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Chapter One
Volatility Models
Luc Bauwens
Christian Hafner
Sébastien Laurent
1.1 Introduction
This chapter presents an introductory review of volatility models and some applications. We link our review with other chapters that contain more detailed presentations. Section 1.2 deals with generalized autoregressive conditionally heteroskedastic models, Section 1.3 with stochastic volatility (SV) models, and Section 1.4 with realized volatility.
1.2 GARCH
1.2.1 Univariate GARCH
Univariate ARCH models appeared in the literature with the paper of Engle (1982a), soon followed by the generalization to GARCH of Bollerslev (1986). Although applied, in these pathbreaking papers, to account for the changing volatility of inflation series, the models and their later extensions were quickly found to be relevant for the conditional volatility of financial returns observed at a monthly and higher frequency (Bollerslev, 1987), and thus to the study of the intertemporal relation between risk and expected return (French et al., 1987; Engle et al., 1987). The reason is that time series of returns (even if adjusted for autocorrelation, typically through an ARMA model) have several features that are well fitted by GARCH models. The main stylized feature is volatility clustering: “large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes” (Mandelbrot, 1963). This results in positive autocorrelation coefficients of squared returns, typically with a relatively slowly decreasing pattern starting from a first small value (say, < 0.2). Said differently, volatility, measured by squared returns, is persistent, hence to some extent predictable even if it is noisy. Another stylized property of financial returns that was known long before ARCH models appeared is that their unconditional probability distributions are leptokurtic, that is, they have fatter tails and more mass around their center than the Gaussian distribution (Mandelbrot 1963). In this and later papers (e.g., Fama, 1963, 1965; Mandelbrot and Taylor, 1967), the returns are modeled as independently and identically distributed (i.i.d.) according to a stable Paretian distribution. But clearly, if squared returns are autocorrelated, they are not independent. A great advantage of GARCH models is that the returns are not assumed independent, and even if they are assumed Gaussian conditional to past returns, unconditionally they are not Gaussian, because volatility clustering generates leptokurtosis.
We illustrate the stylized facts with the percentage daily returns of the S&P 500 index, that is, the returns (yt) are computed as 100(pt − pt−1), where pt = logPt and Pt is the closing price index value adjusted for dividends and splits (available at http://finance.yahoo.com) and t is the time index referring to trading day t. The sample period starts on January 3, 1950 and ends on July 14, 2011 for a total of 15,482 returns. Table 1.1 contains descriptive statistics of the original and demeaned returns, the latter being the residuals of an AR(2) model fitted to the original returns. The descriptive statistics of the two series hardly differ and the large excess kurtosis coefficients confirm their leptokurtosis. Figure 1.1 displays the full sample series of returns (a) and the series for the last five years (2006/07/14–2011/07/14) (b). Figure 1.2 shows the full series of absolute demeaned returns (a), the sample ACF of the corresponding squared series until lag 100 (b), and the absolute demeaned returns or the last five years (c). The squared demeaned returns are positively autocorrelated: their ACF starts at 0.15, has a peak of 0.20 at lag 5, and decreases rather slowly. Volatility clustering is clearly visible on the top and bottom graphs of both figures. The leptokutrosis of the estimated density of the demeaned returns, shown over a truncated support—see maximum and minimum values in Table 1.1—is visible on Figure 1.3, where a Gaussian density with the same mean (0) and standard deviation (0.969) is drawn for comparison. The negative skewness coefficients reported in Table 1.1 illustrate that large negative returns are more probable than large positive ones. This is also a widespread feature, by no means universal, of financial return series, which we discuss below.
Table 1.1 Descriptive Statistics for S&P 500 Returns
| Observations | 15,482 | 15,480 |
| Mean | 0.02818 | 0 |
| Standard deviation | 0.97078 | 0.96897 |
| Skewness | −1.0567 | −1.0738 |
| Kurtosis | 32.035 | 31.623 |
| Minimum | −22.900 | −22.856 |
| Maximum | 10.957 | 10.571 |
