New Directions in Fractal Modeling: Estimation, Filtering, and Applications

点击翻译按钮获取中文摘要

Fractals and multifractals are of major importance in mathematics and many areas of engineering and the natural sciences. The first multifractal measures were developed and applied in geology (de Wijs, 1951) and the modelling of turbulence (Kolmogorov, 1962; Mandelbrot, 1972, 1974). Subsequent applications include astronomy, genetics, hydrology, meteorology, medicine, network traffic modeling, and seismology.  In finance, my own research developed the first multifractal stochastic processes, based on time-deformed Brownian motions (Calvet, Fisher, and Mandelbrot, 1997). Subsequent research provides moment-based inference (Calvet and Fisher, 2002), exact filtering and maximum-likelihood estimation (Calvet and Fisher, 2001); particle filtering (Calvet, Fisher, and Thompson, 2006); extensions using equilibrium theory, including multifractal jump-diffusions (Calvet and Fisher, 2007, 2008); and applications to interest rates (Calvet, Fisher, and Wu, 2013) and option pricing (Calvet, Fisher, Fearnley, and Leippold, 2014). The proposed research advances multifractal tools and develops new applications. To date, the Markov-switching Multifractal (MSM), based on discrete-valued state variables, has been a key building block for multifractal applications in finance.  Recent particle filtering techniques permit consideration of a broader set of models with diffusive components. This should in particular prove advantageous in option pricing applications, which show good results relative to standard benchmarks.  Application and further development of these particle filtering methods will be a key area of research. The research program will also develop simpler computational methods by adaptating Mixed Data Sampling methods (“MIDAS”, Ghysels, Santa-Clara, and Valkanov, 2006) to multifractal processes. The research program will also seek to determine the causes of multifractality in financial data by investigating the transmission of information across stocks, where stocks are modeled as a network. Coauthors and I have shown evidence of slow information diffusion, best captured by a hyperbolic decay, in daily stock returns (Boguth, Carlson, Fisher, and Simutin, 2014). The proposed research will examine information transmission at much higher frequencies, using a relatively recent news database called RavenPack. News are sourced from professional blogs, newspapers and newswires time stamped to the milliseconds. To process this news database such as extracting news for a specific firm on multiple years requires parallel data processing because of the size of the data. This research will permit precise tracking of the transmission of news across stocks, identification of the relevant transmission network, identification of the link to volatility, and to fractal properties in asset returns.