Higher Dimensional Multi-fractal Processes: Filtering via Simulation

One important contribution has been made by Benoit Mandelbrot, the famous father of fractals who proposed a multi-fractal model of asset returns (MMAR), a theory which inherits all the hallmarks of Mandelbrot’s earlier work that has emerged since the 1970s. As a new formalization of stochastic models for the volatility dynamics of asset prices, it preserves the hierarchical multiplicative structure of volatility components with different characteristic time scales, which are in harmony with the stylized facts of financial markets. In this paper, we concentrate on developing parsimonious multivariate multi-fractal processes, and implement their estimation via maximum likelihood approaches. We then apply our multivariate multi-fractal processes to empirically relevant topics, such as portfolio risk measurement and management. Keyword: Multi-fractal, multivariate, likelihood inference, particle filter, Value-at-Risk, expected shortfall. JEL Classification: C11, C13, G15