In a cascade process¹, energy or an analogous quantity is transferred from large to small scales through an independent, self-similar, random factor called cascade variable. Cascade processes were first observed in Fully Developed Turbulence, but such a behaviour is quite ubiquitous in nature and has been proposed also in stock market series and other complex systems.

While the distribution of the cascade variable characterizes the global self-similarity properties of a system, it is possible to achieve a local dynamical description with its optimal wavelet². The optimal wavelet of the system forms the representation basis of an effective dynamics, based on the cascade, that can be used to forecast future events.

Wavelet transforms separate the details of a signal that are relevant at given scales and thus they are especially appropriate bases to express cascade processes. As wavelet transforms are invertible, then the cascade-based representation can completely reconstruct the signal. Given a signal s(t), the wavelet transform at scale r is defined as α_r(t) = ∫dt s(t-t) ψ(t/r), where ψ is certain fast-decaying waveform called wavelet.

In a cascade process, the wavelet transform values at two scales r, L with r<L are multiplicatively related by the cascade variable:

α_r = η_r/L α_L

where η_r/L and α_L are mutually independent. The symbol ≗ means that both sides equal in distribution, a distribution that is the same with almost any wavelet.

With the optimal wavelet, the previous cascade relation holds not only in distribution but also geometrically, i.e., at each point t of the signal. This allows to infer the wavelet transform values at small scales from the larger ones. Then, inverting the wavelet transform not only reconstructs the signal but can even infer future points.

We will analyse Spanish IBEX-35 stock market series and show that the cascade process is observed in both the logarithm of the price and the volatility with related optimal wavelets. The cascade process models the distribution of future returns and tells how the value-at-risk evolves over time. In addition, we will forecast future volatilities with different wavelets and check that the optimal wavelet attains the highest accuracy of forecasting; this accuracy improves on average that of a neural seasonal ARIMA.

¹Benzi et al., Physica D 65 (1993)

²Turiel and Parga, PRL 85 (2000)

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