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`
`5.6 AUTOCORRELATION AND SEASONALITY
`
`167
`
`TABLE 5.13 Correlation coefficients for activity measures.
`Correlation coefficients computed for the different intraday analyses for the USO against
`DEM, JPY, CHF, and GBP and XAU (gold) against USO. Sampling period: from January 1,
`1987, to December 31, 1993.
`
`E(lrl)-ticks
`E(lrl)-spread
`Ticks-spread
`
`DEM
`
`+0.540
`-0.220
`-0.693
`
`JPY
`
`+0.421
`-0.485
`-0.018
`
`GBP
`
`+0.779
`-0.570
`-0.881
`
`CHF
`
`+0.755
`-0.704
`-0.707
`
`XAU
`
`+0.885
`-0.287
`-0.450
`
`coefficients between the intradaily histograms of Figure 5.12 are also positive, as
`explicitly shown in the first line of Table 5.13. We conjecture that both variables
`are positively correlated to a third one, the worldwide intraday transaction volume,
`which is not known for the FX market. Transaction volume figures are, however,
`available for the stock market; their positive correlation to squared returns (and
`hence the volatility) has been found by Harris (1987) and other authors. Recently,
`Hasbrouck (1999) examined the data of the New York Stock Exchange and found
`similar correlations as in Table 5.13 for his transaction data, but the correlations
`did not uniformly increase when the data were aggregated.
`The statistics show that an analysis of return distributions that neglects the large
`differences between the hours of a day and the days of the week is inappropriate.
`In Chapter 6, we will introduce a new time scale to solve this problem.
`
`5.6.3 Seasonal Volatility: U-Shaped for Exchange Traded
`Instruments
`Intradaily seasonalities were also found in the stock markets by Ghysels and Jasiak
`(1995), Andersen and Bollerslev (1997b) and Hasbrouck (1999). Unlike the FX
`market, stock exchanges and money market exchanges are active less than 24 hr a
`day. Thus the shape of the seasonality is different. It is called the U-shape because
`the high volatility of the opening is followed by a decrease, which is in tum
`followed by an increase of volatility just before closing. Ballocchi et al. (1999b)
`study the Eurofutures markets and find the expected intraday seasonality. For all
`contracts traded on LIFFE the hourly tick activity displays the U-shape with its
`minimum around 11 a.m. to 1 p.m. (GMT) and a clustering of activity around the
`beginning and the end of the trading day. There are differences among Eurofutures
`between the levels and widths of the peaks and the level of the minimum. The
`Eurodollar (a contract type traded on CME, see section 2.4.1) displays similar
`behavior but the activity in the first half of the working day, which takes place
`when the European markets are still open, is higher than during the second half of
`the day, when European markets have already closed and Asian markets are not
`yet open.
`
`0266
`
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`168
`
`CHAPTER 5 BASIC STYLIZED FACTS
`
`.4
`
`.2
`
`.0-
`
`.8
`
`.6-
`
`.4
`
`0+-"=F='l-'----4--'---'--\...LL-f-L-4-~--i
`6
`12
`18
`0
`Hour of the Day (GMT)
`
`24
`
`--------
`
`.2
`0 - 11111
`TITTT
`·~
`12
`18
`6
`0
`24
`Hour of the Day (GMT)
`
`FIGURE 5.13
`lntraday analysis of Short Sterling in position two. The intraday tick
`activity (left histogram) displays the average number of ticks occurring in each hour of
`the day whereas the intraday volatility (right histogram) shows the mean absolute return.
`Both plots display similar U-shapes, the only difference being that the minimum appears
`one hour later for intraday returns. The time scale is GMT (not UKT, the local time used
`by LIFFE in London). The sampling period starts on January 1, 1994, and ends on April 15,
`1997. The total number of ticks 1s 184,360.
`
`Intraday returns follow a pattern similar to that presented by intraday tick
`activity. In general, opening hours show the highest price variation (the difference
`with respect to the average of the other hours is around one basis point); only in
`some cases does the largest return occur toward closing time (usually in the last
`positions). Differences occur in some positions 28 for Short Sterling, Eurolira, and
`Three-Month Ecu,29 which display the minimum of the U-curve 1 hour later than
`in the tick-activity case. This can be seen in Figure 5.13, which displays intraday
`tick activity and intraday returns for Short Sterling in position two. Note that the
`U-shapes in this figure are blurred by the fact that Greenwich Mean Time (GMT) is
`used. The observations do not only cover winter months but also summers where
`the time scale used by LIFFE in London is shifted by 1 hour ( daylight saving time).
`If the time scale was local time (UKT) instead of GMT, the U-shapes would be
`more pronounced with clearer peaks at opening and closing.
`The first two positions of the Euromark display less regularity in the intraday
`return behavior. This behavior is confirmed also by correlation results: on the
`whole, the correlation between hourly tick activity and hourly returns is above
`0.96; only Euromark for the first two positions and Three Month-Ecu for the
`fourth position show a lower correlation around 0.90. In general, for Eurodollar,
`Euromark and Short Sterling, hourly returns tend to increase from position 1
`
`28 For an explanation of the word "position," see Section 2.1.2.
`29 Short Sterling, Eurolira, Three Month-Ecu, and Euromark are names ofLIFFE contracts, all with
`an underlying 3-month deposit. Ecu is the European Currency Unit that preceded the Euro.
`
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`

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`5.6 AUTOCORRELATION AND SEASONALITY
`
`169
`
`16~------,--------,--------,-------~
`
`r C ....
`i 12- .....
`
`.5
`
`4"+---,----,----;l---r---r---+l---r----r---+1----,----,---+-"I
`90
`0
`
`180
`Time to Maturity (Days)
`
`270
`
`360
`
`FIGURE 5.14 Volatility as a function of time to expiry. The volatility values are daily
`averages over 36 contracts (9 for Eurodollar, 9 for Euromark, 9 for Short Sterling, and 9 for
`Eurolira). The abscissa corresponds to the time to expiry: the farther on the right-hand
`side, the farther away from expiry.
`
`to position 4; Eurodollar and Short Sterling display a decrease for some hours
`in position 4.
`Looking at intraweek tick activity, there is evidence of a day-of-the-week
`effect. In general, the level of activity displays a minimum on Monday and a
`maximum on the last two working days of the week, usually on Thursday for
`LIFFE contracts and on Friday for CME contracts. The difference is definitely
`significant for the Eurodollar; in fact, for positions 1 and 2 the tick activity on
`Friday is almost double that on Monday and it becomes more than double for
`positions 3 and 4. In general, there is a gradual increase from Monday to Friday.
`
`5.6.4 Deterministic Volatility in Eurofutures Contracts
`Ballocchi et al. (2001) provide evidence that the volatility of futures prices sys(cid:173)
`tematically depends on the time interval left until contract expiry. We call these
`systematic volatility patterns deterministic, as opposed to the also existing stochas(cid:173)
`tic :fluctuations of volatility. In order to probe the existence of a seasonality related
`to contract expiry, a sample consisting of many futures contracts is needed. For
`several Eurofutures contact type (Eurodollar, Euromark, Short Sterling, and Eu(cid:173)
`rolira) and for each contract expiry, we build a series of hourly returns using linear
`interpolation. Then we compute daily volatilities taking the mean absolute value
`of hourly returns from 00:00 to 24:00 (GMT) of each working day (weekends and
`holidays are excluded). These daily volatilities are plotted against time to expiry.
`The result is shown in Figure 5 .14. The vertical axis represents the mean volatility
`
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`

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`
`170
`
`CHAPTER 5 BASIC STYLIZED FACTS
`
`computed from all Eurofutures and all contracts together. The horizontal axis
`represents the time left to expiry, as we move towards the left the number of days
`to expiry decreases.
`Figure 5.14 spans a period of about 360 days because only within that period
`we are able to compute our mean volatility based on a full set of contracts. Some
`contracts have bad data coverage for times to expiry exceeding 360 days. The
`results obtained are quite interesting. There is a downward trend in volatility as the
`time to expiry decreases ( moving from right to left in Figure 5 .I 4). This downward
`trend is weak between about 300 and 180 days before expiry but becomes strong as
`we move toward the expiry date. There is also an unexpected behavior consisting
`of oscillatory movements with peaks every 90 days corresponding to rollover
`activities near the ending of contracts. These results are confirmed also by a
`deterministic volatility study on each single Eurofutures type-except Eurolira,
`which displays an increment in volatility as we move toward expiry. Eurodollar,
`Euromark, and Short Sterling show a decreasing volatility at least for the last 300
`days before expiry. All Eurofutures display oscillatory movements with peaks
`around expiry dates (this appears particularly evident for Short Sterling).
`A possible explanation for this effect is that these markets are all "cash settled"
`and therefore have no "delivery risk"; this means there is no risk of holding these
`futures on expiry day. Due to transaction costs, it is cheaper to take the cash at
`In other
`expiry than to close the position and realize the cash the day before.
`future markets such as the Deutsche Termin-Borse or the commodity markets,
`people who hold long positions to expiry actually take physical delivery of the
`underlying commodity or bond. There is a risk as expiry approaches as to which
`bond or type of commodity will be delivered. This may cause an increase in
`volatility as expiry approaches-a behavior opposite to that of Figure 5.14.
`
`5.6.5 Bid-Ask Spreads
`The bid-ask spread reflects many factors such as transaction costs, the market
`maker's profit, and the compensation against risk for the market maker, see ( Glass(cid:173)
`man, 1987; Glosten, 1987). The subject of the intraday and intraweek analysis is
`the relative spreads i (Equation 3.12). It is usually below or around 0.1 %, and
`its distribution is not symmetric. Negative changes are bounded as spreads are
`always positive, but the spread can exceed 0.5% in times oflow market activity.
`The arithmetic mean of s i weights these low-activity spreads too strongly and
`therefore we choose the geometric mean as a more appropriate measure:
`
`Si= n Si,j
`
`(
`
`1=1
`
`n,
`
`) t;
`
`(5.39)
`
`The index i indicates the hour ofa day ( or a week) or the day of the week, depending
`on the analysis. The total number of ticks that belong to the i th interval is ni. j is
`the index and s;,J the spread of these ticks.
`
`0269
`
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`

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`
`5.6 AUTOCORRELATION AND SEASONALITY
`
`171
`
`TABLE 5.14 Average spreads.
`Geometric average of the relative spread for each day of the week (including weekends) for
`the USD against DEM, JPY, CHF, and GBP and XAU (gold) against USD; for the period from
`January 1, 1987, to December 31, 1993. The relative spread figures have to be multiplied
`by 10- 4.
`
`Monday
`Tuesday
`Wednesday
`Thursday
`Friday
`Saturday
`Sunday
`
`DEM
`
`4.57
`4.52
`4.57
`4.64
`4.79
`7.69
`5.28
`
`JPY
`
`5.72
`5.64
`5.71
`5.77
`6.00
`17.91
`6.78
`
`GBP
`
`4.82
`4.77
`4.81
`4.84
`4.99
`17.32
`9.60
`
`CHF
`
`6.32
`6.28
`6.32
`6.38
`6.49
`18.02
`10.99
`
`XAU
`
`12.58
`12.51
`12.49
`12.62
`12.59
`13.26
`14.04
`
`Mi.ill er and Sgier (1992) analyze in detail the statistical behavior of the quoted
`spread. Here we shall present their main conclusions. First, it is important to
`remember that all the statistical analyses are dominated by one property of quoted
`FX spreads, which is the discontinuity of quoted values (see Section 5.2.2). This
`data set contains price quotes rather than traded prices. The banks that issue these
`price quotes are facing the following constraints:
`is, 1.6755
`• Granularity: FX prices are usua1ly quoted with five digits-that
`(USD-DEM) or 105.21 (USD-JPY). The lowest digit sets the granularity
`and thus the unit basis points.
`• Quoted spreads are wider than traded spreads as they include "safety mar(cid:173)
`gins" on both sides of the real spread negotiated in simultaneous real
`transactions. These margins allow the FX dealers, when called by a cus(cid:173)
`tomer during the lifetime of the quote, to make a fine adjustment of the
`bid and ask prices within the range given by the wide quoted spread. They
`can thus react to the most recent market developments.
`• FX dealers often have biased intentions: while one of the prices, bid or
`ask, is carefully chosen to attract a deal in the desired direction, the other
`price is made unattractive by increasing the spread.
`• Because quoted spreads are wider than traded spreads, they do not need
`the high precision required in the direct negotiation with the customer on
`the phone. Hence, there is a tendency to publish formal, "even" values of
`quoted spreads as discussed in Section 5.2.2.
`The strong preference for a few formal spread values, mainly 5 and IO basis points,
`clearly affects every statistical analysis.
`The results are shown in the middle histograms of Figure 5.12 and in
`Table 5.14. The general behavior of spreads is opposite to those of volatility
`and tick frequency. Spreads are high when activity is low, as already noticed by
`
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`
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`

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`
`172
`
`CHAPTER 5 BASIC STYLIZED FACTS
`
`0999
`099'!
`0995
`0990
`0980
`
`0950
`
`0900
`
`0800
`
`0500
`
`0200
`
`0100
`
`0050
`
`0999
`0998
`
`0995
`0990
`0980
`
`0950
`
`0900
`
`0800
`
`0500
`
`0200
`
`0100
`
`0050
`
`0020
`0020
`0010
`0010
`0005
`0005
`oa
`o=
`0 001 +--'--r-,--~-,----,---,-----1
`0001 +'-,---,..-,---,..-,--,--,--t--_,---,,-------1
`-80
`-70
`-80
`Q QOOQ Q QQ1Q Q QQ2Q Q OQ3Q O Q04Q Q QQ50 Q QQ6Q
`w9 0
`Log(Relative Spread)
`Relative Spread
`
`-80
`
`FIGURE 5.15 Cumulative distributions of relative spreads (left) and logarithm of the
`relative spread (right) shown against the Gaussian probability on the y-axis. The distribu(cid:173)
`tion 1s computed from a time series of linearly interpolated spread sampled every 10 min
`for USO-DEM. The sample runs from March 1, 1986, to March 1, 1991.
`
`Glassman (1987). FX spreads on Saturdays and Sundays can have double and
`more the size of those on weekdays and, as in Table 5.12, Sundays differ slightly
`less from working days than Saturdays. Sunday in GMT also covers the early
`morning of Monday in East Asian time zones. Unlike the volatilities, the average
`FX spreads exhibit a clear weekend effect in the sense that the Friday figures are
`higher, though still much lower than those of Saturday and Sunday. The spreads of
`gold vary less strongly, but they have double the size of the FX spreads on working
`days. The FX rate with the smallest spreads, USD-DEM, was the most traded one
`according to all the BIS studies (until it was replaced by EUR-USD in 1999). The
`histograms in Figure 5.12 have intraday patterns that are less distinct than those
`of volatility, but still characteristic. We analyze their correlations with both the
`volatilities and the numbers of quoted ticks. All the correlation coefficients on the
`second line of Table 5.13 and most of them on the third line are negative, as one
`would expect. The FX rates have different spread patterns. For USD-CHF, for
`instance, there is a general spread increase during the European afternoon when
`the center of market activity shifts from Europe to America, while the USD-JPY
`spreads decrease on average at the same daytime. This indicates that American
`traders are less interested in Swiss Francs and more in Japanese Yens than other
`traders. Hartmann (1998) uses the spreads to study the role of the German Mark
`and the Japanese Yen as "vehicle currencies," as compared to the USD.
`An analysis of the empirical cumulative distribution function of the relative
`spreads is shown in the left graph of Figure 5.15 forUSD-DEM and forlns in the
`
`0271
`
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`

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`
`5.6 AUTOCORRELATION AND SEASONALITY
`
`173
`
`right graph of Figure 5 .15. The resulting cumulative distribution functions have
`the following properties:
`1. They are not Gaussian, but convex (s strongly, Ins slightly), indicating a
`positive skewness and leptokurticity (of the tail on the positive side).
`2. They look like a staircase with smooth comers. For the nominal spread
`in basis point, Snom, we would expect a staircase with sharp comers, the
`vertical parts of the staircase function indicating the preferred "even" val(cid:173)
`ues such as 10 basis points. Although sis a relative spread c~ Snom/ Pbid),
`where the bid price Pbid fluctuates over the 5-year sample, and although we
`use linear interpolation in the time series construction (see Section 3.2.l),
`the preferred "even" Snom values are still visible.
`
`0272
`
`

`

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`
`6
`
`MODELING SEASONAL
`VOLATILITY
`
`INTRODUCTION
`6.1
`The intradaily and intraweekly seasonality of volatility is a dominant effect that
`overshadows many further stylized facts of high-frequency data. In order to con(cid:173)
`tinue the research for stylized facts, we need a powerful treatment of this season(cid:173)
`ality.
`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 Sat(cid:173)
`urdays, Sundays, and holidays. With the t-scale we extend this concept to the
`intraday domain, thereby allowing us to tackle a fundamental source of seasonality
`originating from the cyclical nature of the 24-hr hour trading around the globe in
`different geographical locations.
`There are, therefore, three main motivations for our model:
`• To provide a tool for the analysis of market prices by extending the concept
`of business time scale to intraday prices
`• To make a first step toward formulating a model of market prices that also
`covers the intraday movements
`• To gain insight into the interactions of the main market centers around the
`world and their relevance to each particular foreign exchange (FX) rate
`
`174
`
`0273
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`

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`
`6.2 A MODEL OF MARKET ACTIVITY
`
`175
`
`A number of papers such as Andersen and Bollerslev (1997b, 1998b), Taylor
`and Xu (1997), and Beltratti and Morana (1999) propose alternative approaches
`for dealing with volatility seasonalities. They are based on a factorization of
`the volatility into an essentially deterministic seasonal part and a stochastic part,
`which is (more or less) free of seasonalities. The former is then modeled by a set
`of smooth functions. Cutting out the inactive periods of the time series and gluing
`together the active parts, Andersen and Bollerslev (1997b) succeeded in applying
`their method also to the S&P 500 index. This procedure is not fully satisfactory for
`a number of reasons: time series have to be preprocessed, there is no treatment of
`public holidays and other special days, the model fails when the opening or closing
`time of the market changes, and it is not adequate for instruments with a complex,
`hybrid volatility pattern. Gen~ay et al. (2001a) use the wavelet multiresolution
`methods for dealing with volatility seasonalities which is studied in Section 6.4.
`
`6.2 A MODEL OF MARKET ACTIVITY
`6.2.1 Seasonal Patterns of the Volatility and Presence of Markets
`if it exhibits a periodic pattern
`The behavior of a time series is called seasonal
`in addition to less regular movements. In Chapter 5 we demonstrated daily and
`weekly seasonal heteroskedasticity ofFX prices. This seasonality of volatility has
`been found in intradaily and intra weekly frequencies. In the presence of seasonal
`heteroskedasticity, autocorrelation coefficients are significantly higher for time
`lags that are integer multiples of the seasonal period than for other lags. An
`extended autocorrelation analysis is studied in Chapter 7.
`As studied in Chapter 5, the intra week analysis indicates that the mean absolute
`returns are much higher over working days than over Saturdays and Sundays, when
`the market agents are hardly present. The intraday analysis also demonstrates that
`the mean absolute hourly returns 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 returns is found.
`Chapter 5 also presents evidence of a strong correlation between market pres(cid:173)
`ence and volatility such that the intraday price quotes are positively correlated to
`volatility when measured with mean absolute hourly returns. Market presence is
`related to worldwide transaction volume which cannot be observed directly. In the
`literature, a number of papers present substantial evidence in favor of a positive
`correlation between returns and volume in financial markets, see the survey of
`(Karpoff, 1987).
`The correlation of market presence and volatility requires us to model and
`explain the empirically found seasonal volatility patterns with the help of funda(cid:173)
`mental 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
`
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`

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`176
`
`CHAPTER 6 MODELING SEASONAL VOLATILITY
`
`overlap, market activity must be attributed to their cumulative presence; it is im(cid:173)
`possible 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
`a high-frequency database (such as the O&A database), which also contains the
`originating locations of the quoted prices.
`In many of the approaches cited in the introduction, in particular in Baillie
`and Bollerslev (1990) where the seasonality of volatility is modeled by dummy
`variables, no further explanation of this seasonal pattern is given. We consider
`it advantageous to try to identify at every moment of the day which markets are
`responsible for the current volatility.
`
`6.2.2 Modeling the Volatility Patterns with an Alternative Time
`Scale and an Activity Variable
`Before relating the empirically observed volatility to the market presence, we
`introduce a model of the price process, which will be used for describing and
`analyzing the seasonal volatility patterns. A return process with strong intraday
`and intraweek volatility patterns may not be stationary. Our model for the seasonal
`volatility fluctuations introduces a new time scale such that the transformed data
`in this new time scale do not possess intraday seasonalities.
`The construction of this time scale utilizes two components: the directing pro(cid:173)
`cess, {} (t ), and a subordinated price process generated from the directing process.
`Let x(t) be the tick-by-tick financial time series that inherits intraday seasonal(cid:173)
`ities. The directing process, {}(t) : R -+ R, is a mapping from physical time
`to another predetermined time scale. Here, it is defined such that it contains the
`intraday seasonal variations. 1 iJ(t), when used with the subordinated price gen(cid:173)
`erating process x(t) = x*[{}(t)],
`leads to the x* process, which has no intraday
`seasonalities. Although this is not the only possible model to treat the observed
`seasonality, other traditional deseasonalization techniques are not applicable as
`the volatility is seasonal, not the raw time series.
`In the literature, a variety of alternative time scales have been proposed, in
`different contexts. In the early 1960s, Allais (see, for instance, Allais, 1974) had
`proposed the concept of psychological time to formulate the quantity theory of
`money. Mandelbrot and Taylor (1967) suggested to cumulate the transaction vol(cid:173)
`ume to obtain a new time scale which they call the transaction clock. Clark (1973)
`suggested a similar approach. Stock (1988) studied the postwar U.S. GNP and
`interest rates and proposed a new time scale to model the conditional heteroskedas(cid:173)
`ticity exhibited by these time series. Here we propose to use a new time scale to
`account for the seasonality.
`
`1 The{} (t) process can assume different roles in different filtering environments. If, for instance, the
`interest is to simply filter out certain holiday effects from the data, then {} (t) can be defined accordingly.
`Under such a definition, the transformation will only eliminate the specified holiday effects from the
`underlying x (t) process. The {} type time transformations are not limited to seasonality filtering. They
`can also be used within other contexts such as the modeling of intrinsic time or transaction clock.
`
`0275
`
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`

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`
`6.2 A MODEL OF MARKET ACTIVITY
`
`177
`
`Because the iJ-scale fully accounts for the seasonality of x, x* has no seasonal
`volatility patterns. The process x* may however have nonseasonal volatility pat(cid:173)
`terns; it may be conditionally heteroskedastic. No attempt is made in this chapter
`to determine its exact nature. The time scale iJ(t) is a strictly monotonic :function
`of physical time t. Any time interval from t1 to t2(> t 1) corresponds to a iJ-time
`iJ1. The new activity variable a is defined as the
`interval of the positive size iJ2 -
`ratio of the interval sizes on the different scales,
`
`(6.1)
`
`This activity reflects the seasonal volatility patterns. Its relation to other "activ(cid:173)
`ity" variables such as market presence or transaction volume was mentioned in
`Section 6.2.1 and is discussed below.2
`
`6.2.3 Market Activity and Scaling Law
`The volatility-based activity defined by Equation 6.1 can be computed with the
`empirical scaling law (see Chapter 5) for returns, which relates (for p = l)
`the mean absolute returns over a time interval to the size of this inter(cid:173)
`(\L'lx\),
`val, 1',.t,
`
`(6.2)
`
`where E is the expectation operator, c is a constant depending of the specific
`time series. D is the drift exponent, which determines the scaling properties of
`the underlying process across different data frequencies. The drift exponent D
`is about 0.6 for major FX rates, whereas the pure Gaussian random walk model
`would imply D = 0.5. The scaling law expressed in Equation 6.2 holds for all
`time series studied and for a wide variety of time intervals ranging from 10 min to
`more than a year.
`The scaling law is applied to subsamples in a so-called intraweek 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).
`Here, we choose a sampling granularity of !::,.t = l hr. The week is subdivided
`to Sunday 23:00 - 24:00 (Greenwich Mean
`into 168 hr from Monday 0:00-1:00
`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
`index i. The 168 subsamples together constitute the full 4-year sample. The sam(cid:173)
`ple pattern is independent of bank holidays and daylight saving time. A typical
`intraday and intraweek pattern across the 168 hr of a typical week is shown in
`Figure 6.1.
`
`2 In skipping Saturdays and Sundays, other researchers use an implicit activity model with zero
`activity on the weekends.
`
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`178
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`CHAPTER 6 MODELING SEASONAL VOLATILITY
`
`40
`
`30
`
`2.0
`
`1,0
`
`00
`
`lntraweek Hourly Index
`
`FIGURE 6.1 Histogram of the average hourly activity (as defined in Equation 6.4 for
`a statistical week (over 4 years) for the USO-DEM rate.
`
`The scaling law, Equation 6.2, is applied to the i th hourly subsample instead
`of the full sample and mathematically transformed to
`
`'6.ih ::::: ( E~:;I])
`
`1/D
`
`(6.3)
`
`From Chapter 5, we know that r; can strongly vary for the different hours of a
`1 hr (for the hourly sampling) is nevertheless
`week. The time interval b.t
`constant. Therefore, it is replaced by the interval ,6. if; on the new time scale if.
`The size of M}; is no longer constant, but reflects the typical volatility of the i th
`hour. The constant c* is essentially the c of Equation 6.2, but can differ slightly
`as it is calibrated by a normalization condition presented later.
`The activity of the i th hourly subsample directly follows from Equation 6.1,
`1 (E[lr;l]) 1
`/D
`--
`astat i = -
`c*
`flt
`'
`
`, !::.t ::::: 1 hr
`
`(6.4)
`
`This is the volatility-based activity definition used in the following analysis. The
`constant c* is calibrated to satisfy the following, straightforward normalization
`condition:
`
`168
`
`168 L Gstat, i
`
`i=l
`
`(6.5)
`
`6.2.4 Geographical Components of Market Activity
`In Figure 6.1, the histogram of the average hourly activity defined by Equation 6.4
`is plotted for the USD-DEM rate. Although the activity definition is based only
`
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`6.2 A MODEL OF MARKET ACTIVITY
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`179
`
`on return statistics, the histogram exhibits clear structures where there is very low
`activity 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 f:..t granularity
`of this analysis from 15 min to 4 hr 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 4 years of the total sample. The strong
`relation between return 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 activity of the market during different market
`hours that sometimes overlap.
`Goodhart andFigliuoli (1992) have explored the geographical nature of the FX
`market to look for what they call the island hypothesis. They studied the possibil(cid:173)
`ity that the price bounces back and forth from different centers when special news
`occurs before finally adjusting to it. 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 GAR CH model, investigate 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 autocorrela(cid:173)
`tion. 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 6.1 and the results reported in Chapter 5
`suggest that the worldwide market can be divided into three continental compo(cid:173)
`nents: East Asia, Europe, and America. The grouping of the countries appearing
`on the Reuters pages in our three components can be found in Table 6.1. This divi(cid:173)
`sion into three components is quite natural and some empirical evidence supporting
`it will be presented in Chapter 7.
`The model activity of a particular geographical component k is called ak(t);
`the sum of the three additive component activities is a(t):
`3
`a(t) = Lak(t)
`k=l
`This total activity should model the intraweekly pattern of the statistical activity
`astat,i as closely as possible. Unlike astat, which has re