New Advances in Dynamic Risk Models

The central theme of the entire thesis is to explore new ways of modelling the time-varying conditional distributions of financial asset returns. Connected by this central theme, the thesis is separated into three main parts. The first part is on modelling the time-varying variances of financial returns, where the idea of flexibly modelling the news impact curve in a GARCH model is extended to build a more general functional coefficient semiparametric volatility model. It is shown that most existing GARCH models can be written as special cases of the new functional coefficient model. The coefficient function is approximated by a regression spline. An adaptive MCMC algorithm is developed to simulate from the joint posterior of knot configurations and the spline coefficients. The second part is on modelling insurance loss using flexible Tukey family of quantile distributions, such as the g-and-h and g-and-k. The key contribution here is to propose a new estimator for the parameters of the Tukey family of distributions using the idea of L-moments. The estimator is shown to be more statistically efficient and requires less computing time compared to previously proposed methods. This second part serves as an important prerequisite for last part of the thesis; two important concepts introduced in this part are fundamental to the ideas developed in the final part, namely, the g-and-h quantile function and the method of L-moments. In the final part of the thesis, a functional time series model is developed and applied to model the quantile functions of high frequency financial returns; the model can also be viewed as a time-varying model for symbolic data, where each symbolic observation is a quantile function. A key advantage (and contribution) of this symbolic model is that the likelihood function of high frequency returns is allowed to be efficiently constructed via the g-and-h distribution and L-moments. This further allows a forecast model to be built for the entire quantile function, e.g., for the one-minute returns of an asset in one trading day. An efficient adaptive MCMC algorithm is developed for parameter estimation. A novel mixture distribution for modelling positive random variables is also proposed, which is called the Apatosaurus distribution