Paper: arXiv 2512.23640

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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

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  2. A model for stock returns and volatility (2014)
  3. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  4. Asymmetry in Distributions of Accumulated Gains and Losses in Stock Returns — → 2503.24241
  5. Distributions of Historic Market Data – Stock Returns (2019)
  6. Universal behavior of extreme price movements in stock markets (2009)
  7. Stochastic volatility and leverage effect (2003)
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  10. Probability distribution of returns in the Heston model with stochastic volatility (2002)
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1 of 21 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.