Bayesian inference with an adaptive proposal density for GARCH models

We perform the Bayesian inference of a GARCH model by the Metropolis-Hastings algorithm with an adaptive proposal density. The adaptive proposal density is assumed to be the Student's t-distribution and the distribution parameters are evaluated by using the data sampled during the simulation. We apply the method for the QGARCH model which is one of asymmetric GARCH models and make empirical studies for for Nikkei 225, DAX and Hang indexes. We find that autocorrelation times from our method are very small, thus the method is very efficient for generating uncorrelated Monte Carlo data. The results from the QGARCH model show that all the three indexes show the leverage effect, i.e. the volatility is high after negative observations

Bayesian Semi-parametric Realized-CARE Models for Tail Risk Forecasting Incorporating Range and Realized Measures

A new framework named Realized Conditional Autoregressive Expectile (Realized- CARE) is proposed, through incorporating a measurement equation into the conventional CARE model, in a framework analogous to Realized-GARCH. The Range and realized measures (Realized Variance and Realized Range) are employed as the dependent variables of the measurement equation, since they have proven more efficient than return for volatility estimation. The dependence between Range & realized measures and expectile can be modelled with this measurement equation. The grid search accuracy of the expectile level will be potentially improved with introducing this measurement equation. In addition, through employing a quadratic fitting target search, the speed of grid search is significantly improved. Bayesian adaptive Markov Chain Monte Carlo is used for estimation, and demonstrates its superiority compared to maximum likelihood in a simulation study. Furthermore, we propose an innovative sub-sampled Realized Range and also adopt an existing scaling scheme, in order to deal with the micro-structure noise of the high frequency volatility measures. Compared to the CARE, the parametric GARCH and the Realized-GARCH models, Value-at-Risk and Expected Shortfall forecasting results of 6 indices and 3 assets series favor the proposed Realized-CARE model, especially the Realized-CARE model with Realized Range and sub-sampled Realized Range

Likelihood-based estimation of latent generalised ARCH structures

GARCH models are commonly used as latent processes in econometrics, financial economics and macroeconomics. Yet no exact likelihood analysis of these models has been provided so far. In this paper we outline the issues and suggest a Markov chain Monte Carlo algorithm which allows the calculation of a classical estimator via the simulated EM algorithm or a Bayesian solution in O(T) computational operations, where T denotes the sample size. We assess the performance of our proposed algorithm in the context of both artificial examples and an empirical application to 26 UK sectorial stock returns, and compare it to existing approximate solutions.Bayesian inference; Dynamic Heteroskedasticity; Factor models; Markov chain Monte Carlo; Simulated EM algorithm; Volatility.

LIKELIHOOD-BASED ESTIMATION OF LATENT GENERALISED ARCH STRUCTURES

GARCH models are commonly used as latent processes in econometrics, financial economics and macroeconomics. Yet no exact likelihood analysis of these models has been provided so far. In this paper we outline the issues and suggest a Markov chain Monte Carlo algorithm which allows the calculation of a classical estimator via the simulated EM algorithm or a Bayesian solution in O(T) computational operations, where T denotes the sample size. We assess the performance of our proposed algorithm in the context of both artificial examples and an empirical application to 26 UK sectorial stock returns, and compare it to existing approximate solutions.Bayesian inference; Dynamic Heteroskedasticity; Factor models

"Multivariate stochastic volatility"

We provide a detailed summary of the large and vibrant emerging literature that deals with the multivariate modeling of conditional volatility of financial time series within the framework of stochastic volatility. The developments and achievements in this area represent one of the great success stories of financial econometrics. Three broad classes of multivariate stochastic volatility models have emerged, one that is a direct extension of the univariate class of stochastic volatility model, another that is related to the factor models of multivariate analysis, and a third that is based on the direct modeling of time-varying correlation matrices via matrix exponential transformations, Wishart processes and other means. We discuss each of the various model formulations, provide connections and differences and show how the models are estimated. Given the interest in this area, further significant developments can be expected, perhaps fostered by the overview and details delineated in this paper, especially in the fitting of high dimensional models.

Multivariate stochastic volatility (Revised in May 2007, Handbook of Financial Time Series (Published in "Handbook of Financial Time Series" (eds T.G. Andersen, R.A. Davis, Jens-Peter Kreiss and T. Mikosch), 365-400. Springer-Verlag: New York. April 2009. )

The success of univariate stochastic volatility (SV) models in relation to univariate GARCH models has spurred an enormous interest in generalizations of SV models to a multivariate setting. A large number of multivariate SV (MSV) models are now available along with clearly articulated estimation recipes. Our goal in this paper is to provide the first detailed summary of the various model formulations, along with connections and differences, and discuss how the models are estimated. We aim to show that the developments and achievements in this area represent one of the great success stories of financial econometrics.

Bayesian parametric and semi-parametric financial tail-risk forecasting incorporating range and realized measures

Now we are in a world saturated with data and information, and numerous quantitative methods for financial risk management are proposed and used by many financial research institutions and organizations within recent years. Quantitative financial risk measurement is now a fundamental tool for investment decisions, capital allocation and external regulation. The Global Financial Crisis (GFC) has once again emphasized the importance of accurate risk measurement and prediction for financial organizations, which require accurate volatility estimation and forecasting. The intra-day range has been frequently used in the literature and proven its superiority compared to return in volatility estimation and forecasting. Furthermore, high frequency econometrics has been gaining more popularity in the last decade and has developed into a major area in econometrics, driven by the increasing availability of high frequency data and algorithm-based high frequency trading in seconds or even milliseconds. The data recorded on a high frequency level contain much more information than the conventional daily financial data, and thus the volatility measures calculated based on high frequency data are much more efficient than the daily return and range. In this thesis, we aim to develop a series of volatility and tail risk models employing intra-day and high frequency volatility measures. Firstly, the Realized GARCH framework is extended to incorporate the realized range, as potentially more efficient series of information than realized variance. Furthermore, we propose an innovative sub-sampled realized range and also adopt an existing scaling scheme, in order to deal with the micro-structure noise of the high frequency volatility measures. In addition, a Bayesian estimator is developed for the Realized GARCH type models, and presents favourable results compared to the frequentist estimator. Through empirical studies on various market indices that consider predictive likelihoods as well as 1% VaR and ES forecasting, results clearly indicate that the realized range and sub-sampled realized range in a Realized GARCH framework, with Student-t errors, lead to more accurate volatility and predictive density forecasts. Further, a new framework called Realized Conditional Autoregressive Expectile (Realized CARE) is proposed, through incorporating a measurement equation into the conventional CARE model, in a manner analogous to the Realized GARCH model. The intra-day range and realized measures (e.g. realized variance and realized range, etc.) are employed as the dependent variable in the measurement equation. The measurement equation here models the contemporaneous dependence between the realized measure and the latent conditional expectile. In addition, a targeted search based on a quadratic approximation is proposed, which improves the computational speed of estimation of the expectile level parameter. Bayesian adaptive Markov Chain Monte Carlo methods and likelihood-based frequentist methods are proposed for estimation, whilst their properties are compared via a simulation study. Furthermore, the methods of sub-sampling and scaling are applied to the realized variance and realized range, to help deal with the inherent micro-structure noise of the realized volatility measures. In a real forecasting study applied to 6 market indices and 3 individual assets, compared to the original CARE, the parametric GARCH and Realized GARCH models, one-day-ahead Value-at-Risk and Expected Shortfall forecasting results favor the proposed Realized CARE model, especially the Realized CARE model incorporating the realized range and the sub-sampled realized range. Finally, we propose a new intra-day volatility estimator named signed range, which incorporates open, high and low prices for its calculation. A high frequency simulation study is conducted to analyze the relationship between signed range volatility and return volatility. An adaptive MCMC is developed for the parameter estimation and is compared with the maximum likelihood approach through simulation study. Then we propose the symmetric and asymmetric Conditional Autoregressive Signed Range (CARSR) type models, and the proposed models demonstrate their superiority compared to GARCH and Conditional Autoregressive Range (CARR) models in the 1% VaR and ES forecasting study

Bayesian parametric and semi-parametric financial tail-risk forecasting incorporating range and realized measures

Now we are in a world saturated with data and information, and numerous quantitative methods for financial risk management are proposed and used by many financial research institutions and organizations within recent years. Quantitative financial risk measurement is now a fundamental tool for investment decisions, capital allocation and external regulation. The Global Financial Crisis (GFC) has once again emphasized the importance of accurate risk measurement and prediction for financial organizations, which require accurate volatility estimation and forecasting. The intra-day range has been frequently used in the literature and proven its superiority compared to return in volatility estimation and forecasting. Furthermore, high frequency econometrics has been gaining more popularity in the last decade and has developed into a major area in econometrics, driven by the increasing availability of high frequency data and algorithm-based high frequency trading in seconds or even milliseconds. The data recorded on a high frequency level contain much more information than the conventional daily financial data, and thus the volatility measures calculated based on high frequency data are much more efficient than the daily return and range. In this thesis, we aim to develop a series of volatility and tail risk models employing intra-day and high frequency volatility measures. Firstly, the Realized GARCH framework is extended to incorporate the realized range, as potentially more efficient series of information than realized variance. Furthermore, we propose an innovative sub-sampled realized range and also adopt an existing scaling scheme, in order to deal with the micro-structure noise of the high frequency volatility measures. In addition, a Bayesian estimator is developed for the Realized GARCH type models, and presents favourable results compared to the frequentist estimator. Through empirical studies on various market indices that consider predictive likelihoods as well as 1% VaR and ES forecasting, results clearly indicate that the realized range and sub-sampled realized range in a Realized GARCH framework, with Student-t errors, lead to more accurate volatility and predictive density forecasts. Further, a new framework called Realized Conditional Autoregressive Expectile (Realized CARE) is proposed, through incorporating a measurement equation into the conventional CARE model, in a manner analogous to the Realized GARCH model. The intra-day range and realized measures (e.g. realized variance and realized range, etc.) are employed as the dependent variable in the measurement equation. The measurement equation here models the contemporaneous dependence between the realized measure and the latent conditional expectile. In addition, a targeted search based on a quadratic approximation is proposed, which improves the computational speed of estimation of the expectile level parameter. Bayesian adaptive Markov Chain Monte Carlo methods and likelihood-based frequentist methods are proposed for estimation, whilst their properties are compared via a simulation study. Furthermore, the methods of sub-sampling and scaling are applied to the realized variance and realized range, to help deal with the inherent micro-structure noise of the realized volatility measures. In a real forecasting study applied to 6 market indices and 3 individual assets, compared to the original CARE, the parametric GARCH and Realized GARCH models, one-day-ahead Value-at-Risk and Expected Shortfall forecasting results favor the proposed Realized CARE model, especially the Realized CARE model incorporating the realized range and the sub-sampled realized range. Finally, we propose a new intra-day volatility estimator named signed range, which incorporates open, high and low prices for its calculation. A high frequency simulation study is conducted to analyze the relationship between signed range volatility and return volatility. An adaptive MCMC is developed for the parameter estimation and is compared with the maximum likelihood approach through simulation study. Then we propose the symmetric and asymmetric Conditional Autoregressive Signed Range (CARSR) type models, and the proposed models demonstrate their superiority compared to GARCH and Conditional Autoregressive Range (CARR) models in the 1% VaR and ES forecasting study