The Fourier estimator of spot volatility: Unbounded coefficients and jumps in the price process

ArXiv ID: 2601.09074 “View on arXiv”

Authors: L. J. Espinosa González, Erick Treviño Aguilar

Abstract

In this paper we study the Fourier estimator of Malliavin and Mancino for the spot volatility. We establish the convergence of the trigonometric polynomial to the volatility’s path in a setting that includes the following aspects. First, the volatility is required to satisfy a mild integrability condition, but otherwise allowed to be unbounded. Second, the price process is assumed to have cadlag paths, not necessarily continuous. We obtain convergence rates for the probability of a bad approximation in estimated coefficients, with a speed that allow to obtain an almost sure convergence and not just in probability in the estimated reconstruction of the volatility’s path. This is a new result even in the setting of continuous paths. We prove that a rescaled trigonometric polynomial approximate the quadratic jump process.

Keywords: Malliavin-Mancino estimator, Spot Volatility, Trigonometric Polynomial, Cadlag paths, Quadratic Jump Process, Derivatives

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: The paper is heavily theoretical, featuring advanced stochastic calculus (Itô processes, Burkholder-Davis-Gundy inequality), Fourier analysis (Bohr convolution), and detailed convergence proofs with rates, scoring high on math complexity. However, it lacks any empirical backtesting, real-world data analysis, or implementation details, relying only on brief numerical simulations for illustration, scoring low on empirical rigor.
  flowchart TD
    A["Research Goal:<br>Fourier Estimator of Spot Volatility<br>for Unbounded Coefficients & Jumps"] --> B["Methodology:<br>Fourier Series & Trigonometric Polynomials<br>for Malliavin-Mancino Estimator"]
    B --> C["Inputs:<br>Price Process<br>with Cadlag Paths"]
    C --> D["Computation:<br>Estimate Coefficients &<br>Reconstruct Volatility Path"]
    D --> E["Key Findings:<br>1. Convergence of Trigonometric Polynomial<br>to Volatility Path (Unbounded & Jump Aware)<br>2. Almost Sure Convergence & Rates<br>3. Rescaled Polynomial approximates<br>Quadratic Jump Process"]
    style A fill:#e1f5fe
    style E fill:#e8f5e8