The Bayesian approach has been developed in various areas and has come to be part of main stream statistical research. Markov Chain Monte Carlo (MCMC) methods have freed us from computational constraints for a wide class of models and several MCMC methods are now available for sampling from posterior distributions. However, when data is large and models are complex and the likelihood function is intractable we are limited in the use of MCMC, especially in evaluating likelihood function. As a solution to the problem, researchers have put forward approximate Bayesian computation (ABC), also known as a likelihood-free method. In this report I introduce the ABC algorithm and show implementation for a stochastic volatility model (SV). Even though there are alternative methods for analyzing SV models, such as particle filters and other MCMC methods, I show the ABC method with an SV model and compare it, based on the same data and the SV model, to an approach based on a mixture of normals and MCMC. / text

This dissertation studies volatility measurement and modeling issues when asset prices are subject to price limits based on Bayesian approaches. Two types of estimators are developed to consistently estimate integrated volatility in the presence of price limits. One is a realized volatility type estimator, but using both realized asset prices and simulated asset prices. The other is a discrete sample analogue of integrated volatility using posterior samples of the latent volatility states. These two types of estimators are first constructed based on the simple log-stochastic volatility model in Chapter 2. The simple log-stochastic volatility framework is extended in Chapter 3 to incorporate correlated innovations and further extended in Chapter 4 to accommodate jumps and fat-tailed innovations. For each framework, a MCMC algorithm is designed to simulate the unobserved asset prices, model parameters and latent states. Performances of both type estimators are also examined using simulations under each framework. Applications to Chinese stock markets are also provided. / Thesis (Ph.D, Economics) -- Queen's University, 2014-01-22 10:29:12.507

Estimation of stochastic volatility (SV) models is a formidable task because the presence of the latent variable makes the likelihood function difficult to construct. The model can be transformed to a linear state space with non-Gaussian disturbances. Durbin and Koopman (1997) have shown that the likelihood function of the general non-Gaussian state space model can be approximated arbitrarily accurately by decomposing it into a Gaussian part (constructed by the Kalman filter) and a remainder function (whose expectation is evaluated by simulation). This general methodology is specialised to the estimation of SV models. A finite sample simulation experiment illustrates that the resulting Monte Carlo likelihood estimator achieves full efficiency with minimal computational effort. Accurate values of the likelihood function allow inference within the model to be performed by means of likelihood ratio tests. This enables tests for the presence of a unit root in the volatility process to be constructed which are shown to be more powerful than the conventional unit root tests. The second part of the thesis consists of two empirical applications of the SV model. First, the informational content of implied volatility is examined. It is shown that the in- sample evolution of DEM/USD exchange rate volatility can be accurately captured by implied volatility of options. However, better forecasts of ex post volatility can be constructed from the basic SV model. This suggests that options implied volatility may not be market's best forecast of the future asset volatility, as is often assumed. Second, the regulatory claim of a destabilising effect of futures market trading on stock market volatility is critically assessed. It is shown how volume-volatility relationships can be accurately modelled in the SV framework. The variables which approximate the activity in the FT100 index futures market are found to have no influence on the volatility of the underlying stock market index.

In this thesis we consider estimation of the tail index for heavy tailed stochastic volatility models with long memory. We prove a central limit theorem for a Hill estimator. In particular, it is shown that neither the rate of convergence nor the asymptotic variance is affected by long memory. The theoretical findings are verified by simulation studies.

In this thesis we consider estimation of the tail index for heavy tailed stochastic volatility models with long memory. We prove a central limit theorem for a Hill estimator. In particular, it is shown that neither the rate of convergence nor the asymptotic variance is affected by long memory. The theoretical findings are verified by simulation studies.

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Title: Volatility bursts and order book dynamics Author: Jana Plačková Department: Department of Probability and Mathematical Statistics Supervisor: Dr. Jan M. Swart Supervisor's e-mail address: swart@utia.cas.cz Abstract: The presented paper studies the dynamics of supply and demand through the electronic order book. We describe and define the basic rules of the order book and its dynamics. We also define limit and market orders and describe the differences between them and how they influenced the evolution of ask, bid price and spread. Next part of the paper is dedicated to the de- scription and definition of volatility and its basic models. The brief overview about volatility clustering and its modeling by economists and physicists can be found in the following part. In the last part we introduce a simple model of order book in which we observe ask, bid price and spread. Then we study the empirical distribution of spread and try to find its probability distribu- tion. The volatility clustering is then observed through the relative returns of spread. In the last part we introduce some possible improvement of the model. Keywords: volatility clustering, order book, limit orders, market orders 1

I perform comprehensive comparison of the standard realised volatility estimators including a novel wavelet time-frequency estimator (Barunik and Vacha 2012) on wide variety of assets: crude oil, gold and S&P 500. The wavelet estimator allows to decompose the realised volatility into several investment horizons which is hypothesised in the literature to bring more information about the volatility time series. Moreover, I propose artificial neural networks (ANN) as a tool for forecasting of the realised volatility. Multi-layer perceptron and recursive neural networks typologies are used in the estimation. I forecast cumulative realised volatility on 1 day, 5 days, 10 days and 20 days ahead horizons. The forecasts from neural networks are benchmarked to a standard autoregressive fractionally integrated moving averages (ARFIMA) model and a mundane model. I confirm favourable features of the novel wavelet realised volatility estimator on crude oil and gold, and reject them in case of S&P 500. Possible explanation is an absence of jumps in this asset and hence over-adjustment of data for jumps by the estimator. In forecasting, the ANN models outperform the ARFIMA in terms of information content about dynamic structure of the time series.

Master of Science / Department of Agricultural Economics / Ted C. Schroeder / In recent years, the agricultural markets have been subject to increased prices and unusual levels of elevated volatility. One likely driver of this is the mandated ethanol expansion in the Energy Policy Act of 2005. Previous research has identified relationships in market prices and variability between the energy and grain markets, but little has been done to evaluate volatility spillover across a broader spectrum of agricultural commodities. Additionally, few studies have assessed causal linkages across market implied volatilities. This research examines implied volatility spillover in futures markets across major agricultural commodities and energies. The analysis also determines the time path and magnitude of volatility translation across the markets and compares the causal relationships between pre-ethanol boom and post-ethanol boom time periods. Granger causality tests are conducted using multivariate and bivariate vector autoregressive modeling techniques, and impulse response functions are employed to obtain time paths of the reactions. Overall, results indicate that strong implied volatility spillover relationships exist between the grain markets and between the live cattle and feeder cattle markets. The analysis also finds that the agricultural markets have evolved from lean hogs being the primary volatility leader in the pre-ethanol boom era to corn being the primary volatility leader in the post-ethanol boom era. Despite a high correlation between crude oil and corn volatilities in the post-ethanol boom time period, the causal linkage between the two commodities’ volatilities may not be as definite as other literature suggests.

In this study, I examine the transmissions of volatility spillovers during the subprime crisis in the U.S between Vietnam and other Asian financial markets (Japan, Korea, China, Hong Kong, and Taiwan). I attempt to explore the level and magnitude of volatility spillover effects of other Asian markets on the Vietnam stock market by applying a multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model. It is found that the level of the volatility effect of the selected financial markets on the Vietnamese stock market’s return from 2006 to August - 2009 increases over time. Particularly, the level of volatility transmissions and spillover effect of two developed markets, Hong Kong and Japan onto the Vietnamese market are relatively higher and more consistent than other markets during the 2006-2009 period. Also, the Vietnamese financial market seems to perform better than other markets during my 2006-2009 sample, including the financial crisis period in 2007.