VOLATILITY PROCESSES

8.1 introduction

One of the many challenges posed by the study of high-frequency data in finance is to build models that can explain the empirical behavior of the data at any frequency at which they are measured from minutes to months. We are now going to examine how conventional models perform when confronted with this problem. In the previous chapter, we discussed the rich structure of the volatility dynamics. We need to introduce new types of volatility models to account for this structure, leading to a higher predictive power.

Many statistical processes proposed in the literature can be described by the following general formula for equally spaced returns rt:

r, = cr, e, (8.1)

where et is an identically and independently distributed (i.i.d.) random variable1 with zero mean and variance 1. In this chapter, t denotes the index of a homogeneous time series rather than time itself. "Homogeneous" means equally spaced on any chosen time scale. We usually choose #-time as introduced in Chapter 6, so

In this chapter, normal and Student-f distributions of s, are studied.

the model appropriately accounts for seasonalities. The volatility o> is the square root of the variance of the return rt.

Many models are based on Equation 8.1, but they largely differ in the modeling of the volatility variable cr,. We distinguish three main types of volatility modeling:

1. ARCH-type models. These autoregressive conditional heteroskedastic models define the variance a} of the return r, as a function of past returns. This function can be simple or rather complicated. In the GARCH process, for example, cr, also depends on its own past values, but there is always an equivalent formulation that defines cr, as a function of past returns only. The volatility 07 is a model variable that cannot be directly observed, but it can be computed if a sufficiently long series of past return values up to r,-\ is known. All the statistical processes discussed in the following sections of this chapter belong to the ARCH type.

2. Stochastic volatility models. In stochastic volatility models, the volatility variable cr; does not depend on past returns. Instead, it depends on its own past values. The volatility variable 07 is neither observable nor directly computable from past returns. As a consequence, it is more difficult to estimate the parameters of stochastic volatility models than those of ARCH-type models. The statistical process of trt has a memory, so an autoregressive conditional heteroskedastic behavior can be obtained also with stochastic volatility models. There are different types of stochastic volatility models as noted in Taylor (1994); Ghysels and Jasiak (1995), and Ghysels et al. (1996). It is possible to model heterogeneous market behavior in the framework of stochastic volatility; a modern example is the cascade model of Ghashghaie et al. (1996) and Breymann et al. (2000) where volatility modeling is inspired by turbulence models.

3. Models based on realized volatility. Rather than modeling o>, (Andersen et al., 2000) propose to define 07 as the realized volatility computed at index t - 1. This realized volatility is computed with high-frequency data, with return intervals of, for example, 30-min, in order to keep stochastic errors low. The time interval of the main model (i.e., the interval between the indices t - 1 and t) is usually much larger (e.g., 1 working day). This means using realized volatility at t - 1 as a predictor of the volatility between t - 1 and t by relying on the volatility clustering. This model has the advantage of using empirical data instead of model assumptions that might be wrong. However, it has some disadvantages:

Realized volatility is biased if computed at high frequency (see Section 7.2). A bias correction method such as Equation 7.4 would improve the model.

Realized volatility computed at high frequency (fine volatility) lags behind coarse volatility in the lead-lag analysis (see Section 7.4.2). This lag leads to suboptimal forecast quality when predicting the volatility of the next step of the model.

In general, realized volatility at t - 1 may not be the best predictor of volatility between t - 1 and t. It should be replaced by a more sophisticated forecast of realized volatility at t.

In most of these statistical processes of a}, it is possible to add some terms modeling external (exogeneous) influences. If volume figures at t - 1 are available, for example, they may be a piece of information to predict the volatility o>. The processes discussed here are not of this type, they are univariate.

In the remainder of this chapter, we stay within the framework of ARCH-type modeling and compare different models. The ultimate quality criterion of a model is its predictive power. Therefore there are some volatility forecast tests in Section 8.4. Forecasting is further discussed in Chapter 9 where it is the main subject.

8.2 intraday volatility and garch models

The Autoregressive Conditional Heteroskedastic (ARCH) model of Engle (1982) and its generalized version (GARCH) by Bollerslev (1986) are widely used, not only in the foreign exchange (FX) literature (see, for a review, Bollerslev et al, 1992) but also as the basic framework for empirical studies of the market microstructure such as the impact of news (Goodhart and Figliuoli, 1991; Goodhart et al., 1993) and central bank interventions (Goodhart and Hesse, 1993; Peiers, 1997), or inter and intramarket relationships in Engle et al. (1990) and Baillie and Bollerslev (1990). The main assumption behind this class of models is the relative homogeneity of the price discovery process among market participants at the origin of the volatility process. In other words, the conditional density of one GARCH process is assumed to adequately capture all the information and the news. In particular, GARCH parameters for the weekly frequency theoretically derived from daily empirical estimates are usually within the confidence interval of weekly empirical estimates (Drost and Nijman, 1993).

However, we have already seen in this book that several empirical facts are at odds with this homogeneous view of the market. First, the long memory of the volatility (Section 7.3.2) indicates the presence of several market components corresponding to several time horizons. Note that this property of the volatility has already been successfully incorporated in the GARCH setting as the fractionally integrated GARCH (Baillie et al, 1996). Second, at the intradaily frequency, round-the-clock time series reveal seasonal patterns that reflect, among others, the geographical dispersion of the traders, concentrated in three main geographical areas: Asia, Europe, and America. Although the first investigations of the effect of these different geographical locations seemed to indicate that news would uniformly spread out around the world (the so-called meteor shower hypothesis in Engle et al, 1990), we saw traces of heat wave effects in the previous chapter. Third, exchange rates movements are not necessarily related to the arrival of news when inspected at the intraday frequency, Goodhart (1989), reflecting the fact that intraday traders may have other constraints and objectives than, for