[Paper Review] Scaling transformation and probability distributions for financial time series

Brachet, Marc-Etienne, Taflin, Erik+1 more|arXiv (Cornell University)|May 12, 1999

Complex Systems and Time Series Analysis22 citations

TL;DR

This paper introduces a nonlinear group-theoretical framework to analyze scaling transformations in financial time series, demonstrating that price increment distributions exhibit multifractal behavior through spectral concavity. Using foreign exchange (DM/$) and CAC 40 index data, it derives a family of probability distributions for price increments across time scales via moment-based inversion, showing strong agreement with empirical data over wide ranges of time horizons and price changes.

ABSTRACT

The price of financial assets are, since Bachelier, considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works. In this letter we investigate the question of scaling transformation of price processes by establishing a new connexion between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.

Motivation & Objective

• To investigate scaling transformations in financial time series using nonlinear group representations.
• To establish a connection between group theoretical methods and multifractal analysis in financial data.
• To derive explicit probability distributions for price increments across different time scales.
• To test whether financial data exhibit multifractal scaling behavior through spectral concavity of moment scaling exponents.
• To validate the model by comparing the derived distributions with empirical data from FX and equity indices.

Proposed method

• Uses a scaling group action T on the set of price increment distributions, defined via time dilation τ → aτ.
• Transforms probability distributions into moment vectors Sr(τ) = ∫x^r q(x)dx for r ∈ ℝ⁺.
• Applies inverse moment transformation (S⁻¹) to reconstruct probability densities from moments using formulas from [8][9][16].
• Employs a linear approximation U_a^(1)(m) = (a^{ζ_r} m_r)_r for small scaling parameters a.
• Analyzes the spectrum {ζ_r} via linear regression of ln s_r(τ) vs. ln τ to extract scaling exponents.
• Validates the model by comparing reconstructed distributions (M.A.M.) with empirical densities across τ = 8 and 512 minutes.

Experimental results

Research questions

• RQ1 Does the scaling transformation of financial price increments admit a nonlinear group representation that can be approximated linearly for small scales?
• RQ2 Is the spectrum of scaling exponents ζ_r concave, indicating multifractal behavior in financial time series?
• RQ3 Can a family of probability distributions for price increments be explicitly constructed from empirical moments across time scales?
• RQ4 How well does the moment-based inverse method reproduce empirical price increment distributions?
• RQ5 Do different financial instruments (e.g., FX vs. equity indices) exhibit similar scaling and multifractal properties?

Key findings

• For FX DM/$, the linear scaling approximation holds for 11 ≤ τ ≤ 2896 minutes and 1 ≤ r ≤ 10, with ζ_r extracted as slopes of ln s_r(τ) vs. ln τ.
• For CAC 40, the linear approximation holds over a broader range: 1 ≤ τ ≤ 2048 minutes and 1 ≤ r ≤ 10.
• The spectrum ζ_r is non-trivially concave in both datasets, confirming multifractal behavior by definition [7].
• The reconstructed probability distributions (M.A.M.) match empirical data well across a wide range of price increments for τ = 8 and 512 minutes.
• The model outperforms standard distributions (e.g., normal, Lévy, log-normal) in fitting empirical densities, especially in the tails.
• The inverse moment method provides an explicit, data-driven construction of the family {p_w(τ)}_τ>0 for financial return processes.