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

Paper: arXiv 2601.09074

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.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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

Cited References (14)

1. Estimation of volatility in a high-frequency setting: a short review (2019) — link
2. Fourier transform methods for pathwise covariance estimation in the presence of jumps (2013) — link
3. A Fourier transform method for nonparametric estimation of multivariate volatility (2009) — link
4. Fourier series method for measurement of multivariate volatilities (2002) — link
5. Classical Fourier Analysis , volume 249 of Graduate texts in Mathematics (2014)
6. Brownian Motion and Stochastic Calculus (2008) — link
7. The Fine Structure of Asset Retums : An Empirical Investigation ’ (2006)
8. Fourier Analysis: An Introduction (2003)

0 of 14 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.