Fractals

Laurent E. Calvet (1)

1 GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC

Fractals have become increasingly useful tools for the statistical modelling of financial prices. While early research assumed invariance of the return density with the time horizon, new processes have recently been developed to capture nonlinear changes in return dynamics across frequencies. The Markov-switching multifractal (MSM) is a parsimonious stochastic volatility model containing arbitrarily many shocks of heterogeneous durations. MSM captures the outliers, volatility persistence and power variation of financial series, while permitting maximum likelihood estimation and analytical multi-step forecasting. MSM compares favourably with standard volatility models such as GARCH(1,1) both in- and out-of-sample.

Keywords: Bayesian filtering; Brownian motion; continuous time valuation; fractals; fractional Brownian motion; Lévy-stable processes; long memory models; Mandelbrot; B.; Markov-switching multifractal (MSM); maximum likelihood; multifractal model of asset returns (MMAR); multifractality; regime switching; self-similarity processes

Laurent E. Calvet. Fractals. Steven N. Durlauf and Lawrence E. Blume Eds. The New Palgrave Dictionary of Economics (2nd edition), Palgrave, pp.NC, 2008, ⟨10.1057/9780230226203.0598⟩. ⟨hal-00671878⟩