Random cascade model in the limit of infinite integral scale as the exponential of a non-stationary $1/f$ noise. Application to volatility fluctuations in stock markets

TL;DR

This paper introduces a non-stationary Gaussian process model for financial volatility that mimics cascade models and explains the empirical correlation between integral scale and sample length in stock market data.

Contribution

The paper presents a novel non-stationary Gaussian process model that captures volatility fluctuations and explains the empirical relationship between integral scale and sample length.

Findings

Model reproduces stylized facts of financial time series

Quantitatively matches empirical integral scale correlations

Applicable to daily stock index data

Abstract

In this paper we propose a new model for volatility fluctuations in financial time series. This model relies on a non-stationary gaussian process that exhibits aging behavior. It turns out that its properties, over any finite time interval, are very close to continuous cascade models. These latter models are indeed well known to reproduce faithfully the main stylized facts of financial time series. However, it involve a large scale parameter (the so-called "integral scale" where the cascade is initiated) that is hard to interpret in finance. Moreover the empirical value of the integral scale is in general deeply correlated to the overall length of the sample. This feature is precisely predicted by our model that turns out, as illustrated on various examples from daily stock index data, to quantitatively reproduce the empirical observations.