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.