Some contributions to filtering theory with applications in financial modelling

Abstract

Two main groups of filtering algorithms are characterised and developed. Their applicability is demonstrated using actuarial and financial time series data. The first group of algorithms involved hidden Markov models (HMM), where the parameters of an asset price model switch between regimes in accordance with the dynamics of a Markov chain. We start with the known HMM filtering set-up and extend the framework to the case where the drift and volatility have independent probabilistic behaviour. In addition, a non-normal noise term is considered and recursive formulae in the online re-estimation of model parameters are derived for the case of students’ t-distributed noise. Change of reference probability is employed in the construction of the filters. Both extensions are then tested on financial and actuarial data. The second group of filtering algorithms deals with sigma point filtering techniques. We propose a method to generate sigma points from symmetric multivariate distributions. The algorithm matches the first three moments exactly and the fourth moment approximately; this minimises the worst case mismatch using a semidefinite programming approach. The sigma point generation procedure is in turn applied to construct algorithms in the latent state estimation of nonlinear time series models; a numerical demonstration of the procedure’s effectiveness is given. Finally, we propose a partially linearised sigma point filter, which is an alternative technique for the optimal state estimation of a wide class of nonlinear time series models. In particular, sigma points are employed for generating samples of possible state values and then a linear programming-based procedure is utilised in the update step of the state simulation. The performance of the filtering technique is then assessed on simulated, highly non-linear multivariate interest rate process and is shown to perform significantly better than the extended Kalman filter in terms of computational time.