Multifractal Models, Intertrade Durations and Return Volatility

This thesis covers the application of multifractal processes in modeling financial time series. It aims to demonstrate the capacity and the robustness of the multifractal processes to better model return volatility and ultra high frequency financial data than both the generalized autoregressive conditional heteroscedasticity (GARCH)-type and autoregressive conditional duration (ACD) models currently used in research and practice. The thesis is comprised of four main parts that particularize the different procedures and the main findings. In the first part of the thesis we first delineate the genesis of multifractal (MF) measures and processes and how one can construct a simple MF measure. We outline the generic properties of the MF processes, mention how they motivate financial time series models, and present the different tools developed for the estimation of the MF models and the forecasting of return volatilities and some empirical results. Second, we give a short overview of both autoregressive conditional duration (ACD) models and Markov switching multifractal duration (MSMD) models. We start with some theoretical microstructure literature that motivate both models. We present ACD and MSMD models and their subsequent extensions. Finally, we cite the different diagnostic tests developed in the literature for assessing their adequacy and provide some prominent empirical studies. The second part deals with the application the Markov-switching multifractal (MSM) model and generalized autoregressive conditional heteroscedasticity (GARCH) type models in forecasting crude oil price volatility. Based on six different loss functions and by means of the superior predictive ability (SPA) test of Hansen (2005) we evaluate and compare their forecasting performance at short- and long-horizons. The results give evidence that none of our volatility models can outperform other models across all six different loss functions. However, the long memory GARCH-type models and the MSM model seem to be more appropriate in terms of fitting and forecasting oil price volatility. We also found that forecast combinations of long memory GARCH-type models and the MSM lead to an improvement in forecasting crude oil price volatility. The third and longest part of the thesis compares the predictive ability of the Markov switching multifractal duration (MSMD) model recently introduced by Chen et al. (2013) to those of the standard ACD (cf. Engle and Russell, 1998), Log-ACD (cf. Bauwens and Giot, 2000), and fractionally integrated ACD (FIACD) (cf. Jasiak, 1998) models. We assume that innovations in the ACD and Log-ACD models follow Weibull, Burr, generalized gamma and Lognormal distributions. For FIACD we only consider the case where the innovation is standard exponentially distributed. We assess the forecasting performance of the models using density forecasts evaluation methodologies proposed by Diebold et al. (1998) and the likelihood ratio test of Berkowitz (2001). We complement these methodologies with Kolmogorov-Smirnov and Anderson-Darling distances (cf. Rachev and Mittnik, 2000). Empirically, results are quite nice and speak for the MSMD model. In fact, the MSMD model can better capture the long memory and the fat tails observed in trade and price duration data, and therefore, outperforms both the FIACD, ACD and Log-ACD models. We also found that certain distributional assumptions for the innovations strongly enhance the forecasting performance of the ACD and Log-ACD models. In line with the last result, we want to know to what extent different distributional assumptions for the innovation in the MSMD model may influence the model’s forecasting performance. So, we assume that the innovation in the MSMD model follows generalized gamma or Burr distribution. To compare and select the model that provides better fit to the empirical data (trade, price and volume durations) we make use of the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the likelihood ratio test. Surprisingly, both distributional assumptions for the innovation do not much affect the predictive ability of the model. It seems that the ability of the MSMD model to fit financial duration data largely stems from the multifractal processes. Third, we generalize the univariate MSMD model to a bivariate one. The bivariate MSMD model is substantially an adaptation of the bivariate Markov switching multifractal (MSM) process proposed by Calvet et al. (2006) to high frequency financial data. We apply the bivariate MSMD model to analyze the co-movement between the bid-ask spreads of different stocks. The results indicate that bid-ask spreads of sector-specific or cross-sector stocks may be simultaneously affected by arrival of information in the market. Fourth, we apply the standard MSMD and the generalized gamma ACD (GGACD) models to forecast irregularly spaced intra-day value-at-risk (ISIVaR) in a semi-parametric framework. We assess the performance of both models to produce accurate irregularly spaced intra-day VaR via the generalized moments method (GMM) duration-based test developed by Candelon et al. (2011). The results show that the MSMD model outperforms the GGACD model and can be used in practice to manage market risk. The last part summarizes the main findings of the thesis and presents some outlooks for future research.


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