Multiple imputation is a commonly used approach to deal with missing values. In this approach, an imputer repeatedly imputes the missing values by taking draws from the posterior predictive distribution for the missing values conditional on the observed values, and releases these completed data sets to analysts. With each completed data set the analyst performs the analysis of interest, treating the data as if it were fully observed. These analyses are then combined with standard combining rules, allowing the analyst to make appropriate inferences which take into account the uncertainty present due to the missing data. In order to preserve the statistical properties present in the data, the imputer must use a plausible distribution to generate the imputed values. In data sets containing variables with different measurement scales, e.g. some categorical and some continuous variables, Multivariate Imputation by Chained Equations (MICE) is a commonly used multiple imputation method. However, imputations from such an approach are not necessarily drawn from 1 a proper posterior predictive distribution. We propose a method to multiply impute missing values in such data sets by modelling the joint distribution of the variables in the data through a sequence of generalised linear models, and use data augmentation methods to draw imputations from a proper posterior distribution using Markov Chain Monte Carlo (MCMC). We compare the performance of our method with MICE using simulation studies and on a genuine data set taken from a breast feeding study.

Variable annuities play an important role in protecting one's future income. The pricing of such annuities remain a big challenge to the annuities' issuers due to the presence of embedded options. Many variable annuity pricing methodologies employ constant volatility assumption for the movement of the underlying asset, which is not the case in real market. In this study, we propose a new pricing model that can account for stochastic volatility, by incorporating the Heston model into our pricing framework. Heston model is one of the commonly used methods to model volatility. Using this new pricing framework, we evaluate the mortality and expenses (M&E) fee charged by the issuers for a group of male and female aged 50-69. The M&E fee evaluated based on the constant volatility pricing model will be compared to the results obtained using our framework. Mathematics Subject Classification: 91G80

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This study aims to propose an improved modelling framework for high frequency volatitliy in financial stock market. Extended heterogeneous autoregressive (HAR) and fractionally integrated autoregressive moving average (ARFIMA) models are introduced to model the S&P500 index using various realized volatility measures that are robust to jumps. These measures are the tripower variation volatility, and the realized volatities integrated with the nearest neighbor truncation (NNT) approach, namely the minimum and the median realized volatilities. In order to capture volatility clustering and the asymmetric property of various realized volatilities, the HAR and ARFIMA models are extended with asymmetric GARCH threshold specification. In addition, the asymmetric innovations of various realized volatilities are characterized by a skewed student-t distribution. The empirical findings show that the extended model returns the best performance in the insample and out-of-sample forecast evaluations. The forecasted results are used in the determination of value-at-risk for S&P500 market.

This study explores the multipower variation integrated volatility estimates using high frequency data in financial stock market. The different combinations of multipower variation estimators are robust to drastic financial jumps and market microstructure noise. In order to examine the informationally market efficiency, we proposed a rolling window estimate procedures of Hurst parameter using the modified rescale-range approach. In order to test the robustness of the method, we have selected the S&P500 as the empirical data. The empirical study found that the long memory cascading volatility is fluctuating across the studied period and drastically trim down after the subprime mortgage crisis. This time-varying long memory analysis allow us to understand the informationally market efficiency before and after the subprime mortgage crisis in U.S.