Polynomial-Preserving Processes and Volatility Modelling in Discrete Time

Paper: arXiv 2611.10928

Authors: Maciej Augustyniak, Alexandru Badescu, Zekun Zhang

Abstract

This paper develops a discrete-time framework for volatility modelling and derivative pricing based on polynomial preservation. The framework provides exact moments of arbitrary order and serial cross-moments of variance and return processes over any finite horizon through finite-dimensional matrix representations. This analytical tractability is maintained while accommodating flexible leverage and variance-of-variance dynamics and non-Gaussian innovations. We introduce two novel classes of stochastic volatility models that nest widely used discrete-time specifications and study their statistical properties. Under two distinct scaling regimes, we establish weak convergence to polynomial diffusions and jump-diffusions and convergence of polynomial moments, linking discrete-time moment recursions to continuous-time matrix-exponential formulas. An empirical analysis of S&P~500 returns and a large panel of options spanning 1996–2023 shows that the proposed specifications improve return fit and reduce mean weekly implied volatility root mean squared errors by approximately 8–41% relative to their nested benchmarks, depending on the variance dynamics and innovation distribution.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical framework for discrete-time volatility modeling, involving advanced concepts like polynomial preservation and weak convergence. It is strongly supported by extensive empirical analysis on S&P 500 returns and a large options panel, demonstrating significant improvements over benchmarks. The combination of deep theoretical development and robust empirical validation places it firmly in the ‘Holy Grail’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop Discrete-Time Volatility Framework & Models] --> B{Methodology: Polynomial Preservation, Matrix Representations};
    B --> C[Model Development: Two Novel Classes of Stochastic Volatility Models];
    C --> D{Data/Inputs: S&P 500 Returns, Large Panel of Options (1996-2023)};
    D --> E[Computational Processes: Statistical Analysis, Empirical Evaluation, Weak Convergence];
    E --> F[Key Findings: Improved Return Fit, Reduced Implied Volatility RMSE (8-41%)];
    E --> G[Outcomes: Exact Moments, Flexible Dynamics, Link to Continuous-Time Models];