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The topic of volatility measurement and estimation is central to financial and more generally time-series econometrics. In this chapter, we begin by surveying models of volatility, both discrete and continuous, and then we summarize some selected empirical findings from the literature. In particular, in the first sections of this chapter, we discuss important developments in volatility models, with focus on time-varying and stochastic volatility as well as nonparametric volatility estimation. The models discussed share the common feature that volatilities are unobserved and belong to the class of missing variables. We then provide empirical evidence on “small” and “large” jumps from the perspective of their contribution to overall realized variation, using high-frequency price return data on 25 stocks in the DOW 30. Our “small” and “large” jump variations are constructed at three truncation levels, using extant methodology of Barndorff-Nielsen and Shephard (2006), Andersen, Bollerslev, and Diebold (2007), and Aït-Sahalia and Jacod (2009a, 2009b, 2009c). Evidence of jumps is found in around 22.8% of the days during the 1993–2000 period, much higher than the corresponding figure of 9.4% during the 2001–2008 period. Although the overall role of jumps is lessening, the role of large jumps has not decreased, and indeed, the relative role of large jumps, as a proportion of overall jumps, has actually increased in the 2000s.

In this chapter, I examine the properties of four realized correlation estimators and model their jumps. The correlations are between the three main FTSE indices of the Athens Stock Exchange. Using intraday data I first construct four state-of-the-art realized correlation estimators which I then use in testing for normality, long memory, asymmetries and jumps and also in modelling for jumps. Jumps are detected when the realized correlation is higher than 0.99 and lower than 0.01 in absolute values. Then the realized correlation is modelled with the simple heterogeneous autoregressive (HAR) model and the HAR model with jumps (HAR-J). This is the first time, to the best of my knowledge, that the realized correlation between the three indices for the Greek equity market is examined.

This research studies private information from extreme price movements or jumps. The authors calculate the private information using a reduced form model from the stochastic volatility jump process and use several statistical robustness tests as well as several frequencies to improve our consistency. This study reveals that private information is significant in explain the existence of jumps in capital markets in Southeast Asia, whereas macroeconomic events cannot explain them. The authors determine empirically that private information in Malaysia, Singapore, Thailand, and Indonesia are not persistent and its value gradually decreases when we use the lower frequency. Based on the Fama–Macbeth regression, this study shows that private information in the capital market has a strong positive relationship with individual returns in Indonesia’s capital market and Thailand’s capital market for all frequencies.

The purpose of this paper is to consider that the model of volatility characteristics is more reasonable and the description of volatility is more explanatory.

This paper analyzes the basic characteristics of market yield volatility based on the five-minute trading data of the Chinese CSI300 stock index futures from 2012 to 2017 by Hurst index and GPH test, A-J and J-O Jumping test and Realized-EGARCH model, respectively. The results show that the yield fluctuation rate of CSI300 stock index futures market has obvious non-linear characteristics including long memory, jumpy and asymmetry.

This paper finds that the LHAR-RV-CJ model has a better prediction effect on the volatility of CSI300 stock index futures. The research shows that CSI300 stock index futures market is heterogeneous, means that long-term investors are focused on long-term market fluctuations rather than short-term fluctuations; the influence of the short-term jumping component on the market volatility is limited, and the long jump has a greater negative influence on market fluctuation; the negative impact of long-period yield is limited to short-term market fluctuation, while, with the period extending, the negative influence of long-period impact is gradually increased.

This paper has research limitations in variable measurement and data selection.

This study is based on the high-frequency data or the application number of financial modeling analysis, especially in the study of asset price volatility. It makes full use of all kinds of information contained in high-frequency data, compared to low-frequency data such as day, weekly or monthly data. High-frequency data can be more accurate, better guide financial asset pricing and risk management, and result in effective configuration.

The existing research on the futures market volatility of high frequency data, mainly focus on single feature analysis, and the comprehensive comparative analysis on the volatility characteristics of study is less, at the same time in setting up the model for the forecast of volatility, based on the model research on the basic characteristics is less, so the construction of a model is relatively subjective, in this paper, considering the fluctuation characteristics of the model is more reasonable, characterization of volatility will also be more explanatory power. The difference between this paper and the existing literature lies in that this paper establishes a prediction model based on the basic characteristics of market return volatility, and conducts a description and prediction study on volatility.

Prior studies use two measures of firm-specific return variation (FSRV): idiosyncratic volatility in absolute and relative terms, the latter of which is also termed stock price nonsynchronicity. Whereas most research focuses on investigating the idiosyncratic volatility puzzle, the authors carry out comparison of these two measures and further investigate which of the two constituents of nonsynchronicity explain the association between FSRV and stock returns, emphasising the importance of assessing which component drives stock returns.

The authors use the US individual stock returns from 1925 to 2016 and define the two measures of FRSV based on the Fama and French (1993) model. Specifically, the authors decompose the relative measure into two components: (i) absolute idiosyncratic volatility and (ii) systematic volatility. The authors conduct various tests based on high-minus-low, zero-investment quintile portfolio sorts and perform the Fama–MacBeth analysis by singling out each component.

The authors find a positive return on the portfolio sorted on relative idiosyncratic volatility or on systematic volatility, but find a negative return sorted on absolute idiosyncratic volatility. The results are robust after controlling for size, BM and other risk characteristics using a double-sorting approach. The Fama–MacBeth regression results show that a positive association between the relative measure and stock returns is driven primarily by the low-systematic-volatility anomaly across firms. The findings are robust to controlling for return residual momentum, skewness, jumps and information discreteness.

Extant research posits the idiosyncratic volatility puzzle and the low-volatility anomaly. The authors emphasize the importance of integrating these two streams of research. This study enhances the understanding of the driving force underlying the relationship between FSRV and cross-sectional stock returns.

Motivated by empirical features that characterize cryptocurrency volatility data, the authors develop a forecasting strategy that can account for both model uncertainty and heteroskedasticity of unknown form. The theoretical investigation establishes the asymptotic optimality of the proposed heteroskedastic model averaging heterogeneous autoregressive (H-MAHAR) estimator under mild conditions. The authors additionally examine the convergence rate of the estimated weights of the proposed H-MAHAR estimator. This analysis sheds new light on the asymptotic properties of the least squares model averaging estimator under alternative complicated data generating processes (DGPs). To examine the performance of the H-MAHAR estimator, the authors conduct an out-of-sample forecasting application involving 22 different cryptocurrency assets. The results emphasize the importance of accounting for both model uncertainty and heteroskedasticity in practice.

This study aims to explore the risk spillover effects among different sectors of the Chinese stock market after the outbreak of COVID-19 from both Internet sentiment and price fluctuations.

The authors develop four indicators used for risk contagion analysis, including Internet investors and news sentiments constructed by the FinBERT model, together with realized and jump volatilities yielded by high-frequency data. The authors also apply the time-varying parameter vector autoregressive (TVP-VAR) model-based and the tail-based connectedness framework to investigate the interdependence of tail risk during catastrophic events.

The empirical analysis provides meaningful results related to the COVID-19 pandemic, stock market conditions and tail behavior. The results show that after the outbreak of COVID-19, the connectivity between risk spillovers in China's stock market has grown, indicating the increased instability of the connected system and enhanced connectivity in the tail. The changes in network structure during COVID-19 pandemic are not only reflected by the increased spillover connectivity but also by the closer relationships between some industries. The authors also found that major public events could significantly impact total connectedness. In addition, spillovers and network structures vary with market conditions and tend to exhibit a highly connected network structure during extreme market status.

The results confirm the connectivity between sentiments and volatilities spillovers in China's stock market, especially in the tails. The conclusion further expands the practical application and theoretical framework of behavioral finance and also lays a theoretical basis for investors to focus on the practical application of volatility prediction and risk management across stock sectors.

This paper aims to present the results of further investigating the Polimenis (2012) stochastic model, which aims to decompose the stock return evolution into positive and negative jumps, and a Brownian noise (white noise), by taking into account different noise levels. This paper provides a sensitivity analysis of the model (through the analysis of its parameters) and applies this analysis to Google and Yahoo returns during the periods 2006-2008 and 2008-2010, by means of the third central moment of Nasdaq index. Moreover, the paper studies the behavior of the calibrated jump sensitivities of a single stock as market skew changes. Finally, simulations are provided for the estimation of the jump betas coefficients, assuming that the jumps follow Gamma distributions.

In the present paper, the model proposed in Polimenis (2012) is considered and further investigated. The sensitivity of the parameters for the Google and Yahoo stock during 2006-2008 estimated by means of the third (central) moment of Nasdaq index is examined, and consequently, the calibration of the model to the returns is studied. The associated robustness is examined also for the period 2008-2010. A similar sensitivity analysis has been studied in Polimenis and Papantonis (2014), but unlike the latter reference, where the analysis is done while market skew is kept constant with an emphasis in jointly estimating jump sensitivities for many stocks, here, the authors study the behavior of the calibrated jump sensitivities of a single stock as market skew changes. Finally, simulations are taken place for the estimation of the jump betas coefficients, assuming that the jumps follow Gamma distributions.

A sensitivity analysis of the model proposed in Polimenis (2012) is illustrated above. In Section 2, the paper ascertains the sensitivity of the calibrated parameters related to Google and Yahoo returns, as it varies the third (central) market moment. The authors demonstrate the limits of the third moment of the stock and its mixed third moment with the market so as to get real solutions from (S1). In addition, the authors conclude that (S1) cannot have real solutions in the case where the stock return time series appears to have highly positive third moment, while the third moment of the market is significantly negative. Generally, the positive value of the third moment of the stock combined with the negative value of the third moment of the market can only be explained by assuming an adequate degree of asymmetry of the values of the beta coefficients. In such situations, the model may be expanded to include a correction for idiosyncratic third moment in the fourth equation of (S1). Finally, in Section 4, it is noticed that the distribution of the error estimation of the coefficients cannot be considered to be normal, and the variance of these errors increases as the variance of the noise increases.

As mentioned in the Findings, the paper demonstrates the limits of the third moment of the stock and its mixed third moment with the market so as to get real solutions from the main system of equations (S1). It is concluded that (S1) cannot have real solutions when the stock return time series appears to have highly positive third moment, while the third moment of the market is significantly negative. Generally, the positive value of the third moment of the stock combined with the negative value of the third moment of the market can only be explained by assuming an adequate degree of asymmetry of the values of the beta coefficients. In such situations, the model proposed should be expanded to include a correction for idiosyncratic third moment in the fourth equation of (S1). Finally, it is noticed that the distribution of the error estimation of the coefficients cannot be considered to be normal, and the variance of these errors increases as the variance of the noise increases.

Emerging financial markets have largely proven resilient to the consequences of the Global Financial Crisis. While this owes much to the bitter experience and economic strategies developed and implemented following the Asian Financial Crisis of 1997–1998, providence also played a hand in that relatively few of its financial institutions were exposed to the complex structured products that underpinned the demise of many financial intermediaries in the United States and Europe. The objective of this volume is to investigate and assess the impact and response to the crisis in emerging markets from a number of perspectives. These include asset pricing, contagion, financial intermediation, market structure and regulation. Our hope is that the assembled chapters offer clear insights into the complex financial arrangements that now link emerging and developed financial markets in the current economic environment. The volume spans four dimensions: first, a series of background studies offer explanations of the causes and impacts of the crisis on emerging markets more generally; then, implications are considered. The third and final sections provide insights from regional and country-specific perspectives.