Scaling and shape of financial returns distributions modeled as conditionally independent random variables

ArXiv ID: 2504.20488 “View on arXiv”

Authors: Hernán Larralde, Roberto Mota Navarro

Abstract

We show that assuming that the returns are independent when conditioned on the value of their variance (volatility), which itself varies in time randomly, then the distribution of returns is well described by the statistics of the sum of conditionally independent random variables. In particular, we show that the distribution of returns can be cast in a simple scaling form, and that its functional form is directly related to the distribution of the volatilities. This approach explains the presence of power-law tails in the returns as a direct consequence of the presence of a power law tail in the distribution of volatilities. It also provides the form of the distribution of Bitcoin returns, which behaves as a stretched exponential, as a consequence of the fact that the Bitcoin volatilities distribution is also closely described by a stretched exponential. We test our predictions with data from the S&P 500 index, Apple and Paramount stocks; and Bitcoin.

Keywords: Conditional Independence, Volatility Clustering, Power-Law Tails, Stretched Exponential, Distributional Modeling, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced statistical physics and probability theory to model returns as conditionally independent sums, involving heavy integral transforms and scaling laws. It tests these predictions on real-world data (S&P 500, Apple, Paramount, Bitcoin) with systematic fitting and data collapse analysis, though it lacks executable backtests or code.

  flowchart TD
    A["Research Goal: Model financial return distributions<br>using conditionally independent random variables"] --> B["Methodology: Assumption of Conditional Independence<br>returns are independent given time-varying volatility"]
    B --> C["Computational Process: Model Sum of Conditionally<br>Independent Variables & Derive Scaling Form"]
    C --> D{"Data Input: Empirical Asset Returns"}
    D --> E["Asset 1: S&P 500 Index"]
    D --> F["Asset 2: Apple & Paramount Stocks"]
    D --> G["Asset 3: Bitcoin"]
    E --> H["Key Findings & Outcomes"]
    F --> H
    G --> H
    subgraph H ["Outcomes"]
        I["Power-law tails in equities<br>explained by power-law volatility distribution"]
        J["Stretched exponential in Bitcoin<br>explained by stretched exponential volatility distribution"]
        K["General scaling form derived<br>for return distributions"]
    end