System identification for complex financial system

The mam purpose of this thesis focuses on the investigation of major financial volatility models including the relevant mean model used in the context of volatility estimation, and the development of a systematic nonlinear identification methodology for these problems. Financial volatility is one of the key aspects in financial economics and volatility modelling involves both the mean process modelling, and the volatility process modelling. Although many volatility models have been derived to approximate the volatility process, linear mean models are almost always used and to the best of our knowledge there is no application of fitting the mean process using a nonlinear model with selected structure. Based on the fact that nonlinearity has been observed in many financial market return data sets, the Non linear AutoRegression Moving Average with eXogenous input (NARMAX) modelling methodology with the term selection algorithm Orthogonal Forward Regression (OFR) is proposed to approximate the nonlinear mean process during volatility modelling. However, the assumption of a constant variance is usually violated in financial market return data. A new Weighted OFR algorithm is therefore proposed to correct for the impact of heteroskedastic noise on the term selection of the nonlinear mean model based on the assumption that the variance process is modelled by a Generalized AutoRegressive Conditional Heteroskedastic (GARCH) model. Because the weights to use are unknown, an iterative refined procedure is developed to learn the weights and to simultaneously improve the parameter estimates of both the mean and the volatility models. New validation methods are proposed to validate the nonlinear selected mean model and the volatility model. During the validation, the assumptions associated with the mean model are tested using a correlation method and the assumptions of the volatility model are tested using a Brock-Dechert-Scheinkrnan (80S) independent and identically distributed (i.i.d.) testing method. The prediction performance of the mean and volatility models is evaluated using a hold out Cross Validation (CV)method. A departure in the prediction of the volatility for the linear mean model, when using nonlinear simulated data, is successfully identified by the new validation methods and the nonlinear selected mean model passes the test. Another application of the NARAMX model, in the very new field of modelling mortality rate, is introduced. A quadratic polynomial mortality rate model selected by the OFR algorithm is developed based on the LifeMetrics male deaths and exposures data for England & Wales from the Office of National Statistics. Comparing the long term prediction of the new model with the Cairns-Blake-Dowd (CSO) statistical mortality rate model indicates the better prediction performance of the quadratic polynomial models. A back-testing method is applied to indicate the robustness of the selected NARMAX type mortality rate models. The term selection, parameter estimation, validation methods and new identification procedures proposed in this thesis open a new gateway to apply the NARMAX modelling technique in the financial area, and for mortality rate modelling to provide a new empirical practice of the NARMAX modelling method.