MODELING SEASONAL VOLATILITY

6.1 introduction

The intradaily and intraweekly seasonality of volatility is a dominant effect that overshadows many further stylized facts of high-frequency data. In order to continue the research for stylized facts, we need a powerful treatment of this seasonality.

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 -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

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. Gencay 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

The behavior of a time series is called seasonal if it exhibits a periodic pattern in addition to less regular movements. In Chapter 5 we demonstrated daily and weekly seasonal heteroskedasticity of FX prices. This seasonality of volatility has been found in intradaily and intraweekly 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 intraweek 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 presence 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 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

overlap, 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 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 process, &(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 seasonalities. 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 &(t), when used with the subordinated price generating 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 volume 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-ticity exhibited by these time series. Here we propose to use a new time scale to account for the seasonality.

1 The i? (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? (r) can be defined accordingly. Under such a definition, the transformation will only eliminate the specified holiday effects from the underlying x(t) process. The t? 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.