6 - Multifractal Diffusions Through Time Deformation and the MMAR

Abstract

Time deformation provides a powerful tool to construct multifractal processes out of general multifractal measures. The first example of this technique in the literature is the Multifractal Model of Asset Returns (MMAR), which incorporates the outliers and volatility persistence exhibited by many financial time series, as well as a rich pattern of local variations and moment-scaling properties. This chapter illustrates that multifractal diffusions can be created by compounding a standard Brownian motion. The resulting price process is a semimartingale with a finite variance, which precludes investors from making arbitrage profits. The time deformation implies that the moments of returns scale is a power function of the frequency of observation. The time deformation approach is used to define the first multifractal diffusion with uncorrelated increments, the MMAR, which is also reviewed. In the MMAR, the multifractal time deformation is the cumulative distribution function of a random multiplicative cascade. The construction produces the moment-scaling, thick tails, and long-memory volatility persistence exhibited by many financial time series. It substantially improves on traditional fractal specifications and accommodates flexible tail behaviors with the highest finite moment taking any value greater than two. The model captures the nonlinear changes in the unconditional distribution of returns at various sampling frequencies, while retaining the parsimony and tractability of fractal approaches. The MMAR provides a fundamentally new class of stochastic processes for financial applications.