Essays on financial volatitily modelling.

Correlation, volatility, and covariance are three important metrics of financial risk. They are key quantities in many areas of empirical finance, including portfolio selection, asset pricing, and hedging. For instance, low correlation among assets is the starting point of portfolio diversification strategies. An essential characteristic of asset returns is volatility clustering. Mandelbrot (1963) noted that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes." This led to the development of stochastic volatility models (Clark 1973) and (G)ARCH models (Engle 1982 and Bollerslev et al. 1988). Furthermore, asset return correlations also vary over time. For instance, they tend to decline in bull markets and to rise in bear markets (Longin and Solnik 2001), and they are typically higher in periods of high volatility. Accounting for the time variation in asset return correlations, volatilities, and covariances is important. It allows us, for instance, to compute time-varying hedge ratios, estimate time-varying beta coefficients, and calculate the value-at-risk of a portfolio. This thesis contributes to the GARCH literature, with a focus on two aspects: multivariate GARCH in relatively high dimensions, and the treatment of overnight returns combined with high-frequency returns. In Chapter 1, we propose sparse versions of multivariate GARCH models that allow for volatility and correlation spillover effects across assets. The proposed models are generalizations of existing diagonal DCC and BEKK models, yet they remain estimable for high-dimensional systems of asset returns. To cope with the high dimensionality of the model parameter spaces, we employ the $L_1$ regularization technique to penalize the off-diagonal elements of the coefficient matrices. A simulation experiment for the parameter structure can be uncovered reasonably well. In an application to daily market returns for 23 countries using data from 1988 to 2015, we find that the sparse DCC model outperforms the standard DCC and the diagonal DCC models in and out of sample. Likewise, the sparse BEKK model outperforms the diagonal BEKK model. In Chapter 2, we study international asset pricing in a large-dimensional multivariate GARCH-in-mean framework. We examine different estimation methods and find that the two-step estimation method proposed by \cite{BaliEngle2010} tends to underestimate the risk-return coefficient and the corresponding standard error. We also show that the estimate is improved by one-step estimation and by increasing the cross-sectional dimension. Using stock index returns for up to 24 markets and 4 major currencies over the period of 2001--2015, one-step estimation gives a market risk-return coefficient of around 6. The estimate is robust to variations in model specification, data frequency, and the number of stock markets considered. In Chapter 3, we introduce and evaluate mixed-frequency multivariate GARCH models for forecasting low-frequency (weekly or monthly) multivariate volatility based on high-frequency intra-day returns (at five-minute intervals) and on the overnight returns. The low-frequency conditional volatility matrix is modelled as a weighted sum of an intra-day and an overnight component, driven by the intra-day and the overnight returns, respectively. The components are specified as multivariate GARCH(1,1) models of the BEKK type, adapted to the mixed-frequency data setting. For the intra-day component, the squared high-frequency returns enter the GARCH model through a parametrically specified mixed-data sampling (MIDAS) weight function or through the sum of the intra-day realized volatilities. For the overnight component, the squared overnight returns enter the model with equal weights. Alternatively, the low-frequency conditional volatility matrix may be modelled as a single-component BEKK-GARCH model where the overnight returns and the high-frequency returns enter through the weekly realized volatility (defined as the unweighted sum of squares of overnight and high-frequency returns), or where the overnight returns are simply ignored. All model variants may further be extended by allowing for a non-parametrically estimated slowly-varying long-run volatility matrix. The proposed models are evaluated using five-minute and overnight return data on four DJIA stocks (AXP, GE, HD, and IBM) from January 1988 to November 2014. The focus is on forecasting weekly volatilities (defined as the low frequency). The mixed-frequency GARCH models are found to systematically dominate the low-frequency GARCH model in terms of in-sample fit and out-of-sample forecasting accuracy. They also exhibit much lower low-frequency volatility persistence than the low-frequency GARCH model. Among the mixed-frequency models, the low-frequency persistence estimates decrease as the data frequency increases from daily to five-minute frequency, and as overnight returns are included. That is, ignoring the available high-frequency information leads to spuriously high volatility persistence. Among the other findings are that the single-component model variants perform worse than the two-component variants; that the overnight volatility component exhibits more persistence than the intra-day component; and that MIDAS weighting performs better than not weighting at all (i.e., than realized volatility). In Chapter 4, we propose a semi-parametric coupled component GARCH model for intraday and overnight volatility that allows them to have different properties. To capture the very heavy tail of the overnight return, we adopt a dynamic conditional score model. The large sample property of our model is provided, and a multivariate version is also described. We apply our model to the study of the component stocks of the Dow Jones industrial average over the period 1991-2016. We show that actually overnight volatility has increased in importance during this period. In addition, our model provides better intraday volatility forecast since it takes account of the full dynamic consequences of the overnight shock and previous ones.nrpages: 147status: publishe