The Scale-Invariant Brownian Motion Equation and the Lognormal Cascade in the Stock Market
Stephen H.T. Lihn
affiliation not provided to SSRN
June 24, 2008
A continuous-time scale-invariant Brownian motion (SIBM) stochastic equation is developed to investigate the dynamics of the stock market. The equation is used to solve the fat tail distribution of the stock universe and the DJIA time series. It is also used to model the volatility clustering in the DJIA time series. The equation is transformed from the Langevin equation into a fractal expression involving an infinite array of random walk. It predicts an elegant way of generating the skew form of the lognormal cascade distribution (Kolmogorov and Mandelbrot), which describes the static log-return distribution in the financial market as well as the velocity distribution in Largrangian turbulence. The higher order randomness (HORN) hypothesis is introduced as the stochastic source of the cascade distribution. A leakage term from HORN is introduced to model the covariance between large volatility and large negative return. A volatility model based on two SIBM processes is built to model the volatility autocorrelation. The volatility half-times of 20 days and 300 days are extracted from the DJIA data. The model generates the static log-return distributions from 10 days to 320 days that match the DJIA data satisfactorily. It also predicts an alternative interpretation of the volatility smile/skew observed in the options market. The relation between the SIBM model and the multifractal random walk model is examined, which yields a simplified SIBM model that could be quite useful in finance.
Number of Pages in PDF File: 49
Keywords: lognormal cascade, fat tail, volatility clustering, autocorrelation, multifractal
JEL Classification: G11, G12working papers series
Date posted: June 25, 2008
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