Scaling and Multiscaling in Financial Indexes: a Simple Model

We propose a simple stochastic model for time series which is analytically tractable, easy to simulate and which captures some relevant stylized facts of financial indexes, including scaling properties. We show that the model fits the Dow Jones Industrial Average time series in the period 1935-2009 with a remarkable accuracy. Despite its simplicity, the model has several interesting features. The volatility is not constant and displays high peaks. The empirical distribution of the log-returns (increments of the logarithm of the index) is non-Gaussian and may exhibit heavy tails. Log-returns corresponding to disjoint time intervals are uncorrelated but not independent: the correlation of their absolute values decays exponentially fast in the distance between the time intervals for large distances, while it has a slower decay for moderate distances. Finally, the distribution of the log-returns obeys scaling relations that are detected on real time series, but are not satisfied by most available models.