Hidden Markov Models (HMMs) are popular tools to interpret, model and forecast financial data. In these models, the return dynamics on a financial asset evolve according to a non-observed variable, a Markov chain, which generally represents the volatility of the asset. This volatility is notoriously difficult to reproduce with statistical models as it is very persistent in time. HMMs allow the volatility to vary according to the states of a Markov chain. Historically, these models are estimated with a very small number of regimes (states), because the number of parameters to be estimated grows quickly with the number of regimes and the optimization becomes difficult. The objective of this thesis is to propose a general framework to construct HMMs with a richer state space and a higher level of volatility persistence. In the first part, this thesis studies a general class of high-dimensional HMMs, called factorial HMMs, and derives its theoretical properties. In these models, the volatility is linked to a high-dimensional Markov chain built by multiplying lower-dimensional Markov chains, called components. We discuss how previously proposed models based on two-dimensional components adhere to the factorial HMM framework. Furthermore, we propose a new process---the Multifractal Discrete Stochastic Volatility (MDSV) process---which generalizes existing factorial HMMs to dimensions larger than two. The particular parametrization of the MDSV model allows for enough flexibility to reproduce different decay rates of the autocorrelation function, akin to those observed on financial data. A framework is also proposed to model financial log-returns and realized variances, either separately or jointly. An empirical analysis on 31 financial indices reveals that the MDSV model outperforms the realized EGARCH model in terms of fitting and forecasting performance. Our MDSV model requires us to pre-specify the number of components and assumes that there is no uncertainty on that number. In the second part of the thesis, we propose the infinite Factorial Hidden Markov Volatility (iFHMV) model as part of a Bayesian framework to let the data drive the selection of the number of components and take into account the uncertainty related to the number of components in the fitting and forecasting procedure. We also develop an algorithm inspired by the Indian Buffet Process (IBP) to estimate the iFHMV model on financial log-returns. Empirical analyses on two financial indices and two stocks show that the iFHMV model outperforms popular benchmarks in terms of forecasting performance.

This master’s thesis is centered on the Hidden Markov Models, a family of models in which an unobserved Markov chain dictactes the behaviour of an observable stochastic process through which a noisy version of the latent chain is observed. These bivariate stochastic processes that can be seen as a natural generalization of mixture models have shown their ability to capture the varying dynamics of many time series and, more specifically in finance, to reproduce the stylized facts of financial returns. In particular, we are interested in discrete-time Markov chains with finite state spaces, with the objective of studying the contribution of their hierarchical formulations and the relaxation of the homogeneity hypothesis for the transition matrix to the quality of the fit and predictions, as well as the capacity to reproduce the stylized facts. We therefore present two hierarchical structures, the first allowing for new interpretations of the relationships between states of the chain, and the second allowing for a more parsimonious parameterization of the transition matrix. We also present three non-homogeneous models, two of which have transition probabilities dependent on observed explanatory variables, and the third in which the probabilities depend on another latent variable. We first analyze the goodness of fit and the predictive power of our models on the series of log returns of the S&P 500 and the exchange rate between canadian and american currencies (CADUSD). We also illustrate their capacity to reproduce the stylized facts, and present interpretations of the estimated parameters for the hierarchical and non-homogeneous models. In general, our results seem to confirm the contribution of hierarchical and non-homogeneous models to these measures of performance. In particular, these results seem to suggest that the incorporation of non-homogeneous dynamics to a hierarchical structure may allow for a more faithful reproduction of the stylized facts—even the slow decay of the autocorrelation functions of squared and absolute returns—and better predictive power, while still allowing for the interpretation of the estimated parameters.

The problem of estimating the parameters of stochastic volatility models by direct maximisation of the likelihood is addressed. To this end, we present an algorithm that numerically approximates the optimal filter from the methodology proposed by Kitagawa (1987) for solving the filtering problem in non-linear and/or non-Gaussian systems. We show that this algorithm corresponds to running the Hamilton filter (the Hamilton filter offers an optimal solution to the filtering problem for a hidden Markov model on a finite state space) on a discretization of the continuous latent variable. The proposed solution significantly improves the computation time and produces results at least as good as stateof-the-art approaches in the field. Subsequently, we present and demonstrate a generalization of this algorithm in the case of stochastic volatility models incorporating leverage and jumps. Several Monte Carlo and empirical studies are conducted to evaluate the quality of the approach for approximating the log-likelihood and estimating the parameters. We also present a comparison of this approach to the continuous particle filter approach.

The Markov-switching GARCH model is the foundation of this thesis. This model offers rich dynamics to model financial data by allowing for a GARCH structure with time-varying parameters. This flexibility is unfortunately undermined by a path dependence problem which has prevented maximum likelihood estimation of this model since its introduction, almost 20 years ago. The first half of this thesis provides a solution to this problem by developing two original estimation approaches allowing us to calculate the maximum likelihood estimator of the Markov-switching GARCH model. The first method is based on both the Monte Carlo expectation-maximization algorithm and importance sampling, while the second consists of a generalization of previously proposed approximations of the model, known as collapsing procedures. This generalization establishes a novel relationship in the econometric literature between particle filtering and collapsing procedures. The discovery of this relationship is important because it provides the missing link needed to justify the validity of the collapsing approach for estimating the Markov-switching GARCH model. The second half of this thesis is motivated by the events of the financial crisis of the late 2000s during which numerous institutional failures occurred because risk exposures were inappropriately measured. Using 78 different econometric models, including many generalizations of the Markov-switching GARCH model, it is shown that model risk plays an important role in the measurement and management of long-term investment risk in the context of variable annuities. Although the finance literature has devoted a lot of research into the development of advanced models for improving pricing and hedging performance, the approaches for measuring dynamic hedging effectiveness have evolved little. This thesis offers a methodological contribution in this area by proposing a statistical framework, based on regression analysis, for measuring the effectiveness of dynamic hedges for long-term investment guarantees.

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