Self-similarity of price fluctuations and market dynamics

Self-similarity of volatilities for different time-scales is present in speculative markets. It is a phenomenon in strongly correlated regime ranging from minutes to months (volatility clustering). By directly evaluating Cramer function for volatilities, we show an invariant scaling relation with one parameter in a real data of high-frequency stock prices (NYSE TAQ database). This approach is shown to have a close relation to the multifractal approach and the scaling-of-moments method. Our result clearly shows discrepancy from "log-normal" approximation of Cramer function in the large deviation region, in which the function's behavior is compatible with the so-called "cubic" law in the PDF of price fluctuations. Those statistical properties of long-memory and self-similarity in volatility and spectral whiteness of fluctuation imply that the market has on-off intermittency dynamics.