Broken Symmetry of Stock Returns – a Modified Jones-Faddy Skew t-Distribution

ArXiv ID: 2512.23640 “View on arXiv”

Authors: Siqi Shao, Arshia Ghasemi, Hamed Farahani, R. A. Serota

Abstract

We argue that negative skew and positive mean of the distribution of stock returns are largely due to the broken symmetry of stochastic volatility governing gains and losses. Starting with stochastic differential equations for stock returns and for stochastic volatility we argue that the distribution of stock returns can be effectively split in two – for gains and losses – assuming difference in parameters of their respective stochastic volatilities. A modified Jones-Faddy skew t-distribution utilized here allows to reflect this in a single organic distribution which tends to meaningfully capture this asymmetry. We illustrate its application on distribution of daily S&P500 returns, including analysis of its tails.

Keywords: Stochastic Volatility, Skewness Distribution, Stock Returns, Tail Risk, Jones-Faddy Skew t-Distribution, Equities

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring numerous stochastic differential equations, modified Jones-Faddy distributions, and analytical derivations of statistical moments. However, the empirical component is limited to fitting a single distribution (modified Jones-Faddy) to S&P 500 daily returns without any backtesting, performance metrics, or data implementation heavy analysis beyond descriptive fitting.

  flowchart TD
    Start(["Research Goal: <br>Model asymmetry in stock returns"]) --> Method["Start with SDEs for<br>stock returns & volatility"]
    Method --> Split["Split distribution into<br>Gains vs Losses based on<br>stochastic volatility parameters"]
    Split --> Mod["Apply Modified<br>Jones-Faddy Skew t-Distribution"]
    Mod --> Compute["Fit distribution to<br>S&P 500 daily returns data"]
    Compute --> Findings["Key Findings:<br>- Captures negative skew<br>- Models asymmetric volatility<br>- Analyzes tail risk"]
    
    style Start fill:#e1f5fe,stroke:#01579b,stroke-width:2px
    style Findings fill:#f1f8e9,stroke:#33691e,stroke-width:2px