Dynamical Deseasonalization in OTC and Localized Exchange-Traded Markets

Transcription

1 Dynamical Deseasonalization in OTC and Localized Exchange-Traded Markets Wolfgang Breymann, Gilles Zumbach, Michel M. Dacorogna, and Ulrich A. Müller May 12, Introduction Due to the recent advances in information technology, high-frequency tick-by-tick data are becoming widely available. In principle, this better source of information can be used to improve the tracking, modeling and forecasting of financial time series, as for example in forecasting the next day volatility for risk assesment. The large obstacle to use this better data is the pronounced daily and weekly seasonalities of the volatility (see (Gwilym and Sutcliffe, 1999, chap. 5) for an overview of the literature). In order to uncover the fine structure in the data, deseasonalization is essential. Yet, there is no commonly agreed-upon procedure. Among the first to tackle the deseasonalization problem were (Baillie and Bollerslev, 1990), who modeled hourly volatilities by a GARCH(1,1) process with additional dummy variables for each hour of the day, and (Pecen et al., 1995; Baestaens and Van den Bergh, 1995) who used a quote-based operational time. A more sophisticated method for FX data was suggested by (Dacorogna et al., 1993), who fully recognized the importance of deseasonalization for modeling and forecasting. Their method consists of a time transformation onto a volatility-driven business time scale. It is defined by means of a market activity which is related to the volatility through a scaling law. In addition, the total market activity is modeled as a sum of three regional components: an East Asian, an American and a European market. The new time scale, called ϑ-time, runs faster the higher the hourly mean volatility conditioned on a given interval of the week. However, designed for delocalized over-the-counter (OTC) markets with round-the-clock activity, the ϑ-time algorithm is less performing for localized, exchange-traded instruments, and does not account for changes in the market structure, as, for instance, the extension of the daily trading period. The German stock exchange, for example, extended its floor trading hours from less than 3h to about 6h in a period of two years. This, of course, strongly affect the volatility pattern of the DAX. Recently, a number of other authors as (Andersen and Bollerslev, 1997; Andersen and Bollerslev, 1998), (Beltratti and Morana, 1998), and (Taylor and Xu, 1997) proposed 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, 1997) succeeded in applying their method also to the S&P500 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 as, e.g., XAU/USD with a complex, hybrid volatility pattern (see Fig. 1). The aim of this paper is to present a deseasonalization procedure that overcomes these shortcomings and is universally applicable to arbitrary high-frequency financial time series. We present here a comparative study of the intra-day and intra-week volatility patterns of a large variety of instruments and markets, Olsen & Associates, Research Institute for Applied Economics, Seefeldstrasse 233, CH 8008 Zürich, Switzerland and Institut für Physik, Klingelbergstrasse 82, CH 4056 Basel, Switzerland. Olsen & Associates, Research Institute for Applied Economics, Seefeldstrasse 233, CH 8008 Zürich, Switzerland. 1

2 2.0 ytilitalov 51.0spot FX yadartni usd/jpy nretap nretap ytilitalov yadartni USD/PLZ FX spot ytilitalov tx future oil yadartni heat 3m future (liffe), 30y1 DEM (cbot), yadartniusd future yadartni IR nretap nretap 3m DEM, yadartni deposit, nretap nretap ytilitalov yadartni Nikkei225 (qt) future yadartni SP nretap nretap ytilitalov 0 yadartni XAU/USD nretap yadartni GDAXI Figure 1: Daily volatility pattern for some selected instruments, averaged over the years 1997/1998. The time on the abscissa is given in hours GMT. Full, red line: summer time. Dashed, green line: winter time. Notice that the averaging period starts after the extension of the floor trading hours for the German DAX (lower right). 2

3 an adaptive algorithm which takes into account the seasonalities of the volatility for arbitrary financial instruments and that can easily be extended to a multivariate setting, a study of the deseasonalization properties of the algorithm and of the statistical properties of the deseasonalized time series, and applications to volatility forecasts and gap detection. 2 Comparative study of volatility patterns The daily and weekly volatility patterns have been computed for a large number of financial instruments from filtered tick-by-tick data of the Olsen & Associates database. The following instruments have been studied: FX spote rates (quotes) 1m, 3m, 6m, 9m, and 1y deposits 1m and 3m Eurofutures (quotes and transaction prices) 2y, 5y, 10y, and 30y bond futures (quotes and transaction prices) equity indices and equity index futures (quotes and transaction prices) commodities (quotes) and commodity futures (quotes and transaction prices). The patterns display a variety of different shapes, depending on the nature of the financial asset (FX, bonds, equities, commodities), the kind of instruments (spot, future, etc.) and the organization of the market (OTC vs. localized, exchange-traded markets, open outcry vs. computerized trading). The extreme patterns are (i) a rather smooth pattern with broad maxima, which is typical for delocalized OTC markets with round-the-clock activity as in the case of major FX spot rates, and (ii) the well-known U-shape pattern typical for localized, exchange-traded markets, which is mainly due to the well-defined opening and closing time. Between these two extreme cases there exists a variety of transition patterns. A small sample of intra-day volatility patterns computed for the years 1997/98 is shown in Fig. 1. In addition, a number of other features will be discussed: the influence of Japanese lunch break, daylight saving time (DST) effects (cf. the shift between the full and the dashed lines during active hours in Europe and America), day-of-theweek effects, public holidays, and the slow changes of the pattern over long periods. 3 A general adaptive volatility-driven business time scale The deseasonalization algorithms existing in the literature have typically been applied to only a small number of time series, 1 and their application to a large class of many different instruments is limited by a lack of flexibility as well as time-consuming fitting procedures. Our goal is to design a flexible, adaptive algorithm that is universally applicable to arbitrary high-frequency financial instruments and yields better deseasonalization than the existing algorithms. The basic method used in this work is to compute an appropriate time scale driven by the volatility observed for a given security. 1 An exception is the algorithm suggested in (Dacorogna et al., 1993) which is used in a real time system for over hundred FX spot rates. Even though this algorithm is not designed for localized markets, it is also successfully used for a number of exchanged-traded instruments in the fixed income market. 3

4 The time transformation The transformation from physical time to the volatility-driven business time is given by (Dacorogna et al., 1993) ϑ(t 1, t 2 ) = t2 t 1 a(t)dt, (1) where the activity a(t) 0 is a quantity measuring the passing of events on a financial market, typically the volatility or the tick rate. Different activity measures can be chosen, provided that they are positive and that they scale appropriately with aggregation. In this paper the intra-week volatility pattern is computed from 15 min. absolute returns, and the relation between activity and volatility uses the scaling properties of the ordinary diffusion process. For normal days, namely when there is no public holiday, the activity a(t) equals the observed weekly activity a obs (t). The observed activity is computed by moving averages conditional to the time in the week. Moreover, to take into account the Daylight Saving Time (DST), the observed activity is computed separately for Winter and Summer times. For public holidays, roughly 20 days per year, the activity is given by the observed weekly activity a obs (t) discounted by the market share of the closed market(s). This procedure requires a decomposition of the total activity into regional market components. The main components of the algorithm are described in more details in the following subsections. Activity histograms The activity histogram is computed by a moving average conditional to the time in the week, using the technique described in (Zumbach and Müller, 2000). A memory of a few months is used for the moving average. The weekly histograms automatically account for day-of-the-week effects. The computation of the weekly seasonality pattern is both fully adaptive and very flexible and does not depend on instrumentspecific parameters. In order to properly deal with DST effects, the histograms are computed separately for DST and non-dst periods. Holidays and daylight saving time Public holidays and daylight saving time are separately accounted for each regional market. In addition, we introduce the treatment of days with reduced market activity (fuzzy holidays) as, for example, the working days between Christmas and New Year. During a public or fuzzy holiday in one component, the activity of this component is set to zero or reduced by a given factor, respectively. Holidays and daylight saving periods are configured in advance from publicly available information. Weights of market components The market share of a regional component is modeled by smoothed step functions, parametrized by the opening and closing hours of this market. Furthermore, the relative weights of the different components need to be given in order to separate the individual components when several markets are simulataneously active. Only the weight parameters may differ from one instrument to the other. The number of market components to be considered depends on the instrument. For most FX spot rates, a decomposition into an East-Asian, an American, and a European market component, as in (Dacorogna et al., 1993), is accurate enough. However, for the USD/JPY spot rate, the introduction of an Australian market as 4th component improved the treatment of public holidays. On the other hand, for a typical localized, exchange-traded instrument only a single market component corresponding to its geographical location is needed. The implementation The above algorithm is implemented in C++. It works with arbitrary time series provided that they are long enough, that the frequency is sufficiently high, and that the data quality is good (clean data is important, as the empirical volatility can be badly affected by outliers). Our choices ensure both good modeling of the daily seasonal heteroskedasticity and adaptivity to slow changes of the seasonal volatility pattern on a time scale of months or years. 4

5 4 Properties of deseasonalized time series The studies listed in this section have been executed for different time scales, namely Physical time (clock time) Usual business time (only working days, no weekend) Polynomial-based volatility-driven business time proposed in (Dacorogna et al., 1993). Universal adaptive volatility-driven business time proposed in this paper. The following properties have been investigated: Residual seasonality for absolute returns and squared returns. Autocorrelation function of absolute returns and squared returns. 4.1 Deseasonalization quality The performance of the algorithm proposed here is studied for different asset classes and compared to the performance of the algorithm proposed in (Dacorogna et al., 1993). We report the remaining intra-week seasonality of the price volatility after time transformation, relative to the original seasonality. This ratio is a quality measure of the deseasonalization and thus an indicator of the adequacy of the proposed time transformation: it is one in physical time and zero in a time scale with perfect deseasonalization. The results show the superior performance of the dynamical deseasonalization, in particular for exchangetraded instruments time [days] USD/JPY FX spot time [days] XDAX Figure 2: ACF for (a) USD/JPY (1/ /1999) and (b) German DAX (1/ /1998). Full red lines: dynamical time scale. Dashed green lines: physical time scale. 4.2 Autocorrelation function of volatility Autocorrelation functions (ACF) of absolute and squared returns are also used to detect seasonalities. The comparison of the ACFs for different instruments and different time scales show the excellent quality of the proposed deseasonalization techniques. ACFs computed in physical time scale exhibit periodic patterns with strong daily and weekly seasonalities that completely hide the long-range dependence present in the data (Dacorogna et al., 1993; Ding et al., 1993). For localized instruments there are only very few remaining seasonalities in the ACF, and the long-range pattern becomes clear. For both cases an example is shown in Fig. 2. For FX spot rates the ACF computed after dynamical deseasonalization exhibits significantly less remaining seasonalities than after deseasonalization the ϑ-time scale defined in (Dacorogna et al., 1993). This points to the absence of heat-wave effects for these instruments. 5

6 5 Application to volatility forecasts and data gap detection Because of its sensitivity to remaining seasonality, volatility forecasts provide a good test for deseasonalization algorithms. We are currently estimating the parameters of several time series models (GARCH(1,1), HARCH (Müller et al., 1997), etc.) and then using this model for volatility forecasting. As 2nd application that requires very good deseasonalization we present a data gap detector for high frequency data source, that is able to automatically detect gaps of sizes down to about one hour. References Andersen T. G. and Bollerslev T., 1997, Intraday periodicity and volatility persistence in financial markets, Journal of Empirical Finance, 4(2-3), Andersen T. G. and Bollerslev T., 1998, Deutsche Mark-Dollar volatility: intraday activity patterns, macroeconomic announcements, and longer run dependencies, the Journal of Finance, 53(1), Baestaens D. J. E. and Van den Bergh W. M., 1995, The marginal contribution of news to the DEM/USD swap rate, Neural Network World, 5(4), Baillie R. T. and Bollerslev T., 1990, Intra day and inter market volatility in foreign exchange rates, Review of Economic Studies, 58, Beltratti A. and Morana C., 1998, Computing value at risk with high frequency data, Paper to be presented at the second High Frequency Data in Finance Conference (HFDF), Zürich, April 1-3, Dacorogna M. M., Müller U. A., Nagler R. J., Olsen R. B., and Pictet O. V., 1993, A geographical model for the daily and weekly seasonal volatility in the FX market, Journal of International Money and Finance, 12(4), Ding Z., Granger C. W. J., and Engle R. F., 1993, A long memory property of stock market returns and a new model, Journal of Empirical Finance, 1, Gwilym O. and Sutcliffe C., 1999, High-Frequency Financial Market Data,. Risk Books, London. Müller U. A., Dacorogna M. M., Davé R. D., Olsen R. B., Pictet O. V., and von Weizsäcker J. E., 1997, Volatilities of different time resolutions analyzing the dynamics of market components, Journal of Empirical Finance, 4(2-3), Pecen L., Rame sová N., Pelikán E., and Beran H., 1995, Application of the GUHA method on financial data, Neural Network World, 5(4), Taylor S. J. and Xu X., 1997, The incremental volatility information in one million foreign exchange quotations, the Journal of Empirical Finance, 4(4), Zumbach G. O. and Müller U. A., 2000, Operators on inhomogeneous time series, to be pubilshed in International Journal of Theoretical and Applied Finance. 6