Journal of’ International Money and Finance (1993), 12, 413438 A geographical model for the daily and weekly seasonal volatility in the foreign exchange market
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`MICHEL M. DACOROGNA,
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`ULRICH A. MUELLER, ROBERT J. NAGLER, RICHARD B. OLSEN, AND OLIVIER V. PICTET* Olsen & Associates, Research Institute jbr Applied Economics, Seefeldstrasse 233, 8008 Zurich, Swlitzerland The daily and weekly seasonality of foreign exchange volatility is modeled by introducing an activity variable. This activity is explained by a simple model of the changing and sometimes overlapping market presence of geographical components (East Asia, Europe, and America). Integrating this activity over time results in the new 9 time scale, characterized by non-seasonal volatility. This scale, applied to dense datastreams of absolute price changes, succeeds in removing most of the seasonal heteroscedasticity in an autocorrelation study. Unexpectedly, the positive autocorrelation is found to decline hyperbolically rather than exponentially as a function of the lag. The foreign exchange (FX) market is the largest financial market, encompassing billions of dollars traded daily by thousands of actors, the microanalysis of which is complicated by the decentralizing (and destandardizing) effects of far-flung trading centers and disparate time zones. FX prices are characterized by daily and weekly seasonal heteroscedasticity, which we showed in a recent paper (Miiller et al., 1990). In this paper we follow with a geographical model that not only resolves seasonality through an activity-based time scale that we call ,9-scale, but also relates activity to physical location. The seasonality is strikingly well explained by the changing presence and the partially overlapping business hours of the main markets in the world. Deliberately, we do not try to model the complex intra-day behavior with a
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`.fidl
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`model of the generation process. Our ambition is to analyze, explain, and model the seasonal heteroscedasticity. Seasonal heteroscedasticity affects the results of most statistical studies so it must be treated as a first priority. Many researchers who study daily time series implicitly use, as a solution, a business time scale that differs from the physical scale in its omission of Saturdays, Sundays, and holidays. With the 9-scale we * This paper has benefited from comments by Casper G. de Vries, James R. Lothian. and two anonymous referees. The authors would also like to thank J. Robert Ward and Cindy L. Gauvreau for a careful reading of the manuscript. 0261-5606/93,!04/0413-26 IC:I 1993 Butterworth-Heinemann Ltd
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`GAIN CAPITAL - EXHIBIT 1030
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`414 GeogrupiCul rnodel~fhr duily and wecklJ3 seusonul aolutilit~~ extend this established concept to the intra-day domain, thereby allowing us to tackle a fundamental source of seasonality, the revolution of the earth. There are, therefore, three main motivations for our model:
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`l
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`l
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`to provide a tool for the analysis of market prices by extending the concept of business time scale to intra-day prices;
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`to make a first step towards formulating a model for the generating process of market prices that also covers the intra-day movements; . to gain an insight into the interactions of the main market centers around the world and their relevance to each particular FX rate. The importance of intra-day prices lies both in the large number of independent observations, which enhances the significance of a statistical study, and in the increased ability to analyze finer details of the behavior of different market participants. The few authors who have reported intra-day analysis (Feinstone, 1987; Goodhart and Figliuoli, 1988, 1991; Ito and Roley, 1987; Wasserfallen and Zimmermann, 1985; Wasserfallen, 1989) limited themselves to certain periods of the day, generally the most active ones for a particular market center, so the problem of daily and weekly seasonality was avoided. The paper by Goodhart (1990) is an exception; its subject is the interaction of different market centers around the world with different opening hours. Moreover, the type of data he analyzes is very similar to the one used in this study but for a shorter period. Another paper by Baillie and Bollerslev (1989) treats the same data as Goodhart and reports a behavior of the intra-day market volatility similar to that in Miiller rt Cl/. (1990). The nature of the intra-day data analyzed in this paper is presented in Section I. High density data require automatic filtering because of the enormous number of prices to be treated. Our approach to it is described in the Appendix at the end of the paper. In Section II, the seasonal heteroscedasticity is attributed to the active presence of traders on the FX markets, followed by the introduction of a new time scale, the ,3-scale. In this time scale, price changes have a non-seasonal volatility. We call its derivative against physical time the
`
`wtkity.
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`used
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`This new variable measures, for each time t, the active presence of traders on the FX market through the price changes they induce. The activity of the worldwide FX market is then decomposed into three submarkets geographically centered in East Asia, Europe, and North America. We show how the model parameters can be computed from the data and discuss some interesting results for particular FX rates. The definition of the market activity is discussed and compared with alternative definitions.’ The activity model is
`in Section III to construct the ,%scale as the time integral of worldwide activity. This scale, when used for the analysis of time series with high-frequency data, yields results that are no longer overshadowed by seasonality. Adopting the base requirement of zero seasonality, a statistical measure of the &scale quality is also proposed. Some inevitable complications of the activity model and the &scale construction, such as business holidays and daylight saving time, are discussed. To illustrate the usefulness of this approach, a comparative study of auto- correlation for absolute price changes in twenty minute intervals is presented in Section IV. We show that the autocorrelation peaks indicating seasonality vanish when the analysis is done under N-scale instead of physical time, demonstrating
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`MICHEL M. DACOROGNA
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`415
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`the success of this new time scale. The remaining autocorrelation reflects the conditional heteroscedasticity which can be studied with enhanced accuracy thanks to the high data density. We show that the behavior of the autocorrelation function on the 9-scale is more accurately described by a hyperbolic rather than a conventionally expected exponential decline. Our conclusions are given in Section V.
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`I. Data
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`/.A. The FX market and the duta sources The bid and ask offers of major financial institutions are conveyed to customers’ screens by large data suppliers such as Reuters, Telerate, or Knight Ridder and the deals are negotiated over the telephone. The FX market has no business hours limitations. Any market maker can submit new bid/ask prices; many larger institutions have branches worldwide so that trading is continuous. Nevertheless, the bid/ask prices do emanate from particular banks in particular locations and the deals are entered into dealers’ books in particular institutions. In this study, mainly Reuters data is analyzed. It is the same as that described in Miiller et al. (1990) so our description here is limited to certain aspects that were not mentioned in the earlier paper. Data collected from Knight Ridder is also used to study the dependence of the results on the data supplier. The interbank spot exchange rates are published by Reuters in multiple contributor pages (FXFX, FXFY, and ASAQ). These three Reuters pages contain spot prices of 26 time series, including 24 FX rates against the US Dollar (USD) and two commodities, gold (XAU) and silver (XAG).’ In Table 1, the list of the 12 most frequently updated time series collected on the FXFX, FXFY, and ASAQ pages of Reuters are shown together with their tick frequency statistics. Besides the normal business day tick frequencies those for Saturday are also included, because they are so much lower than the average. Sunday frequencies are not so low, as late Sunday evening (GMT) is already Monday morning in East Asia. The number of ticks counted on a Reuters page is related to the market shares of the particular FX rates, but it depends on the actual market coverage of the Reuters information system. The four or five mujor FX rates are the most frequently updated. Our database now contains more than 12 million ticks for the 26 time series. It covers almost every day of the year except for rare failures of our system or of the data supplier which lead to dutu holes, over which we use linear interpolation, as in Miller et al. (1990). Data filtering is necessary; our approach is described in detail in the Appendix. The results presented here are computed over a sample of four full years, starting March 3, 1986 and ending March 3, 1990. The main variable studied is the logarithmic middle price xj, as in Miiller rt al. (1990),
`
`log Pbid,j+“g
`
`Pask,j
`
`xj E
`
`where Pbid,j and Pask,j are the bid and ask prices.
`
`et al.
`2
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`and ASAQ pages from Reuters. Q 2 Q Mean ;; ‘5 business Mean s- First K. Exchanged Expressed day Saturday tick k rate in frequency frequency 3 recorded g US Dollar US Dollar US Dollar British Pound US Dollar Australian Dollar Gold US Dollar US Dollar US Dollar European Curr. Unit US Dollar Deutsche Mark Japanese Yen Swiss Franc US Dollar French Franc US Dollar US Dollar Netherlands Guilder Italian Lira Canadian Dollar US Dollar Spanish Peseta USD-DEM 3100 49 USD-JPY 1900 42 USD-CHF 1550 45 GBP-USD 1500 49 USD-FRF 1000 38 AUD-USD 900 37 XAU-USD 800 70 USD-NLG 600 30 USD-ITL 550 39 USD-CAD 480 0 ECU-USD 460 0 USD-ESP 350 0 01.02.86 ‘< 31.01.86 : 01.02.86 $ 01.02.86 2 31.08.86 > 27.11.86 01.02.86 & 27.11.86 g 27.11.86 $ 27.11.86 p 10.01.89 x 27.11.86 s
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`FXFX, FXFY,
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`TABLE
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`1. Characterization of the 12 most updated time series collected on the
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`MICHEL M. DACOROGNA
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`417
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`sf
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`I.B. Multiple contributor effect on very short-term price variations The database prices used for our analysis are quoted prices and not actual trading prices.j FX traders we have interviewed find that the FXFX prices lag behind the real market prices and the quoted spreads are about twice as large as those actually used in the deals. According to their estimates, the lag of FXFX prices varies between a few seconds in periods of low volatility and one minute or more in highly volatile periods. Goodhart and Figliuoli (1991) have reported a negative first order auto- correlation of price changes with a lag of one minute. We attribute these short-term oscillations (with amplitudes of about half the spread) to a multiple contributor efltict: many market makers have current preferences for either selling or buying and publish ‘new prices’ attracting traders to make a deal in the desired direction. The sequence of quoted prices originates from market makers with dzfltirent and changing preferences. A proper treatment of this problem would deserve a special study. We have not tried to make a model for the actual traded prices. In order to avoid the multiple contributor effect in our analysis, we do not use samples with resolutions finer than 10 minutes. II. A model for the FX market activity II.A. Season& patterns
`the volatility and presence oj’markets The behavior of a time series is called seasonal if it shows a periodic structure in addition to less regular movements. In Mtiller et al. (1990) we demonstrated daily and weekly seasonal heteroscedasticity, a seasonal behavior of FX price volatility rather than of FX prices themselves. The seasonality has been found in a study with intra-daily and intra-weekly sampling as well as in an autocorrelation analysis. Autocorrelation coefficients are significantly higher for time lags that are integer multiples of the seasonal period than for other lags. An extended autocorrelation study is included in this paper, in Section IV. The intra-week analysis in Miiller et al. (1990) shows that mean absolute price changes are much higher over working days than over Saturdays and Sundays, when the market actors are hardly present. The intra-day analysis in the same paper shows that the mean absolute hourly price changes have distinct seasonal patterns. These patterns are clearly correlated to the changing presence of main market places of the worldwide FX market. The lowest market presence outside the weekend happens during the lunch hour in Japan (noon break in Japan, night in America and Europe); it is at this time when the minimum of mean absolute hourly price changes is found. Further evidence of a strong correlation between market presence and volatility is provided by another result of Miiller et al. (1990): the intra-day behavior of the frequency of price quotes in the Reuters system. This variable obviously reflects the presence of markets and is positively correlated to volatility (in terms of mean absolute hourly price changes). Market presence is related to another variable which cannot be observed directly: the worldwide transaction volume. Many empirical studies give substantial evidence in favor of a positive correlation between price changes and volume in financial markets (see the survey of Karpoff, 1987). The correlation of market presence and volatility leads to a central idea of this
`
`et al.
`

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`418 Geogruphicul rwxi~l,f~w daily und weeki)’ .seusonuI wlutilit~’ paper: to model and to erplain the empirically found seasonal volatility patterns with the help of fundamental information on the presence of the main markets around the world. We know the main market centers (e.g., New York, London, Tokyo), their time zones, and their usual business hours. When business hours of these market centers ot:rrfup, market activity must be attributed to their cumulative presence; it is impossible to assign the market activity to only one financial center at these times. The typical opening and closing times of different markets can be determined from our database, which also contains the originating locations of the quoted prices. In Baillie and Bollerslev’s (1989) approach, the seasonality of volatility is modeled by dummy variables with no further explanation. We consider it advantageous to try to identify at every moment of the day lvhich markets are responsible for the volatility. 1I.B. Modeling the volutility patterns \vith an alternative time scale and an ucticity variable Before relating the empirically observed volatility to the market presence, we introduce a model of the price generation process which will be used for describing and analyzing the seasonal volatility patterns. A generation process of price changes with strong intra-day and intra-week volatility patterns cannot be stationary. Our model for the seasonal volatility fluctuations introduces a new time scale, which, once used as the directing process S(t) of the subordinated price generating process x(t) = .~*[3(t)], will make the process x* non-seasonal and, if possible, stationary. Although a subordinated process is not the only possible model to treat the observed seasonality, other standard techniques of deseasonalization do not apply as the volutility is seasonal, not the raw time series. Similarly a variety of alternative time scales has been proposed, in different contexts, for treating the generation process of time series. In the early sixties, Allais (see, for instance, Allais, 1966) has proposed the concept of psychological time to formulate the quantity theory of money. In 1967, Mandelbrot and Taylor suggested cumulating the transaction volume to obtain a new time scale which they call the transaction clock. Stock (1988) studied postwar US/GNP and interest rates and proposed a new time scale to model the conditional heteroscedasticity exhibited by these time series. Here we propose to use a new time scale to account for the seasonality. Because the &scale fully accounts for the seasonality of x, Y* has no seasonal volatility patterns. The process .Y* may however have non-seasonal volatility patterns; it may be conditionally heteroscedastic. No attempt is made in this paper to determine its exact nature. The time scale s(t) is a strictly monotonic function of physical time t. Any time interval from t1 to t2( > tl) corresponds to a &time interval of the positive size 9,-9,. The new activit?, variable a is defined as the ratio of the interval sizes on the different scales,
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`MICHEL M. DACOROGNA
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`419
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`This activity reflects the seasonal volatility patterns. Its relation to other ‘activity’ variables such as market presence or transaction volume has been mentioned in Section 1I.A and is discussed below. II.C. Activity und scaling 1aM The volatility-based activity defined by equation (2) can be computed with the help of the empirical scaling law, (see Mi.iller et al., 1990) of price changes which relates lAxI, the mean absolute change of the logarithmic middle price over a time interval to the size of this interval, At: (3) IAxl = c AtlIE, where the bar over IAxl indicates the average over a long sample interval (four years in our case) and c is a constant depending on the FX rate. If At is expressed in hours, c is in the order of magnitude of lop3 for the main FX rates against the USD. The drift exponent 1 /E is about 0.6, whereas the pure Gaussian random walk model would imply 1 /E =OS. The scaling law expressed in equation (3) holds for all time series studied and for a wide variety of time intervals ranging from 10 minutes to more than a year. The scaling law is applied to subsamples in a so-called intra-week analysis that allows us to study the daily seasonality (open periods of the main markets around the world) as well as the weekly seasonality (working days-weekend). For this study, we choose a sampling granularity of At = 1 hour. The week is subdivided into 168 hours from Monday O:OO-1:00 to Sunday 23:00-24:00 (Greenwich Mean Time, GMT) with index i. Each observation of the analyzed variable is made in one of these hourly intervals and is assigned to the corresponding subsample with the correct index i. The 168 subsamples together constitute the full four-year sample. The sample pattern is independent of bank holidays and daylight saving time. Any analyzed variable can be plotted as a histogram against the 168 hours of the statistical week which shows the typical intra-day and intra-week patterns of the variable. An example of such an analysis is shown in Figure 1. The scaling law, equation (3), is applied to the ith hourly subsample instead of the full sample and mathematically transformed to (4) From Miiller et al. (1990), we know that Axi can strongly vary for the different hours of the statistical week. The time interval At = 1 hour (for the hourly sampling) is nevertheless constant. Therefore, it is replaced by the interval A,!Ji on the new time scale 9. The size of A9i is no longer constant, but reflects the typical volatility of the ith hour. The constant c* is essentially the c of equation (3), but can differ slightly as it is calibrated by a normalization condition presented below. The mctivity of the ith hourly subsample directly follows from (2), (5) a At = 1 hour.
`
`et al.
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`1.0
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`intra-week hourly index FICUKE 1. Histogram of the average hourly activity for a statistical week (over 4 years) for the USD-DEM rate. This is the volatility-based activity definition used in the following analysis. The constant c* is calibrated to satisfy the following, straightforward normalization condition : (6) II.D. Geogrupphicd components of the actkit?. In Figure 1, the histogram of the average hourly activity defined by equation (5) is shown for the USD-DEM rate. Although the activity definition is based only on price change statistics, the histogram exhibits clear structures reflecting market presence: very low activities over the weekends and strongly oscillating activity patterns on normal business days. The most active period is the afternoon (GMT) when the European and American markets are open simultaneously. We have varied the At granularity of this analysis from l/4 hour to 4 hours and found no systematic deviations of the resulting activity patterns from the hourly ones. Furthermore, the activity patterns are remarkably stable for each of the four years of the total sample. The strong relation between price change based activity and market presence leads to the explanation of activity as the sum of geographical components. Although the FX market is worldwide, the actual transactions are executed and entered in the bookkeeping of particular market centers, the main ones being London, New York, and Tokyo. These centers contribute to the total market activity during different opening hours that sometimes overlap. Goodhart and Figliuoli (1988) have explored the geographical nature of the FX market to look for what they call the island hypothesis. They studied the possibility that the price bounces back and forth from different centers when
`
`420 Geogruphicul rnodel,fbr daily und weekly seusonal colutilit~~
`

`

`et al.
`
`421 special news occurs before finally adjusting to it. They do not report a strong effect of this kind. Along the same idea, Engle et al. (1990), in a study with daily opening and closing USD-JPY prices in the New York and Tokyo markets and a market-specific GARCH model, look at the interaction between markets. They use the terms heat wave hypothesis for a purely market-dependent interaction and meteor shower hypothesis for a market-independent autocorrelation. They find empirical evidence in favor of the latter hypothesis. Both studies have not found peculiar behavior for different markets. This encourages us to model the activity with geographical components exhibiting similar behavior. The activity patterns shown in Figure 1 and the results reported in Miiller et al. (1990) suggest that the worldwide market can be divided into three continental components: East Asia, Europe, and America. The grouping of the countries appearing on the Reuters pages in our three components can be found in Table 2. This division into three components is quite natural and some empirical evidence supporting it will be presented in Section IV. The model activity of a particular geographical component k is called a,(t); the sum of the three additive component activities is a(t): 3 (7) u(t) = c a,(t). k=l This total activity should model the intra-weekly pattern of the statistic& activity a sta,,i as closely as possible. Unlike ustat, which has a relatively complex behavior (see Figure l), the components a,(t) should have a simple form, in line with known opening and closing hours and activity peaks of the market centers. II.E. A model jbr the intra-weekly activity
`
`qf
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`each market Each of the three markets has its activity function &(t). For modeling this, we use quantitative information on market presence. A statistical analysis of number of price quotes4 originating from each of the three markets defined by Table 2 reveals two aspects on market presence: 0 A market has opening times that are longer than those of a particular submarket, e.g., an individual bank in one financial center such as Tokyo, TABLE 2. Grouping of the different countries appearing in the FXFX, FXFY, and ASAQ pages according to the three components of the worldwide market. Index
`Component Countries 1 East Asia Australia, Hong Kong, India, Indonesia, Japan, South Korea, Malaysia, New Zealand, Singapore. 2 Europe Austria, Bahrain, Belgium, Germany, Denmark, Finland, France, Great Britain, Greece, Ireland, Italy, Israel, Jordan, Kuwait, Luxembourg, Netherlands, Norway, Saudi Arabia, South Africa, Spain, Sweden, Switzerland, Turkey, United Arab Emirates. 3 America Argentina, Canada, Mexico, USA.
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`k
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`MICHEL M. DACOROGNA
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`422
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`Geoyruphiml model,fbr daily and weekly seasonal wlutility
`
`union
`of the opening times of all relevant institutions of the market.
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`l
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`Two markets (East Asia and Europe) have a local price quote frequency minimum in the middle of their working day, corresponding to a noon
`
`break.
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`This local minimum is very pronounced in East Asia and moderate in Europe. In America, there is no minimum around noon. These differences reflect the well-known, different business habits concerning lunch breaks. Each of the three markets is modeled to have two basic states, either open or closed. The activity does not completely go to zero when the market is closed because it is defined in terms of price changes. The activity during the closing hours is modeled to stay on a mull constant base level u~.~. During the opening hours, a much stronger, varying, positive activity CL~.~ adds to the base level: The joint base level u0 is regarded as one model parameter. There is no need to analyze components u~,~. The activity during opening hours, u~,~, is modeled with a polynomial with smooth transition to the constant behavior of the closing hours. This choice is arbitrary but mathematically convenient since such functions are easily differentiable and analytically integrable. The number of free parameters of this polynomial is just sufficient to model the smooth transitions, the lunch break found in the quote frequency statistics, and a certain skewness (relative weights of morning and afternoon). In the subsequent analysis, the statistical week is considered from
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`t =0
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`on Monday 0O:OO to t = 168 hours on Sunday 24:00 (GMT), as shown in Figure 1. In order to define the opening and closing conditions of the markets in a convenient form, an atrxil@~ time scale
`is introduced. Essentially, it is GMT time; the following market-dependent shifts of plus or minus 24 hours are only for technical convenience: (9) Tk = [(r + Atk) modulo (24 hours)] - Atk, where At, has the value of 9 hours for East Asia, 0 for Europe, and -5 hours for America. (The result of the modulo operator is the left hand side argument
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`Tk
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`minus
`~~wkend
`condition also depends on the market.” (10) (t+At,) modulo (168 hours) 3 120 hours.
`Now
`the model for an individual market component can be formulated: (11) Ul,kV) = i ; if
`Tk < ok
`Tk > ck
`or weekend (equation (10)) open.k(t) if ok <
`Tk < ck
`
`and not weekend (equation (10)) where ok and ck are the opening and closing hours respectively. The polynomial function is: (l-2) n,,,“,,(r)=o,~f(:ll (Tk-Ok)2(~-C~)2(Tk-.Sk)[(Tk--~)2+dkZ], I ‘I 2 - Sk
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`Paris, or Chicago. The market opening time is the
`the nearest lower integer multiple of the right-hand side argument). The
`or
`

`

`dk
`
`d,
`
`d,
`
`MICHEL M.DACOROGNA eta/. 423 where wk represents the scale factor of the kth market, sk the skewness of the activity curve, mk fixes the place of the relative minimum around the noon break, and
`determines the depth of this minimum. The special form of the first factor is chosen to avoid too strong a dependence of the scale factor on sk. This model applies to all markets. The European and Asian markets (k = 1,2) have finite
`always diverges to very high values. This reflects the missing noon break in this market, which has already been found in the tick frequency statistics. Equation (12) for America thus degenerates to a simpler form with no local activity minimum: (13)
`(~3-~3)'(T3-c3)2(T3-~3). 3 3 F-s3 Some of the model parameters, the opening and closing times, are already known from the quote frequency statistics. For the other parameters, there are constraints. In order to ensure positive activities, a,, and ok must be positive and sk outside the opening hours: (14) a0 > 0, wk > 0, Sk d ok or sk 3 ck. The parameter
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`a open,3(t) = o +y3
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`mk
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`in equation (12) should be within the opening hours as it models the noon break: (15)
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`o,<m,-cc,.
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`a,,,(t)
`a(t)
`a&t).
`must be fitted to the results of the statistics, astat( by minimizing the integral of the weighted square deviation of
`asta,,i
`A continuous function
`a&
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`from ) equation (5). Therefore, the sum over the intra-weekly sample is used instead of the integral: (16) .-ao-~k3=l
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`al,k(ti)l’
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`.
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`in, ri = (i-4) hours.
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`i=l
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`oLr,i
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`The hourly intervals are represented by their middle points in this approximation. There are 11 parameters of the least square fit: three wk’s, three sL’s, two
`d,‘s,
`a,.
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`mk’s,
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`The values of opening, ok, and closing, ck, result from the price quote frequency statistics and have certain random errors. Therefore, these values are allowed to vary
`for adjusting the fit. The minimization problem of equation (16) is non-linear in some of the parameters. It is solved by the Levenberg-Marquardt method (see Press
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`slightly
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`et al.,
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`1986, section 14.4). The main American and European markets observe daylight saving time during summer, whereas the main East Asian markets do not. This fact is ignored for the fitting. Only the GMT scale is used. A posterior daylight saving time correction is proposed in Section 1II.C. The resulting fitted parameters for four major FX rates and gold are presented in Table 3 together with the relative weights of the different markets (to be defined in Section 1II.A). In Figure 2a, the resulting activity model together with the statistical activity for the USD-JPY is shown and in Figure 2b the same quantity
`
`values in the fitting process, but for the American one, the parameter
`The functions
`from
`t is not available but rather the hourly series
`stat,,
`two
`and the base activity
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`2
`2
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`s
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`m
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`c
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`0
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`k
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`00
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`0.0294 1 East Asian 24.1% 1.685 -3:32 8:24 3:33 0.91 -3:33 2 European 38.5% 1.070 5:54 18:39 11:07 2.06 20:21 8 3 American 34.1% 12.46 11:24 23:25 40:44 : 0.0334 1 East Asian 35.4% 1.402 -4:14 8:43 3:35 1.01 -4:17 B 2 European 27.6% 5.369 6155 16:40 11:02 1.51 17:23 3 American 33.4% 18.73 11:48 22:50 34:55 $ 0.0241 1 East Asian 24.3% 1.051 -3:48 8:59 3:40 1.08 -4:02 $-& 2 European 39.1% 0.974 6:00 18:19 11:13 2.85 20:05 $ 3 American 34.0% 13.88 11:24 23:ll 31:43 Q_ f 0.0080 1 East Asian 22.0% 1.123 -4:oo 9:oo 3:40 1.06 -4:oo f 2 European 45.1% 1.037 5:oo 18:00 11:23 2.45 -4145 g 3 American 31.6% 13.71 12:oo 24:00 24:00 0.0162 1 East Asian 9.7% 0.136 -3:43 9:36 4:05 3.17 -4:15 ; B 2 European 54.8% 2.978 5:36 17:19 11:lO 1.54 2142 9 3 American 33.8% 354.9 15:21 21:30 21:32 ” The sum of the market weights is less than 100%; the rest is accounted for by the basic activity a,. The parameters o, c, S, and m are given in GMT (in the sense of E S equation (9)). The residual activity a,, the scale factor o, and the parameter
`TABLE 3. The resulting fitting parameters for the major FX rates and gold with the corresponding market weights. Rate USD-DEM USD-JPY GBP-USD USD-CHF XAU-USD ?
`
`d
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`which measures the depth of the minimum at lunch time are dimensionless numbers. q The values reported for w must be multiplied by 10m4.
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`d
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`Market Weight w
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`MICHEL M. DACOROGNA
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`et al. 425 0
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`E
`lu
`;;i
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`(D
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`z
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`FIGURE 2.
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`The histograms of the average hourly activity for a statistical week (over 4 years) for the USD-JPY (above) and USD-CHF (below) rates plotted together with their respective fitted model activity. for the USD-CHF. In Figures 3a and 3b, the activity model over 48 hours (outside the weekend) with its different components for the same rates is displayed. II.F. Interpretation of the activity modeling results The resulting parameters of the activity model and the Figures 2a, 2b, 3a, and 3b confirm the close relation between market presence and intra-weekly volatility
`
`intra-week hourly index
`intra-week hourly index
`

`

`426
`
`0
`
`6
`
`12
`hourly~kiex
`
`(22&a
`
`30
`
`36
`
`42
`
`48
`
`1.0
`
`0.0
`
`_.._._ ,. .___. &
`
`ast
`
`0
`
`6
`
`12
`hourIy'kiex
`
`FIGURE 3.
`
`30
`
`36
`
`42
`
`48
`
`The model activity decomposed into the three different continental markets over a period of 48 hours during normal business days for the same rates as in Figure 2. The bold curve is the sum of a, and the three market activities. patterns. The market-specific tick frequency analysis and the activity fitting results compare favorably taking into account the Reuters coverage and the limitations of our model. In both cases and for all FX rates, the local minima around noon have the following properties: they are pronounced in East Asia, moderate in Europe, and do not exist in America.