Multifraktale Modelle von Finanzrenditen: Multivariate Erweiterungen, empirische Schätzung und Anwendung im Risikomanagement
Zusammenfassung der Projektergebnisse
Multifractal processes have been originally developed in physics for modeling turbulent fluids and related phenomena. They have recently also attracted attention in empirical finance because of their ability to replicate the major stylized facts of asset returns: varying degrees of long-term correlation of different measures of volatility, and fat tails in the unconditional distribution of price changes. While the multifractal apparatus developed in statistical physics is mainly concerned with combinatorial operations on measures, analogous models for causal multifractal models have been designed for financial applications. Although this literature is still at a very early stage, it has already developed a range of statistical techniques for proper estimation of multifractal models and has demonstrated successful applications in forecasting of volatility. One major task of the first phase of the project was the development of appropriate statistical methodology for multivariate multifractal models. We have investigated the behaviour of various extensions of the baseline multifractal model and have explored the use of multifractal models for risk management. Further research included the analysis of the role of innovations vis-à-vis the intrinsic volatility dynamics as well as the adaptation of the multifractal model to measures of realized volatility. The second phase of the project has extended the earlier work on inference methods for multifractal models and their practical applications by our group and the findings of the first phase of the project. We have addressed four main issues. The first one has consisted in the extension of the methods of statistical inference from time-series models to multiplicative Lognormal cascades, commonly employed in the physics literature. The second issue has been the development of methods for parameter estimation and forecasting of multivariate MSM models together with multivariate best linear forecasts developed in the first phase of the project and its extension to higher dimensions. In addition, we have started to work on the estimation and application of a continuous-time version of the MSM. Lastly, we have pursued new avenues for extracting the parameters of multifractal models from option prices.
