Paper: arXiv 2503.08666

Authors: Xinxin Jiang

Abstract

This paper provides evidence that stock returns, after truncation, might be modeled by a special type of continuous mixtures or normals, so-called $q$-Gaussians. Negative binomial distributions might model the counts for extreme returns. A generalized jump-diffusion model is proposed, and an explicit option pricing formula is obtained.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a solid generalization of Merton’s jump-diffusion model by utilizing q-Gaussians for continuous returns and a negative binomial process for jumps. The empirical analysis on S&P 500 data supports the theoretical distributional choices well, though the truncation method and option pricing derivation completeness could be further elaborated.

Research Flowchart

  flowchart TD
    A[Research Goal: Model Fat-Tailed Stock Returns & Price Options] --> B[Data Input: Historical Stock Returns & Truncated Extremes]
    
    subgraph Methodology & Modeling
        B --> C[Fit q-Gaussians to Truncated Continuous Returns]
        B --> D[Fit Negative Binomial to Discrete Extreme Return Counts]
        C & D --> E[Construct Generalized Jump-Diffusion Model]
    end

    E --> F[Derive Explicit Option Pricing Formula]
    F --> G[Outcome: Enhanced Option Pricing Framework]

Cited References (15)

  1. Stochastic Volatility (1995) — link
  2. Option Pricing Formulas Based on a Non-Gaussian Stock Price Model (2002) — link
  3. Option Pricing Under a Mixed-Exponential Jump Diffusion Model (2011) — link
  4. Option Prices in Merton’s Jump Diffusion Model (2011)
  5. On q-Gaussians and exchangeability (2010) — link
  6. Option Pricing When Underlying Stock Returns Are Discontinuous (1976) — link
  7. The pricing of options and corporate liabilities (1973) — link
  8. Pricing and Inference with Mixtures of Conditionally Normal Processes (2009) — link
and 7 more
  1. The limit distribution in the q-CLT for $q \geq 1$ is unique and can not have a compact support (2016) — link
  2. Tsallis q-triplet and Stock Market Indices: The cases of S&P 500 and TVIX (2015) — link
  3. Dynamical Foundations of Nonextensive Statistical Mechanics (2001) — link
  4. Tsallis’ q-triplet and the ozone layer (2010) — link
  5. Modeling Insurance Surrenders by the Negative Binomial Model (2005)
  6. Econometric models for count data with an application to the patents-R&D relationship (1984) — link
  7. The relation between implied and realized volatility (1998) — link

0 of 15 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.