-day pattern of volatility seems to mimic that of the standardized spread. This evidence does not prima facie support most of the existin information models Glosten and Milrom 985, Kyle 985, Admati and Pfeiderer 99, ust to quote a few in which maretmaers, facin informed and liquidity aents, increase spreads with lower volume and return volatility so that the ains from tradin with the uninformed compensates for the losses to the informed. 7 The results of Table and Fiure suest instead that the bid-as spread increases when there is more uncertainty surroundin the proxy for true currency values, as measured by SVOL for mid-quotes. 8 As lon as hiher volatility induces increasin order imbalance in the dealers inventory, the bid-as spread miht widen, thus confirmin early observations by Madhavan and Smith 993 and Hasbrouc and Sofianos 993 for NYSE stocs. The evidence on mimicin behavior of standardized volatility and bid-as spread is insensitive to the particular time partition 60 minutes we oriinally chose. We repeat the same analysis on an interval that rouhly corresponds to the London tradin day 5:00 a.m. to 9:00 p.m., GMT, and report the results in Table and Fiure 3. 7 In addition, if liquidity traders are allowed to choose when to enter the maret, the resultin order clusterin induces more maret depth, lower transaction costs and increasin competition amon informed traders. For more on this topic, see O Hara The patterns observed at the beinnin of the day appear nonetheless to be consistent with models of asymmetric information, see for example Foster and Viswanathan 994, which predict that tradin costs, hence the bid-as spread, are hih when volatility is hih at the start of the tradin session, i.e. when informed traders are asymmetrically informed. 8

9 Table Mean Values of the Standardized Volatility and Bid-As Spread for DEM/$ Exchane Rate from January 996 to December GMT SVOL^ SSPREAD^ 5: : : : : : : : : : : : : : : : : : : : : : : : : : : : : ^The sample consists of 576 observations on DEM/$ exchane rate on half-hour frequency for reular U.K. business days in 996. The Standardized variable is defined as X it - µ i/s i, where X it is the raw variable, µ i is the mean for the day, and S i is the standard deviation for the day. Aain, the standardized spread pics up simultaneously with an increase of SVOL from 5:00 a.m. to 8:00 a.m., is stable for most of the mornin, rises in the afternoon at the openin of New Yor and reaches the hihest level at around 3:00 p.m. GMT before declinin with the close of the American tradin floors. Standardized DEM/$ Spread & Volatility London Tradin Day SVOL SSPREAD :55 6:07 7:9 8:3 9:43 0:55 :07 3:9 4:3 5:43 6:55 8:07 Fi. 3. Intra-day DEM/$ SSPREAD and SVOL, 996, 30-minutes partition. The lines are smoothed interpolations of the variables estimates. 9

10 -3 Statistical Analysis of Intra-day Volatility The analysis of the last section suests that the behavior of intra-day volatility of the underlyin currency process, as measured by absolute returns, miht play a relevant role in explainin the Reverse U-shaped profile for the bid-as spread in DEM/$ quotes. This section examines more accurately a different measure of volatility, the sample standard deviation of hourly mid-quote returns for the DEM/$ exchane rate in 996, and formally tests the null hypothesis of stable variation within the day. Fiure 4 presents the sample volatility for intra-day hourly mid-quote returns computed as the natural loarithm of the DEM/$ relative value. Sample Period-by-Period Return Volatility on the DEM/$ Exchane Rate for % 0.4% 0.% stdev 0.0% 0.08% 0.06% 0.04% 0:36 3:00 5:4 7:48 0: :36 5:00 7:4 9:48 : Fi. 4. Intra-day DEM/$ sample mid-quote return volatility, 996, one-hour partition. The line is a smoothed interpolation of the variable s estimates. In comparison with the results reported in Fiure, the increase in return volatility appears more pronounced durin the London tradin day, and the evenin decline sharper than in the case in which the return variation is measured by absolute returns. Nonetheless, Fiure 4 is enerally consistent with our previous analysis. There is widespread areement in the empirical literature on exchane rates that currency marets tend to exhibit daily and weely seasonality. Glassman 987 and Bessembinder 994 report that both bid-as spreads and absolute returns, as measures of ris, are usually hiher on Fridays than on any other day of the worin wee. In Table 3 and Fiure 5 we estimate intra-day sample mid-quote return volatility for Mondays to Fridays in 996. No sinificant difference arises from the comparison of the resultin intra-day profiles, all displayin a Reverse U- shape. In particular, the mornin patterns are virtually identical across days. Nonetheless, Tuesdays and Thursdays, not ust Fridays, appear to be characterized by hiher return volatility at the pea of daily currency activity, with the overlap of the New Yor and European marets in the afternoon. 9 9 The recurrence of certain economic events e.. Fed meetins on Tuesdays in specific sub-periods of the day miht explain these results. 0

11 Table 3 Sample Return Volatility for DEM/$ Exchane Rate from January 996 to December GMT^ Measurement Period Total Monday Tuesday Wednesday Thursday Friday 0:00 : % % % 0.085% % 0.087% :00 : % % % % % % :00 3: % % 0.055% % % % 3:00 4: % % % 0.048% % % 4:00 5: % 0.050% % % % 0.094% 5:00 6: % % 0.09% 0.06% % % 6:00 7: % % % % % % 7:00 8: % 0.078% % % % % 8:00 9: % % 0.003% % 0.066% % 9:00 0: % 0.043% % 0.074% 0.086% % 0:00 : % 0.07% 0.089% % 0.097% 0.066% :00 :00 0.4% 0.085% % 0.005% 0.800% 0.084% :00 3: % 0.093% % 0.098% 0.85% 0.00% 3:00 4:00 0.4% % 0.07% 0.046% 0.94% 0.707% 4:00 5:00 0.3% 0.006% 0.50% % 0.9% 0.437% 5:00 6:00 0.4% 0.040% 0.090% 0.045% 0.355% 0.76% 6:00 7: % % 0.779% 0.75% 0.73% 0.468% 7:00 8: % 0.090% 0.40% 0.046% % % 8:00 9: % % 0.469% % 0.038% % 9:00 0: % 0.088% % 0.073% 0.096% % 0:00 : % 0.050% 0.055% % 0.064% % :00 : % 0.088% % 0.004% % 0.098% :00 3: % % % 0.066% 0.043% % 3:00 0: % 0.076% 0.37% % % 0.95% Number of Days ^The sample consists of 688 observations on DEM/$ exchane rate on one-hour frequency for reular U.K. business days in 996. We next test more formally the equality of variances across J intra-day intervals, i.e. the null hypothesis H 0 : σ = σ =. = σ J, usin various non-parametric procedures. The analysis is initially performed across 4 intra-day periods of 6 hours each. Then, results are reported for each sub-period in Table 4. The null hypothesis H 0 is examined first with the Brown-Forsythe 974 modified Levene 960 test statistic, already utilized in a study by Locwood and Linn 990 on stoc maret return volatility. The statistic is computed as: F = D D D i i= = = J = = R n J i n = i= J n D D Di D [ N J J ] = i= M D n i D n i N, [ ]

12 where R i is the return for day i, intra-day period, M is the sample median return for period computed over the n 6 days included in the test, D is the mean absolute deviation from the median for period and D is the rand mean. The statistic in [] is distributed F J-, N-J under the null hypothesis. Sample Standard Deviation for the returns on the DEM/$ Exchane Rate Day-by-Day Decomposition 0.% 0.9% 0.7% 0.5% stdev 0.3% 0.% 0.09% Friday Thursday W ednesday Tuesday Monday 0.07% 0.05% 0.03% 0.0% 0:36 3:00 5:4 7:48 0: :36 5:00 7:4 9:48 : Fi. 5. Intra-day DEM/$ sample mid-quote return volatility, 996, one-hour partition, Mondays to Fridays. The lines are smoothed interpolations of the variables estimates. In Table 4, the null hypothesis H 0 is reected at any conventional sinificance level, indicatin that maret return variance, as measured by the sample mid-quote return proxy, is statistically not constant across the four intra-day periods and is larer durin the London tradin day. In order to identify more precisely the typical intra-day pattern in maret volatility, the D i can be raned and examined for each day in the sample. Patterns in the D i rans allow us to test different expected orderins amon intra-day variances versus a null hypothesis of random orderin, an analysis typically nelected by traditional statistical approaches. Table 4 Modified Levene Statistics for Test of the Equality of Variance across Intra-Day Tradin Periods DEM/$ Exchane Rate Mid-Quote 996 K Measurement Period Mean Bid-As Spread Sample Return Vols # of Days 0:00 to 6: % 6 6:00 to : % 6 3 :00 to 8: % 6 4 8:00 to 0: % 6 F-Test for H 0 : σ =σ =σ 3=σ 4 F 3, = 3.80 F^ = 6.3** ** The F-Statistic is sinificant at the one percent level. For this purpose, we use a sinificance test developed by Pae 963. We define r i throuh r i4 as the rans from the smallest D i to the larest D i in day i. Because we have 4 sub-periods, the number of possible ranins is 4 4!. Under the null hypothesis of random orderin of the r i over 6

13 independent trials, we expect to observe an averae frequency for each orderin of 0.9, with a standard deviation of If we define h i as the ran under a pre-specified alternative hypothesis, the Pae ordered test statistic L is = N J L r i h i i= =. [ ] Thus, the Pae test uses the hypothesized rans h i and the observed rans r i, to produce a statistic that, for any day i, obtains its maximum value when h i, = r i, i.e. 30 for J = 4 and its minimum value when they are instead inversely related e.. 0 in our case with J = 4. Pae 963 provides critical values for the exact distribution of L. Here we instead use a modified version of Pae s test, L*, which is distributed approximately as chi-square with one deree of freedom, [L 3JN J ] L * =. [ 3 ] NJ J J We consider four alternative hypotheses, H A to H A4, for the observed orderin across intra-day subperiods, each of them ex-ante consistent with Fiure 4, i.e. implyin maret volatility risin from the openin in Asia until the European afternoon, and fallin thereafter -3-4-; -4-3-; and respectively. 0 Assumin, consistently with the null hypothesis, a sequence of 6 independent binomial trials, it then follows easily that: E 66 i [# ] = i p p VAR i= 66 [ ] i # = [ i E # ] p p i= i i 6 i 6 i where # is the number of cases in which the orderin is observed, and p is the ex-ante probability to observe # under the null of random orderin /4!. Locwood and Linn 990 report that the Pae test appears to be extremely powerful, especially if an a priori orderin is believed to exist. In this case, the ordinary use of χ tables is equivalent to a two-sided test of the null hypothesis. To implement instead a one-sided test, as in our case, the probability implyin from the chi-square tables needs to be halved. See Pae 963, pp. 0-4, for more details. 3

14 We report in Table 5 the number of days for which each of the alternative orderins is observed in our sample, and the correspondin Modified Pae test statistic. Table 5 Ran Statistics for Test of Random Volatility Ranin across 4 equally spaced Intra-Day Tradin Periods DEM/$ Exchane Rate Mid-Quote 996 Ranins 0:00 to 6:00 to :00 to 8:00 to # of Modified p-value* 6:00 :00 8:00 0:00 Days Pae Test H A ** H A ** H A ** H A ** H 0 : Random Ran RR #^ Observed Ran # Stdevs from RR # ^ Mean Random Frequency for each of the 4 possible ranins over 6 independent trials. stands for the lowest and 4 for the hihest sample return volatility. * The Modified Pae Test is distributed approximately as chi-square with one deree of freedom, and is computed accordin to equation [3] in the text. ** The Null Hypothesis is reected at the one percent sinificance level. H A is observed 8 times, more than 3 standard deviations away from the mean under the null hypothesis of random orderin. Evidence for the other H As is less stron. Nonetheless, the Modified Pae statistic reveals that the null is stronly reected at the one percent level aainst each of the four assumed alternatives, hence confirmin the statistical sinificance of the pattern we previously identified in Fiures and 4 for the maret volatility. The adoption of a finer partition of the day does not affect this conclusion. In Table 6 we apply the Modified Levene test of equation [] to a 3-hour time partition. Table 6 Modified Levene Statistics for Test of the Equality of Variance across Intra-Day Tradin Periods DEM/$ Exchane Rate Mid-Quote 996 K Measurement Period Mean Bid-As Spread Sample Return Vols # of Days 0:00 to 3: % 6 3:00 to 6: % 6 3 6:00 to 9: % 6 4 9:00 to : % 6 5 :00 to 5: % 6 6 5:00 to 8: % 6 7 8:00 to : % 6 8 :00 to 0: % 6 F-Test for H 0 : Homosedasticity F 7, =.64 F^ =.5** across sub-periods ** The F-Statistic is sinificant at the one percent level. Aain, the null hypothesis of Homosedasticity across sub-periods is reected at the one percent level. Finally, we ran, for every day in the sample, each sub-period by the observed maret return 4

15 volatility and report, in Table 7, the number of times for which each of the 8 possible rans has been observed. Ranins Table 7 Intra-Day Tradin Periods classified by Mid-Quote return volatility ranins over successive days DEM/$ Exchane Rate Mid-Quote 996 0:00 to 3:00 to 6:00 to 9:00 to :00 to 5:00 to 8:00 to :00 to 3:00 6:00 9:00 :00 5:00 8:00 :00 0:00 Low Vol Hih Vol χ Independence Test : T^ = 8.3* p-value = * The χ -Statistic is sinificant at the one percent level with 49 derees of freedom, and is computed accordin to footnote 6 in the text. # of Days The resultin matrix reveals a hiher concentration of hih volatility observations durin the London tradin day, and a hiher concentration of low volatility ranins in the Asian marets afternoons. A chi-square Independence test reects at the one percent level the null hypothesis of random orderin for the 64 combinations of volatility ranins and 3-hour sub-periods. 3 To summarize, the results of this section indicate that: a the Bid-As spread for DEM/$ quotes is characterized by a statistically sinificant Reverse U- shaped intra-day pattern; b the observed intra-day spread variation is economically sinificant as well: for a sinle roundtrip $ 00 Mil. transaction at a level of DEM/$.5000, the averae difference in the dealer s fee between the -to-5 pm interval and the Asian mornin hours, , is translated into approximately $ 7300, or about 8% of the noon GMT mean bid-offer spread; c the maret return volatility, as a proxy for maret ris measured over mid-quote intra-day returns, reveals a statistically sinificant seasonal daily pattern that is not explained by random chance. Variation in DEM/$ returns declines around the Asian lunch brea, then increases 3 The test compares the actual number of observations A i for each cell with the expected rane E i under the selected null hypothesis, in this case of random orderin. Hence, the null implies an averae of *6/64 observations per combination. A test statistic is then computed as: A E = 8 8 i i T. i= = Ei Under the null, T is distributed as chi-square with J-I- derees of freedom. 5

16 steadily durin the London mornin tradin hours, peas at the openin of New Yor to then fall with the closin of the European marets. Althouh poinant, the statistical analysis we ust performed does not substitute for an economic interpretation of our findins and eventually does not offer a satisfyin explanation of Fiure 6 below. It remains in fact to clarify a what, if any, is the relationship between the fluctuations of the bid-as spread and the intra-day behavior of the maret volatility; b how much of the maret volatility dynamics is explained by the variation of the underlyin true currency process and how much is instead resultin from fluctuations in tradin costs; c whether the variability of the true underlyin currency process and/or tradin cost considerations, both probably embedded in our estimates of maret variation, help explainin the puzzlin Reverse U-shaped spread we clearly see in Fiure 6. Spread Sam ple Return Volatility and Mean Bid-As Spread on the DEM/$ Exchane Rate for In tra-d ay P erio ds Volatility 0.56% Bid-As Spread 0.06% % % R etu rn V olatility 0.056% :30 4:30 7:30 0:30 3:30 6:30 9:30 : % Fi. 6. Intra-day DEM/$ sample mid-quote return volatility and bid-as spread, 996, three-hour partition. These unanswered questions are investiated in the remainder of this study. 3. The Model 3- Motivation The purpose of this section is to construct, and eventually estimate, a model for the bid and as intra-day return, return volatility, and the bid-as spread for exchane rates that captures the main characteristics of the data we presented in the last section, while incorporatin at the same time some of the features of past and current theoretical contributions to maret microstructure literature. Schwartz 988 identifies four classes of variables that appear to determine bid-as spreads in financial marets: activity, ris, information and competition. As suested by Stoll 989, those 6

17 variables induce three types of costs eventually faced by a dealer: order processin costs, inventoryholdin costs, and adverse information costs. More intense tradin activity may lead to lower spreads because of economies of scale in tradin costs. Researchers have shown that volume, number of shares traded and number of transactions are sinificant determinants of the bid-as spread in stoc marets. 4 Inventory control models, lie in Ho and Stoll 980, 98, 983, show that uncertainty in the arrival of buy and sell orders drives dealers away from their optimal inventory position. As in Amihud and Mendelson 980, when the trader approaches his desired inventory boo, the bid-as spread would be reduced. In short, if economies of scale predominate, then we should observe an inverse relationship between spreads and tradin activity. If we rely on the maret volatility we estimated usin mid-quotes as a proxy for intensity of tradin, then the evidence reported in the last section, especially in Fiures and 6, appears to refute this hypothesis. However, sample return volatility depends on the deree of variation of both the underlyin true currency process and the related tradin costs. Hence, Fiures and 6 alone do not help us disentanle the two effects on the observed bid-as spread Reverse U- shaped intra-day behavior. Other researchers, lie Tinic 97, Stoll 978, and Hamilton 978, describe a direct relationship between the bid-as spread and the ris of holdin a security, whether systematic or unsystematic. Althouh this hypothesis appears more consistent with our empirical findins, the issue of distinuishin properly the ris component related to the true asset price process and the ris component implied by the tradin activity itself however remains. Glosten and Milrom 985 and Hasbrouc 988 show that the spread miht be positively related to the amount of information comin to the maret. The trader s perceived exposure to private information determines how he will respond to lare versus small orders, and to arrivals of maret-enerated and other publicly available information. As the dealer attributes a positive probability to the order bein enerated from informed traders, the spread widens at times of the day durin which substantial informational chanes are more liely, hence when the true asset volatility is hiher. In the context of the lobal currency marets, these intra-day sub-periods correspond to the Asian mornin, the openin of the tradin day in London, and the subsequent activity followin from the U.S. marets. From a different perspective, Saar 000 investiates the role of demand uncertainty, i.e. uncertainty about preferences and endowments of the investors population, in introducin information content to the order flow. He shows that demand uncertainty increases both the bid-as spread and price volatility, consistently with our findins in Tables 4 and 6 and Fiures and 3. 4 For an exhaustive review of empirical and theoretical literature on the components of the bid-as spread, see Stoll 989, McInish and Wood 99 and O Hara

18 Finally, few researchers 5 have reported an inverse relationship between the level of competition and the observed bid-as spreads. This explanation does not appear to be satisfactorily for the case of forein exchane intra-day patterns, iven the evidence presented here. In fact, it is exactly at the pea of the daily activity in the London afternoon, when both European and American dealers are in the maret, that the widest spread is recorded, as clear from Fiure 3. From this discussion, it clearly emeres the need to identify sources of variation in the bid and as intra-day returns, accountin for the inventory costs and information components, and to relate them to the period-by-period behavior of the recorded bid-as spread. This is the tas we reserve to the next section. 3- The Basic Structure We construct a dynamic model for bid and as intra-day returns that incorporates a stochastic process for the underlyin true exchane rate, a random maret-main cost and a specific feature of forein exchane marets, discrete umps. This model will allow us to identify two different possible sources of variation in bid and as intra-day returns, tradin costs due to inventory and/or competition and maret volatility as a proxy for intrinsic ris and/or information asymmetry. The model is very similar in nature to the one of Hasbrouc 999, but distinuishes itself from it in three main aspects. First, we focus our attention on loarithmic returns, instead of levels, in order to pursue a different estimation stratey, GMM instead of Gibbs Samplin. Second, we inore discreteness and clusterin of quotes. Discreteness of quotes arises as a restriction of bid and as prices to a fixed rid. Clusterin is the tendency of quotes and spreads to lie on certain multiples of a basic tic size. Bessembinder 994 and Hasbrouc 999 exhaustively describe both phenomena and document their empirical relevance in currency marets. We observed in the past section that discreteness and clusterin of quotes and spreads miht arise from a tic size bein imposed to the DEM/$ fiures by the particular information system that collects the data. Althouh we do not mae any specific claim on the sinificance of this limitation on the empirical evidence available in the literature on clusterin, we choose to focus this paper on an analysis of the relationship between the bid-as spread and the volatility of bid and as intra-day returns, with the intent to maintain the parsimony of the selected specification. Third, as already mentioned, we introduce the possibility for the true underlyin currency process to be affected by a discrete upward or downward ump. Discrete movements in currency quotes may be induced by the release or the revision of fundamental macro-economic information or the occurrence of maor political events, but also by the purposeful intervention of domestic and forein 5 See Hamilton 976, 978 and Branch and Freed 977, for example. 8

19 central bans. In any case, the introduction of systematic ump ris appears to be one of the most popular and successful tools used by the literature 6 to capture the conditional and unconditional leptourtosis and sewness typically observed in lo-differenced exchane rates. We define the latent variable S t as the efficient or true currency price e.. DEM per dollar arisin at the end of the tradin day conditional on all public information available at that time. In what follows, we indicate any raw variable in upper cases, and the correspondin natural loarithm in lower cases. We assume that s t = LnS t chanes over time accordin to the followin process: s u t = s t µ u t y t t, [ 4a ] iid 0, u t ~ N σ [ 4b ] u y iid λ t ~ Poisson [ 4c ] t = 0 0 with prob p with prob p [ 4d ] where t =,,..,6 stands for a specific day in the sample, = /4,/4,.., is a specific fraction of the day and =,,..,8 is one of the eiht three-hour sub-periods we divide a tradin day in, 7 as in Table 6 and Fiure 6; µ u is a time-homoeneous drift in the exchane rate, while u t reflects updates to the true value of DEM/$ resultin from the release of publicly available information; y t is the number of discrete random ump updates, distributed as a Poisson variate. λ controls the frequency of arrival of events, its mean number and its variance. The binomial variable t determines the direction of the ump, with period-varyin probability p of bein positive. For the purposes of this paper, we consider the probability of a discrete ump to occur and of a ump to be upward as independent from each other and from the public information update u t. 6 See Bates 996 for a review of some of the main modelin solutions adopted in the financial literature to represent the behavior of exchane rates, and for evidence on DEM/$ exchane rate over the period 984 to Hence, t = t /4 corresponds to 0:00 a.m. of day t and sub-period, t 5/4 corresponds to 4:00 a.m. of day t and sub-period, and so on. 9

20 Dealers are assumed to face a non-neative stochastic cost of maret-main, C. As mentioned in section 3-, C incorporates any fixed or marinal i.e. per trade cost incurred by the currency dealer in postin non-bindin quotes. The randomness in the determinants of these costs, whether arisin from inventory considerations or asymmetric information, represents an additional source of variation for bid and as intra-day returns. The cost C is not necessarily symmetric, 8 thus reflectin the fact that in specific maret circumstances or moments of the day quotin a bid price may be more or less expensive than bein on the as side. We assume that a randomly extracted forex dealer from a population of maret-maers posts the followin raw bid and as quotes at the -th fraction of the t-th day, in the -th sub-period: Bid t = S t Cb t [ 5a ] As t = S t Ca t. [ 5b ] Cb < and Ca > represent the mar-up practiced by the maret-maer on the true currency price as a compensation for inventory carryin costs, information asymmetry ris, and maret ris. As mentioned earlier on, no tic and roundin restrictions to the posted quotes are considered here. We model the stochastic nature of these costs by assumin that: 9 t Ln Ca t ca t ca t = µ η Ln Ca ca ca t [ 6a ] t Ln Cb t cb t cb t = µ η Ln Cb cb cb t, [ 6b ] 8 Evidence on asymmetric bid-as spread, based on analysis of daily observations, is reported by Bossaerts and Hillion 99, amon others. 9 This choice is consistent with the dynamic implications of assumin lonormal tradin costs, as in Blume and Stambauh 983, Smith 994 and, more recently, Hasbrouc 999. In fact, for example in the case of bid costs Cb, if we instead assume that Ln[Cb t] ~ iid N[m cb, s cb], it follows that [LnCb t LnCb -t] is distributed as i.i.d. N[I tm cb - m cb, I ts cb s cb- - I ts cb], where I t = 0 for and - in the same interval, and otherwise. Equation [6b] would then ust impose the additional restriction that µ cb = I tm cb - m cb and σ cb = I ts cb s cb- - I ts cb. 0

21 where η η ρ ca cb ca, cb iid t ~ N iid t ~ N, ρ 0, σ ca 0, σ u, ca cb, ρ u, cb 0 [ 6c ] Althouh some of the determinants of maret-main costs are nown to be serially correlated, the random cost innovations are assumed to be i.i.d. Nonetheless, we do not exclude a priori the possibility that updates to public information have an impact on the cost structure ρ u,ca & ρ u,cb 0, not that there miht exists some interaction between the cost of bein on the bid or on the offer side of the maret at any point in time ρ ca,cb 0. If we define: = Bid t rbid t Ln Bid t [ 7a ] = As t ras t Ln As t [ 7b ] as the intra-day bid and as return respectively, our assumptions imply that bid t r t = [ s t s t] [ cb t cb ] [ 8a ] as t r t = [ s t s t] [ ca t ca ] [ 8b ] It also follows from [4], [5] and [6] that the bid-as spread representation provided by the model is: µ = u u t y t t µ ca ηca t µ cb ηcb t Spread t S t e Ca t e Cb t e. [ 9 ]