HMM based scenario generation for an investment optimisation problem

This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2012 Springer-Verlag.The Geometric Brownian motion (GBM) is a standard method for modelling financial time series. An important criticism of this method is that the parameters of the GBM are assumed to be constants; due to this fact, important features of the time series, like extreme behaviour or volatility clustering cannot be captured. We propose an approach by which the parameters of the GBM are able to switch between regimes, more precisely they are governed by a hidden Markov chain. Thus, we model the financial time series via a hidden Markov model (HMM) with a GBM in each state. Using this approach, we generate scenarios for a financial portfolio optimisation problem in which the portfolio CVaR is minimised. Numerical results are presented.This study was funded by NET ACE at OptiRisk Systems

Bayesian Nonparametric Hidden Semi-Markov Models

There is much interest in the Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM) as a natural Bayesian nonparametric extension of the ubiquitous Hidden Markov Model for learning from sequential and time-series data. However, in many settings the HDP-HMM's strict Markovian constraints are undesirable, particularly if we wish to learn or encode non-geometric state durations. We can extend the HDP-HMM to capture such structure by drawing upon explicit-duration semi-Markovianity, which has been developed mainly in the parametric frequentist setting, to allow construction of highly interpretable models that admit natural prior information on state durations. In this paper we introduce the explicit-duration Hierarchical Dirichlet Process Hidden semi-Markov Model (HDP-HSMM) and develop sampling algorithms for efficient posterior inference. The methods we introduce also provide new methods for sampling inference in the finite Bayesian HSMM. Our modular Gibbs sampling methods can be embedded in samplers for larger hierarchical Bayesian models, adding semi-Markov chain modeling as another tool in the Bayesian inference toolbox. We demonstrate the utility of the HDP-HSMM and our inference methods on both synthetic and real experiments

Bayesian Nonparametric Inference of Switching Linear Dynamical Systems

Many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our Bayesian nonparametric approach utilizes a hierarchical Dirichlet process prior to learn an unknown number of persistent, smooth dynamical modes. We additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn SLDS with varying state dimension or switching VAR processes with varying autoregressive order. We develop a sampling algorithm that combines a truncated approximation to the Dirichlet process with efficient joint sampling of the mode and state sequences. The utility and flexibility of our model are demonstrated on synthetic data, sequences of dancing honey bees, the IBOVESPA stock index, and a maneuvering target tracking application.Comment: 50 pages, 7 figure

Simultaneous Learning of Nonlinear Manifold and Dynamical Models for High-dimensional Time Series

The goal of this work is to learn a parsimonious and informative representation for high-dimensional time series. Conceptually, this comprises two distinct yet tightly coupled tasks: learning a low-dimensional manifold and modeling the dynamical process. These two tasks have a complementary relationship as the temporal constraints provide valuable neighborhood information for dimensionality reduction and conversely, the low-dimensional space allows dynamics to be learnt efficiently. Solving these two tasks simultaneously allows important information to be exchanged mutually. If nonlinear models are required to capture the rich complexity of time series, then the learning problem becomes harder as the nonlinearities in both tasks are coupled. The proposed solution approximates the nonlinear manifold and dynamics using piecewise linear models. The interactions among the linear models are captured in a graphical model. By exploiting the model structure, efficient inference and learning algorithms are obtained without oversimplifying the model of the underlying dynamical process. Evaluation of the proposed framework with competing approaches is conducted in three sets of experiments: dimensionality reduction and reconstruction using synthetic time series, video synthesis using a dynamic texture database, and human motion synthesis, classification and tracking on a benchmark data set. In all experiments, the proposed approach provides superior performance.National Science Foundation (IIS 0308213, IIS 0329009, CNS 0202067

Application of Hidden Markov Models and Hidden Semi-Markov Models to Financial Time Series

Hidden Markov Models (HMMs) and Hidden Semi-Markov Models (HSMMs) provide flexible, general-purpose models for univariate and multivariate time series. Although interest in HMMs and HSMMs has continuously increased during the past years, and numerous articles on theoretical and practical aspects have been published, several gaps remain. This thesis addresses some of them, divided into three main topics. 1. Computational issues in parameter estimation of stationary HMMs. The parameters of a HMM can be estimated by direct numerical maximization (DNM) of the log-likelihood function or, more popularly, using the Expectation-Maximization (EM) algorithm. We show how the EM algorithm could be modified to fit stationary HMMs. We propose a hybrid algorithm that is designed to combine the advantageous features of the EM and DNM algorithms, and compare the performance of the three algorithms (EM, DNM and the hybrid). We then describe the results of an experiment to assess the true coverage probability of bootstrap-based confidence intervals for the parameters. 2. A Markov switching approach to model time-varying Beta risk of pan-European Industry portfolios. The motive to take up this topic was the development of a joint model for many financial time series. We study two Markov switching models in a Capital Asset Pricing Model framework, and compare their forecast performances to three models, namely a bivariate t-GARCH(1,1) model, two Kalman filter based approaches and a bivariate stochastic volatility model. 3. Stylized facts of financial time series and HSMMs. The ability of a HMM to reproduce several stylized facts of daily return series was illustrated by Ryden et al. (1998). However, they point out that one stylized fact cannot be reproduced by a HMM, namely the slowly decaying autocorrelation function of squared returns. We present two HSMM-based approaches to model eighteen series of daily sector returns with about 5.000 observations. The key result is that, compared to a HMM, the slowly decaying autocorrelation function is significantly better described by a HSMM with negative binomial sojourn time and Normal conditional distributions.

Growth and volatility regime switching models for New Zealand GDP data

This paper fits hidden Markov switching models to New Zealand GDP data. A primary objective is to better understand the utility of these methods for modelling growth and volatility regimes present in the New Zealand data and their interaction. Properties of the models are developed together with a description of the estimation methods, including use of the Expectation Maximisation (EM) algorithm. The models are fitted to New Zealand GDP and production sector growth rates to analyse changes in their mean and volatility over time. The paper discusses applications of the methodology to identifying changes in growth performances, and examines the timing of growth and volatility regime switching between production sectors. Conclusions to emerge are that, in contrast to the 1980s, New Zealand GDP growth experienced an unusually long period of time in high growth and low volatility regimes during the 1990s. The paper evaluates sector contributions to this 1990s experience and discusses directions for further development.Hidden Markov models; regime switching; growth; business cycles; volatility; production sectors; GDP.

Detecting Turning Points with Many Predictors through Hidden Markov Models

This paper explores the American business cycle with the Hidden Markov Model (HMM) as a monitoring tool using monthly data. It exhibits ten US time series which offer reliable information to detect recessions in real time. It also proposes and assesses the performances of different and complementary “recession models” based on Markovian processes, discusses the most efficient and easiest way of encompassing information through these models and draws three main conclusions: simple HMM are decisive to monitor the business cycle and some series are proved highly reliable; more sophisticated models such as the Dynamic Factor with Markov Switching (DFMS) model or Stock and Watson’s Experimental Recession Index seem not to be more powerful than simple (univariate or pseudo-multivariate) Hidden Markov Models, which remain far more parsimonious; combining information in temporal space seems to work marginally better than in probability space for high frequency data. We conclude about leading and “real time detection” properties related to HMM and give some hints for further research.Business Cycle, Markov Switching, Dynamic Factor, Coincident Indicators