| Given the rapid growth in financial markets and the continual development of new and more complex financial instruments, there is an ever-growing need for theoretical and empirical knowledge of the volatility in financial time series. It is widely known that the daily returns of financial assets, especially of stocks, are difficult, if not impossible, to predict, although the volatility of the returns seems to be relatively easier to forecast. Therefore, it is hardly surprising that financial econometrics, in particular the modeling of financial volatility, has played such a central role in modern pricing and risk management theories.There is, however, an inherent problem in using models where the volatility measure plays a central role. The conditional variance is latent, and hence is not directly observable. It can be estimated, among other approaches, by the (Generalized) Autoregressive Conditional Heteroskedasticity, or (G)ARCH, family of models proposed by Engle (1982) and Bollerslev (1986), stochastic volatility (SV) models (see, for example, Taylor, 1986), or exponentially weighted moving averages (EWMA), as advocated by the Riskmetrics methodology (Morgan, 1996) (see McAleer, 2005 for a recent exposition of a wide range of univariate and multivariate, conditional and stochastic, models of volatility, and Asai et al. (2006) for a review of the growing literature on multivariate stochastic volatility models). However, as observed by Bollerslev (1987), most of the latent volatility models fail to describe satisfactorily several stylized facts that are observed in financial time series.The search for an adequate framework for the estimation and prediction of the conditional variance of financial assets returns has led to the analysis of high frequency intraday data. Merton (1980) noted that the variance over a fixed interval can be estimated arbitrarily, although accurately, as the sum of squared realizations, provided the data are available at a sufficiently high sampling frequency. Inspired by the Heterogeneous Market Hypothesis and by the asymmetric propagation of volatility (between long and short time horizons), Fulvio Corsi propose an additive cascade of different volatility components generated by the actions of different types of market participants. This additive volatility cascade leads to a simple AR-type model in the realized volatility with the feature of considering volatilities realized over different time horizons. We term this model, Heterogeneous Autoregressive model of the Realized Volatility (HAR-RV). In spite of the simplicity of its structure, simulation results, seem to confirm that the HAR-RV model successfully achieves the purpose of reproducing the main empirical features of volatility (long memory, fat tail, self-similarity)in a very simple and parsimoniously way.The article is organized as follows. In Section 1 we will make the general introduction on the base idea of the realized volatility and Measurement errors. In Section 2, we will employ the very high frequency and every five minutes realized volatility estimators developed in Zumbach, Corsi and Trapletti (2002). We conclude the Stylized facts of our stock market as follows: fat tail, long memory and high level of skewness and kurtosis. Section 3,we will briefly introduce the HAR-RV model, the HAR-RV model is a multi-component volatility model with an additive hierarchical structure which will leads to a very simple additive time series model of the realized volatility. The Heterogeneous Market Hypothesis try to explain the empirical observation of a strong positive correlation between volatility and market presence. In fact, in a homogeneous market framework where all the participants are identical, the more agents are presents, the faster the price should converge to its real market value on which all agents agreed. Thus, the volatility should be negatively correlated with market presence and activity. On the contrary in an heterogeneous markets, different actors are likely to settle for different prices and decide to execute their transactions in different market situations, hence they create volatility. In Section 4, we use HAR-RV model to model and forecast our stock market realized volatility. Since there is a Significant ARCH effect in the regression residuals, we propose a HAR-RV-GARCH to achieve more accurate forecast result. In the last Section we use the high-frequency data of the PUFA incorporation and of the SH index to compute the realized beta and use AR(3) model to forecast it. |
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