Empirical scaling laws and the aggregation of non-stationary data

Widely cited evidence for scaling (self-similarity) of the returns of stocks and other securities is inconsistent with virtually all currently-used models for price movements. In particular, state-of-the-art models provide for ubiquitous, irregular, and oftentimes high-frequency fluctuations in volatility (“stochastic volatility”), both intraday and across the days, weeks, and years over which data is aggregated in demonstrations of self-similarity of returns. Stochastic volatility renders these models, which are based on variants and generalizations of random walks, incompatible with self-similarity. We show here that empirical evidence for self-similarity does not actually contradict the analytic lack of self-similarity in these models. The resolution of the mismatch between models and data can be traced to a statistical consequence of aggregating large amounts of non-stationary data. •Stable process is described to accommodate the observations of heavy-tailed returns.•Non-stationarity is formulated into the return process through stochastic volatility models.•Random-walk models for the return sequence are derived from the return process.•The incompatibility between the random-walk model and scaling laws is pointed out.•An aggregation theorem is proposed to offer a resolution of the mismatch problem.