Modelling asset prices with a stochastic differential equation driven by a normal inverse gaussian process

Abstract

This thesis aims to study a new Levy-driven diffusion process, where the random innovations that underlie the realised values are governed by a normal inverse Gaussian (NIG) process. A diffusion process driven by Brownian motion with compound Poisson jumps is often used to capture random jumps in an asset's price. Nonetheless, the resulting jump diffusion often involves many parameters and thus lacks parsimony. This motivates us to investigate a new diffusion process, one that is driven by a more volatile stochastic process than the conventional Brownian motion process. Consequently, the resulting diffusion model has the capacity to capture jumps in an asset's price process without incorporating an additional jump process. In order to position this new Levy-driven diffusion in the existing literature, we present a selected review of several related topics including diffusion processes, Levy processes, several Levy-driven diffusions, and some commonly used methods for modelling exchange rates. We present a comprehensive study of the distributional properties of the NIG distribution to help better understand this heavy-tailed distribution, including the role of each parameter. We discuss the finite-sample performance of maximum likelihood estimation (MLE) of parameters based on random samples, which are simulated from the NIG distribution with pre-specified parameters. We have proved that under some mild conditions, the variance of the new NIG-driven diffusion process is finite. Moreover, we propose to discretise the NIG-driven diffusion process and estimate parameters using MLE. We conduct Monte Carlo simulation studies and find that MLE performs very well in finite samples. Finally, we fit the discretised NIG-driven diffusion model to a sample of the Australian dollar prices, in terms of US dollar, observed at the one-minute frequency level. Results are meaningful in the sense that the new diffusion model is clearly favoured against a competing model under the Akaike information criterion. Therefore, the thesis concludes that MLE can be used to estimate the parameters of the discretised NIG-driven diffusion model, and that such a model is practically useful in modelling high-frequency currency prices. It is anticipated that the NIG-driven diffusion may also be useful for the econometric modelling of other high frequency asset price data.