Transition from lognormal to χ2χ2-superstatistics for financial time series

• For the first time a transition from one superstatistics to another is described, as a function of the time scale considered.
• Relevant example system is financial time series of share price returns on various time scales, good quantitative agreement with data.
• New model interpolating between lognormal and chi-square superstatistics is introduced.

Share price returns on different time scales can be well modelled by a superstatistical dynamics. Here we provide an investigation which type of superstatistics is most suitable to properly describe share price dynamics on various time scales. It is shown that while χ2χ2-superstatistics works well on a time scale of days, on a much smaller time scale of minutes the price changes are better described by lognormal superstatistics. The system dynamics thus exhibits a transition from lognormal to χ2χ2 superstatistics as a function of time scale. We discuss a more general model interpolating between both statistics which fits the observed data very well. We also present results on correlation functions of the extracted superstatistical volatility parameter, which exhibits exponential decay for returns on large time scales, whereas for returns on small time scales there are long-range correlations and power-law decay.