After appropriately defining realized volatility, we can analyze its dynamical behavior through different statistical methods. The fundamental properties of the volatility dynamics are the conditional heteroskedasticity (also called the volatility clustering) and the long memory of the autocorrelation of volatility.2 In this chapter, we also examine the asymmetry of information flow between volatilities computed from returns measured at different frequencies which is a typical property to study with high-frequency data. Financial markets are made of traders with different trading horizons. In the heart of the trading mechanisms are the market makers. At the next level up are the intraday traders who carry out trades only within a given trading day but do not carry overnight positions. Then there are day traders who may carry positions overnight, short-term traders and long-term traders. Each of these classes of traders may have their own trading tool sets consistent with their trading horizon and may possess a homogeneous appearance within their own classes. Overall, it is the sum of the activities of all traders for all horizons that generates the market prices. Therefore, market activity would not exhibit homogeneous behavior, but the underlying dynamics would be heterogeneous with each trading horizon (trader class) dynamically providing feedback across all trader classes. Figure 7.1 illustrates such a heterogeneous market where a low-frequency shock to the system penetrates through all layers reaching the market maker in the middle. The impact of these low-frequency shocks penetrates the entire market. The high-frequency shocks, however, would be short lived and may have no impact outside their boundaries. We will study this heterogeneity-driven asymmetry in this chapter.

This book utilizes the deseasonalization method explained in Chapter 6, and Dacorogna et al. (1993), but a flurry of alternative ways of treating the seasonality have also been proposed: the time-of-day dummy variables, Baillie and Bollerslev (1990); a renormalization of the returns by the seasonal volatility, Taylor and Xu (1997); the flexible Fourier framework to model the seasonal pattern, Andersen and Bollerslev (1997b); time deformation with tick frequency, Pecenef a/. (1995); Baestaens and Van den Bergh (1995); the use of cubic splines, Engle and Russell (1997); models that include both systematic components and stochastic seasonal components, Beltratti and Morana (1998); and the wavelet multiresolution method of Gencay et al. (2001a) in Section 6.4.

7,2 the bias of realized volatility and its correction

Realized volatility plays a key role both for the exploration of stylized facts and for practical applications such as market risk assessment. When computing it,

2 This clustering property was first noted in Mandelbrot (1963) in his study of cotton prices and the long memory in Mandelbrot (1971). These findings remained dormant until the early 1980s for the volatility clustering until Engle (1982) and Bollerslev (1986) proposed the ARCH and GARCH processes. In the early 1990s, a comprehensive study of the long memory properties of the financial markets had started.

Time Horizons (price changes)

FIGURE 7.1 Financial markets are made of traders with different trading horizons. In the heart of the trading mechanisms are the market makers. A next level up are the intraday traders who carry out trades only within a given trading day. Then there are day traders who may carry positions overnight, short-term traders and long-term traders. Each of these classes of traders may have their own trading tool sets and may possess a homogeneous appearance within their own classes. Overall, it is the sum of the activities of all traders for all horizons that generates the market prices. Therefore, market activity is heterogeneous with each trading horizon dynamically providing feedback across the distributions of trading classes.

using Equation 3.8, we can take advantage of high-frequency data by choosing a short time interval At of the analyzed returns. This leads to a large number of observations within a given sample and thus a low stochastic error. At the same time, it leads to a considerable bias in most cases.

In the following bias study, Equation 3.8 is considered in the following form:

v(n)

v(At, n, 2; t-,)

(7.1)

The choice of the exponent p = 2 has some advantages here. In Section 5.5, we found that the empirical drift exponent of v is close to the Gaussian value 0.5 if v is defined with an exponent p = 2. Assuming such a scaling behavior and a fixed

sample of size T = n At, v has an expectation independent of Af:

E[v2{nAt,\,2;ti)] = n E[v2(At,n,2;ti)] = J][r(Af; f, „+y)]2 (7.2)

Thus v2 can be empirically estimated as the sum of all squared returns within T, irrespective of the size of Af. Moreover, the time scale can be changed, such as from #-time to physical time, and the return intervals can be of irregular size. This implies that the estimator is also immune to data gaps within the full sample. If prices are interpolated, previous-tick interpolation (see Equation 3.1) should be used here, because linear interpolation leads to an underestimation of volatility. With all the mentioned modifications, the sum of squared returns remains an estimator for v2, as long as all the return intervals exactly cover the full sample T. These nice properties may have led Andersen et al. (2000) to choose the name "realized volatility" for the sum of squared returns, as on the right-hand side of Equation 7.2.

The empirically found bias violates Equation 7.2, especially if Af is very small. The deviation of the empirical behavior from Equation 7.2 provides a measure of the bias. We choose a large enough time interval Afref = qAt as the bias-free reference case to judge the bias of smaller intervals Af. In practice, a good choice of Afref is between few hours and 1 working day. We define the bias factor B(ti):

where m is the number of analyzed reference intervals of size Afref, and q = Afref I At is an integer number. If the scaling assumption of Equation 7.2 is true, B(tj) converges to 1 for large samples (i.e., large m and q). The bias can be measured in terms of the deviation of B(tj) from 1.

In Figure 7.2, the bias factor B(f() is plotted versus time, for two different markets: the FX rate USD-CHF and the equity index Nikkei-225. The time scale in both cases is a business time: the 49 weekend hr from Friday 8 p.m. GMT to Sunday 9 p.m. GMT are compressed to the equivalent of only 1 hr outside the weekend. The results do not strongly depend on this choice. Similar bias behaviors are obtained when the analysis is done in #-time or physical time. The reference time interval is Afref = 1 working day. The investigated return intervals Af are much shorter and vary between 2 min (q = 720) and 1 hr (q = 24). The number m = 260 of reference intervals is chosen high enough to limit the stochastic error of v( Afref, m,2;ti). This means a bias measurement on a moving sample of about 1 year » 260 working days.

The bias factor distinctly deviates from 1 in Figure 7.2, especially for small values of Af such as 2 min and 5 min. For Af = 1 hr, the bias is still visible but

B(n) =

/q v(At, m q, 2; f,)

ZArjAt; ti-mq+j)]2

(7.3)

u(Afref, m, 2; f,)

\\ 2Z%\\r{\ti-mq+jq)\2