Analysis of market returns using multifractal time series and agent-based simulation

Abstract

Many types of financial time series, most notably market returns, have been found to exhibit long-range memory as well as dramatic day-to-day swings that cannot be adequately represented by light-tailed distributions such as the normal distribution. In particular, this means that for such time series, the usual variance parameter (i.e., the sum of covariances at all time lags) is not defined because the covariance function does not converge to zero fast enough as the time lag increases. Moreover in such time series, often the tails of the marginal density converge to zero so slowly that higher-order marginal moments such as skewness and kurtosis fail to exist. Therefore conventional methods for analyzing simulation-generated time series cannot generally be applied to high-fidelity simulations of financial markets. Building on earlier work in fractal geometry and fractal time series, Mandelbrot et al. proposed the multifractal model of asset returns (MMAR) as an alternative to the ARCH models for analyzing time series exhibiting volatility clustering, long-range dependence, and heavy-tailed returns (Mandelbrot et al. 1997). They defined the multifractal spectrum as the renormalized probability density function of the Hölder exponents observed in the time series; and they used the multifractal spectrum to measure the ability of MMAR to match the statistical properties of real data. In 2002 Kantelhardt et al. formulated multifractal