Essays on Economic Sentiment Dynamics and Asymmetric Multifractal Models of Financial Volatility

In the first chapter of the dissertation an estimation of the continuous-time Markov chain (CTMC) model of experts' sentiment  index is considered in the case of incomplete data. Particularly, three estimation approaches based on a discrete-time sample are presented: the EM algorithm and two versions of the maximum likelihood estimation method. The first approach for the estimation of the considered model is iterative and leads to massive recursive computations of matrices. The most crucial part of the second and third approaches is the numerical computation of the matrix exponential of the intensity matrix. In particular, the second approach is based on the eigendecomposition of the intensity matrix and the corresponding well-known property of matrix exponential for such decomposition. In order to increase the effectiveness of the method in the third approach the fact that the intensity matrix has a lower Hessenberg form is used. All three approaches are based on numerical optimization using the nonlinear conjugate optimizer.

The second chapter is dedicated to the development of the methods of calibration and estimation of the model belonging to the asset price class of models. Two variants of the generalization of the Markov Switching Multifractal (MSM) model,  called the Asymmetric Markov-Switching Multifrequency, are considered.  The modifications are aimed to reproduce such a phenomenon of asset returns as leverage effect. Other features of the model, namely the long memory stylized fact for different frequencies and degrees of persistence, the mean reversion of volatility, and the volatility clustering, are investigated and proven. The option pricing theory based on risk-neutral measure is developed for this model. In-sample and out-of-sample performance are tested.


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