Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Rogemar Mamon

Abstract

We consider a higher-order hidden Markov models (HMM), also called weak HMM (WHMM), to capture the regime-switching and memory properties of financial time series. A technique of transforming a WHMM into a regular HMM is employed, which in turn enables the development of recursive filters. With the use of the change of reference probability measure methodology and EM algorithm, a dynamic estimation of model parameters is obtained. Several applications and extensions were investigated. WHMM is adopted in describing the evolution of asset prices and its performance is examined through a forecasting analysis. This is extended to the case when the drift and volatility components of the logreturns are modulated by two independent WHMMs that are not necessarily having the same number of states. Numerical experiment is conducted based on simulated data to demonstrate the ability of our estimation approach in recovering the “true” model parameters. The analogue of recursive filtering and parameter estimation to handle multivariate data is also established. Some aspects of statistical inference arising from model implementation such as the assessment of model adequacy and goodness of fit are examined and addressed. The usefulness of the WHMM framework is tested on an asset allocation problem whereby investors determine the optimal investment strategy for the next time step through the results of the algorithm procedure. As an application in the modelling of yield curves, it is shown that the WHMM, with its memory-capturing mechanism, outperforms the usual HMM. A mean-reverting interest rate model is further developed whereby its parameters are modulated by a WHMM along with the formulation of a self-tuning parameter estimation. Finally, we propose an inverse Stieltjes moment approach to solve the inverse problem of calibration inherent in an HMM-based regime-switching set-up.


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