8 - Power Variation

Abstract

The multifractal diffusions imply that the moments of returns vary as a power law of the time horizon. This chapter confirms the empirical validity of this property on currency and equity data. Scaling behavior in an equity index and five major U.S. stocks are demonstrated. For Deutsche mark/U.S. dollar (DM/USD) exchange rates, a high-frequency data set of approximately 1.5 million quotes collected over one year, and a 24-year sample of daily prices is used. The exchange rate displays multifractal moment-scaling over a remarkable range of time horizons. The spectrum of local Holder exponents is estimated, and a Markov switching multifractal (MSM) generating mechanism is inferred. An estimate of the multifractal spectrum by a Legendre transform of the moments’ growth rates is obtained. The implied process is simulated and it is confirmed that the multifractal model replicates the moment behavior found in the data. Monte Carlo simulations show that Generalized Autoregressive Conditional Heteroskedasticity (GARCH) and fractionally integrated GARCH model (FIGARCH) are less likely to reproduce these results than a multifractal model. In addition, evidence of scaling in a U.S. equity index and five individual stocks is provided. Results are updated to reflect the strictly stationary MSM specification. Complementary studies confirm that multiscaling is a common feature of many financial time series. Multifractal models such as MSM provide a parsimonious representation of these empirical features of financial data.